Papers from 2026

A major problem in geometric analysis is to understand the behaviour of the Lagrangian mean curvature flow. This has proved to be a challenging problem, in particular, not many explicit examples of flows that exist for all time and converge to a minimal Lagrangian are known. In this paper we construct new examples of Lagrangian mean curvature flows in the Kummer K3 surface that converge to special Lagrangian spheres. We reduce the construction to a scalar perturbation problem that can be solved by a fixed point argument.

We construct an embedded torus and an entire graph in ${\mathbb{R}}^4$ whose normal bundles are initially flat but lose this property instantaneously under mean curvature flow. We also give an example showing that the parallel principal normal condition is not preserved under mean curvature flow, even though the normal bundle remains flat along the flow.

We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.

We formalize the smooth three-dimensional Poincaré conjecture, together with the Moise smoothing theorem, yielding the topological three-dimensional Poincaré conjecture. The smooth proof follows the Hamilton–Perelman route through Ricci flow with surgery and finite-time extinction.

For a convex mean curvature flow in ${\mathbb{R}}^{n+1}$, the corresponding arrival time function solves a degenerate elliptic equation that becomes singular at extinction. We prove that it is $C^{24,\alpha }$ for any $0<\alpha <1$ in the plane, with a logarithmic modulus for its twenty-fourth derivatives. We prove that this bound is sharp combining regularity obstruction in our companion paper. We obtain asymptotic expansions of the rescaled flow to any prescribed order, including derivative estimates for the remainder. Combined with the work of Sesum for $n\ge 2$, this also gives the sharp $C^{2,2/n}$ regularity for convex arrival time functions in higher dimensions.

We study the regularity of the arrival time function associated with mean curvature flow, formulated as a degenerate elliptic equation encoding the singular structure of the flow. We identify a systematic mechanism obstructing higher regularity, arising from higher-order asymptotic expansions near spherical singularities. We construct uncountably many low-regularity arrival time functions in all dimensions. In particular, we settle the question of smoothness in the planar case by showing that even for convex curve shortening flow the arrival time need not be $C^{25}$. Our companion paper proves that every such arrival time is $C^{24,\alpha }$ for any $0<\alpha <1$. Hence the obstruction at order $25$ gives the optimal regularity threshold. Our approach introduces new analytic ingredients, including a precise correspondence between elliptic asymptotic expansions and parabolic long-time asymptotics of the associated rescaled mean curvature flow, a complexification of the arrival time equation, and a method for prescribing higher-order asymptotic expansions.

We prove that at any nondegenerate neck pinch singularity of Lagrangian mean curvature flow of surfaces the mean curvature becomes unbounded, with control on the blow-up rate.

Let S^n+1_+ be the open hemisphere of the unit sphere S^n+1 centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k alpha and the critical regime beta=1+kalpha, where beta is the growth order of the profile at the origin, f(r) almost equals r^beta as r descends to 0. Under the structural condition that f^{1/(1+kalpha)} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_infty>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.

We prove that every complete nonsteady gradient Kähler-Ricci soliton with constant scalar curvature is rigid. This establishes Cao's rigidity conjecture for Kähler-Ricci solitons in arbitrary dimension and gives the corresponding result for expanding solitons, without additional curvature assumptions. The proof uses a rigidity criterion for gradient Ricci solitons with constant scalar curvature, expressed by the vanishing of the Lie derivative of the Ricci tensor along the soliton vector field. In the Kähler case, this vanishing follows from constant scalar curvature, the closedness of the Ricci form, and the classical real holomorphicity of the soliton vector field.

Let H^n+1 be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^betar; we treat both regimes: beta>1+kalpha and beta=1+kalpha. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_infty respectively in two different cases.

In this work we study the long term behavior of the curve shortening flow of asymptotically conical curves with convex projections in ${\mathbb{R}}^3$.

The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.

We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci–Yang–Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a $SU(2)$ bundle over $S^4$.

We study positive mass and related Brown–York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi–Tam type construction to obtain a Brown–York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.

We prove a Positive Mass Theorem for $C^0$-asymptotically flat Riemannian metrics with nonnegative scalar curvature in a weak sense that are sufficiently uniformly close to Euclidean space. More precisely, we show that a $C^0$-asymptotically flat Riemannian metric that is a $C^0$ perturbation of Euclidean space with nonnegative scalar curvature in the sense of Ricci flow has nonnegative mass, where the mass is given by a $C^0$ analog of the classical ADM mass previously introduced by the author.

We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.

In this paper, we study the anisotropic-length-preserving weighted anisotropic curvature flow for smooth convex closed plane curves. For any smooth, embedded, closed, convex initial curve, the flow exists smoothly for all time, and the anisotropic curvature converges smoothly to that of the boundary of the corresponding homothetic Wulff shape as time tends to infinity.

The focus of this paper is the curve shortening flow for closed spacelike curves in pseudo-Euclidean spaces, which has very few results so far. Will they produce singularities where certain tangent line tends to light cone? If not, will such a curve shrink to a circular point? To answer these questions, we establish a dichotomy for planar weighted normalized curve shortening flow with uniformly positive and bounded weights. Applying to closed smooth spacelike curves in pseudo-Euclidean spaces that admit a one-to-one convex projection onto a spacelike plane, at their finite maximal time we will see: either the curve shrinks to a point and becomes asymptotically circular, or the tangent directions subsequentially approach the null cone. Both alternatives occur. In the first case, this proves our previous conjecture that a strong spacelike curve in ${\mathbb{R}}^{2,q}$ with index 1 will converge to a circular point under the usual CSF. In the latter case, a monotone area-bivector defect is found in ${\mathbb{R}}^{2,1}$, which gives a quantitative obstruction to point collapse. Explicit examples of spacelike curves with lightlike tangent limit are given.

We prove that every extremal Kähler–Ricci soliton on a Fano manifold is Kähler–Einstein. This solves the problem of Calamai and Petrecca in full generality.

We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies $\rho =C\omega +nd{d}^{c}f$, and which satisfy $C>0$. With analogous assumptions on the flow, we obtain subconvergence to an $f$-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that $(M,g,f)$ is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.

We introduce a geometric structure – a conformal Killing–Yano Ricci soliton (CKY–RS) – that couples conformal Ricci soliton (CRS) geometry to conformal Killing–Yano (CKY) 2–forms. The soliton field of the CRS geometry is given by the divergence of the CKY 2–form. We introduce a conserved CKY–Cotton current and derive a compatibility identity relating the Cotton tensor, the CRS obstruction tensor, and the CKY 2–form. In 4–dimensional Lorentzian signature, we show that, under non-degeneracy and closedness assumptions on the CKY form, a CKY–RS structure forces the conformal representative to be locally Kerr–NUT–(A)dS. For a closed non-degenerate CKY on a Kerr–NUT–(A)dS background, the conformal deformation is necessarily trivial. For Einstein backgrounds of arbitrary dimension and signature, the conformal factor satisfies an eigenvalue equation and an Obata–type Hessian equation. If the background is also compact or a CKY orbit is periodic, the conformal factor is an invariant of the CKY–flow and we obtain simple spectral obstructions to non-trivial CKY–RS structures. From the Hessian equation, we obtain obstruction and classification results for the non-trivial conformal sector, including product/Brinkmann geometries and a Weyl–aligned branch. Finally, we give explicit constructions for static spherically symmetric geometries and BTZ backgrounds, including a CKY–RS realization with a time-dependent conformally flat representative. These results provide a geometric framework for studying CRS with hidden symmetry structure, with potential applications to exact geometries in general relativity.

We prove linear stability of all steady and expanding gradient Kähler-Ricci solitons. In the expanding case, we prove strict linear stability under very general assumptions. In particular, every asymptotically conical expanding gradient Kähler-Ricci soliton is strictly linearly stable.

The analytic minimal model program proposed in seeks to describe the birational transitions and fibre collapsing of the Kähler–Ricci flow through the geometry of its singularities. A fundamental conjecture of predicts Type I bounds for the scalar curvature and diameter of every fibre contratced by the limiting cohomological class at the finite singular time. This paper is a continuation of that establishes the Type I conjecture for collapsing solutions on Fano bundles with arbitrary smooth Fano fibres. If the fibre admits a smooth Kähler-Einstein metric, the Type I bound holds for the full curvature tensor.

Our first main result is a rigidity theorem for complete self-expanders in the Euclidean space with higher codimension, assuming an integral cur More precisely, any smooth complete self-expander $x:M\to {\mathbb{R}}^{n+p}$($n\ge 3$) that satisfies both $({\int }_M|A{|}^{n}d\mu {)}^{\frac{1}{n}}<K(n)$ and ${\int }_M|A{|}^n{e}^{\frac{|x{|}^2}{2}}d\mu <\infty$ for a positive constant $K(n)$ depending only o isometric to ${\mathbb{R}}^n$. Moreover, we show that the rigidity result persists when the pinching condition is expressed in terms of the trace-free second fundamental form.

Let $(M^n,g)$ be a complete, simply connected Riemannian manifold without boundary, of dimension $n\ge 3$, with curvature operator at least that of the unit sphere. We prove that $${\int }_M\mathrm{scal}(x)d{\mathrm{Vol}}_x\le n(n-1){\omega }_n,$$ where ${\omega }_n$ is the volume of the unit $n$-sphere. Equality holds if and only if $(M,g)$ is isometric to the unit round sphere. In fact, we obtain a stronger bound containing $\mathrm{Vol}(M,g)$. In even dimensions, the proof follows from the Chern-Gauss-Bonnet formula. In odd dimensions, we apply the corresponding boundary formula to Deruelle's Ricci expander filling.

We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.

We prove the linear stability of an embedded mean curvature flow shrinker in ${\mathbb{R}}^4$ with the topology of the Möbius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the Möbius shrinker models the reversal of a real blow-up, suggesting that stable higher-codimension singularities may encode non-trivial topological operations under mean curvature flow.

In [SX25], we introduced the notion of nondegenerate cylindrical singularity using the ideas from dynamical systems. In this paper, we further study the properties of these singularities, showing that they are stable under small perturbations. Furthermore, we show that we can perturb some mean curvature flow with degenerate singularities to produce nondegenerate singularities.

We prove that any finite-time collapsing Kähler-Ricci flow on compact Kähler surfaces develops a Type I singularity. Together with the previous results, this implies that any finite time singularity of the Kähler-Ricci flow on compact Kähler surfaces is of Type I.

We show that a compact mean curvature flow whose singularities have multiplicity-one cylindrical tangent flows admits a smooth Morse resolution. The resolution agrees with the spacetime track outside any prescribed neighborhood of its singular set and all its critical points have the expected index. No nondegeneracy or isolatedness assumption is required. In the mean-convex case the result follows by smoothing the global arrival function.

As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function $f$ admits a heat kernel $H_f$ under the weighted volume measure $e^{-f}dv$. In this paper, we study $H_f$ systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between $H_f$ and the spacetime heat kernel $H(x,t;y,s)$ under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that $\left|\nabla R\right|=o(f^{3/2})$ implies $R=o(f)$.

Let $S$ be a Kato surface and $D$ its maximal reduced divisor of rational curves. On $S\setminus D$ we construct Hermitian metrics which are flat along the leaves of the canonical foliation and study their evolution under the Chern-Ricci flow. For Enoki surfaces, we construct an immortal normalised solution which, on every compact sublevel of the natural exhaustion, converges in the Gromov-Hausdorff sense to the elliptic base endowed with an explicit flat metric. For Kato surfaces of intermediate type, the affine Green model and its finite-index extension give, under the assumption $0<\mu <2$, an explicit normalised solution which, on every compact Green block, collapses in the Gromov-Hausdorff sense to a circle. In the Enoki case the limiting area is $2\pi b_2(S)$; in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.

We prove a late-time monotonicity of the Willmore deficit ${\int }_{\Sigma }{H}^{2}d\mu -16\pi$ along the mean curvature flow of smooth closed embedded strictly convex surfaces in ${\mathbb{R}}^3$,which refining the exponential convergence given by Huisken. As a consequence, the Hawking mass is eventually non–decreasing along mean curvature flow. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.

A $\lambda$-translating soliton is a hypersurface in ${\mathbb{R}}^{n+1}$ satisfying $H=⟨\mathbf{T},\nu ⟩+\lambda$; equivalently, it has constant weighted mean curvature with respect to the log-linear density $e^{⟨T,X⟩}$, and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed $\lambda$-translating soliton with $\lambda >0$ and ${\inf }_{\Sigma }H>\lambda$ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic $\lambda$-translating solitons ($\lambda ⩾0$) with bounded gradient, and two rigidity theorems.

There exist three nonequivalent left invariant Lorentzian metrics on the Heisenberg group $H_{2n+1}$, or equivalently, three nonequivalent Lorentzian inner products on the Heisenberg Lie algebra ${\mathfrak{h}}_{2n+1}$, denoted by $\mu$, $\nu$, and $\phi$. We show that, in a specific case, $\mu$ is an algebraic Ricci soliton that is shrinking. Moreover, $\nu$ is an algebraic Ricci soliton only on the three-dimensional Heisenberg Lie algebra ${\mathfrak{h}}_3$ and it is shrinking. Finally, we show that $\phi$ is a steady algebraic Ricci soliton on ${\mathfrak{h}}_{2n+1}$ for $n>1$. However, for $n=1$, $\phi$ is flat.

For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.

We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the $2$-sphere thus confirming a conjecture of Branding–Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on ${\mathbb{S}}^2$. Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on ${\mathbb{RP}}^2$, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on ${\mathbb{S}}^2$ whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz–Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.

In this paper, we introduce some type vector fields with respect to a semi-symmetric non-metric (SSNM) connection. We investigate several geometric properties of a K-contact manifold equipped with an SSNM connection and provide a concrete example to justify the relation between the scalar curvature of the SSNM connection and Levi-Civita connection that we have obtained in this paper. Furthermore, we have found the nature of Riemann solitons, conformal Ricci solitons and conformal $\eta$-Ricci-Yamabe solitons on K-contact manifolds admitting a SSNM connection.\\ Finally, we determine the necessary and sufficient conditions for such a manifold to be $\text{Tilde}\tau$-semi-symmetric, quasi-conformal-semi-symmetric and pseudo-projective-semi-symmetric.

We study the iterated action of the Lie derivative on the curvature tensor of a Riemannian manifold with rank-one anisotropic curvature, whose Riemann tensor is expressed via the Kulkarni-Nomizu product as $R=\lambda ({\xi }^{♭}\otimes {\xi }^{♭})\text{owedge}g$. First, we examine the conditions under which such a manifold admits a Ricci soliton structure and demonstrate that this property implies the almost $\theta$-Yamabe soliton structure. Furthermore, we show that if the associated potential vector field $X$ is a symmetry of the Ricci tensor of a fixed order $k$ (i.e., ${\mathcal{L}}_X^k\mathrm{Ric}=0$), the geometric problem reduces to solving a partial differential equation of order $k+1$ along the flow. Finally, under the assumption that $X$ is a conformal vector field (${\mathcal{L}}_{X}g=2\phi g$) whose infinitesimal flow preserves the line distribution $\mathcal{D}=\mathrm{Span}\{\xi \}$ (with $[X,\xi ]=a\xi$ for $a\in \mathbb{R}$), we prove that several key geometric problems (such as establishing the relation ${\mathcal{L}}_X^{k}R=R$, determining the minimal order $k$ for $X$ to be a Lie curvature symmetry, or satisfying ${\mathcal{L}}_X^{k+1}R=f{\mathcal{L}}_X^{k}R$ for a continuous function $f$) are equivalent to a scalar differential problem governed by the operator $D_X=X+6\phi +2a$.

This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah–Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais–Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.

In this note, we study volume comparison for 3-manifolds with 2-Ricci curvature lower bound. We establish monotonicity and rigidity for geodesic sphere area and ball volume, and derive the pointed Gromov-Hausdorff precompactness under a uniform lower bound on the conjugate radius for 3-manifolds with 2-Ricci curvature lower bound. We also discuss the almost preservation of 2-Ricci curvature lower bound and the local non-collapsing propagation along Ricci flow. Finally, we point out a topological rigidity result on nonnegative scalar curvature.

We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval $[-3,-\frac{3}{2}]$. The upper bound $-\frac{3}{2}$ is achieved by the complex Heisenberg group ${\mathrm{Heis}}_3(\mathbb{C})$ with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold $N$ has the pinching constant $-\frac{3}{2}$, then $N$ admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant $-3$. This is derived by showing that there is an open neighborhood $U$ of ${\mathrm{Heis}}_3(\mathbb{R})\times {\mathbb{R}}^{n-3}$ in the space of $n$-dimensional 2-step nilmanifolds such that the pinching constant is $-3$ on $U$, and any 2-step nilpotent Lie group $N$ has a metric $g$ such that $(N,g)$ lies in $U$. In fact, if $N$ is not isomorphic to ${\mathrm{Heis}}_3(\mathbb{R})\times {\mathbb{R}}^{n-3}$, then there is a curve $g_t$ of metrics on $N$ with $(N,g_t)\in U$, showing that there are uncountably many left-invariant metrics on $N$ such that the pinching constant is $-3$. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant $-\frac{3}{2}$ is also given, and the pinching constants of various examples are computed.

In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions $6m+d$ for $m>d\ge 1$. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra ${\mathfrak{m}}_2(n)$, $n\ge 5$, admits a generalized nilsoliton if and only if $5\le n\le 8$.

In this article, we study Ricci-Yamabe solitons on the Lie group $\mathrm{Sol}\times {\mathbb{R}}^n$ equipped with a natural left-invariant Riemannian metric, explicitly determining the vector fields that characterize them. We then deduce that, in the case of a Ricci soliton, it is expanding, whereas in the case of a Yamabe soliton, it is shrinking. Finally, we show that if this Lie group is a gradient Ricci-Yamabe soliton, the vector field belongs to $\mathrm{Span}\{{\partial }_{t_1},\dots ,{\partial }_{t_n}\}$, and we explicitly provide the Perelman potential.

We show that on any closed orientable real-analytic Riemannian surface, any smoothly immersed closed immortal curve-shortening flow converges to a unique non-constant closed geodesic.

We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius $R$, the region where the curvature radius is smaller than $R^{-1}$ occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li–Wang.

In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.

In this paper, we study the Kähler–Ricci flow on $\mathbb{C}{P}^m$-bundles over a product of Kähler–Einstein manifolds, starting from an initial metric with Calabi symmetry. We prove that every finite-time singularity arising along the flow must be of Type I.

We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is $n-1$, we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.

We study entire graphical translating solutions of the mean curvature flow, $$\mathrm{div}\left(\frac{\nabla G}{\sqrt{1+|\nabla G{|}^2}}\right)=\frac{1}{\sqrt{1+|\nabla G{|}^2}}\text{in }{\mathbb{R}}^N.$$ Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every $N\ge 8$ there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri–De Giorgi–Giusti (BDG) entire minimal graph in ${\mathbb{R}}^8$ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.

We study the inverse-weight Lin–Lu–Yau Ricci flow on finite trees, and prove the convergence of the curvature on every edge via its diffusion equation. Moreover, the limiting curvature is the unique minimum-norm point of a polyhedron determined by the tree. We characterize possible limits of the normalized edge weights, give criteria for their convergence, and show that different positive initial metrics have equivalent omega-limit sets.

We study the singularity type and models of the Kähler–Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler–Einstein manifolds $N:=N_1\times \cdots \times N_r$, with metric constructed using the ansatz considered in, et. al. In the earlier work by the authors, we considered the "two-bolt" case where both ends of the foliation close with the "bolt" $N$. In this article, we continue our work on the more subtle "nut-bolt" and "two-nut" cases. The former has one end of the interval closes with a nut-type collapse (i.e. $N^{\prime }:=N_2\times \cdots \times N_r$) and the other with a bolt (i.e. $N$). The compactification $\hat{M}$ is then a ${\mathbb{CP}}^{m+1}$-bundle over $N^{\prime }$. The "two-nut" case is one that both ends close with nut-type collapses, necessarily two of the $N_i$'s must be ${\mathbb{CP}}^{m_0}$ and ${\mathbb{CP}}^{m_{\ell }}$, and the compactification $\hat{M}$ is a ${\mathbb{CP}}^{m_0+m_{\ell }+1}$-bundle over ${\prod }_{k\ge 3}{N}_k$. We proved that in all "two-bolt", "nut-bolt" and "two-nut" caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be $({\Sigma }^{m+1},g_{\Sigma }(t))\times ({\mathbb{C}}^k,\text{flat})$ with $m,k\ge 0$, where $\Sigma$ is one of the following: ${\mathbb{CP}}^{m+1}$, $\text{Tot}({\mathcal{L}}^{\oplus (m+1)})$, or a projectivization $\mathbb{P}({\mathcal{O}}^{\oplus (m_0+1)}\oplus {\mathcal{L}}^{\oplus (m_{\ell }+1)})$ with $m_0+m_{\ell }=m$, and $\mathcal{L}$ is a line bundle over the product of some of the $N_1,\cdots ,N_r$ factors. The metric $g_{\Sigma }(t)$ is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.

We prove uniqueness of tangent flows for mean curvature flow with free boundary at singularities modeled on half-cylindrical self-shrinkers. More precisely, if a tangent flow is of the form ${\mathbb{S}}^{n-k}\times {\mathbb{R}}_{+}^k$ or ${\mathbb{S}}_{+}^{n-k}\times {\mathbb{R}}^k$, then it is unique. This provides the free-boundary analogue of the uniqueness theory for cylindrical singularities.

We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle $X^n\to Y^m$, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as $({\text{C}}^m,g_{\mathrm{E}},J_0)\times (Z^{\prime },d^{\prime },J^{\prime })$. If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to $\sqrt{T-t}$. Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder ${\text{C}}^m\times {\text{PP}}^1$.

We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker ${\mathbb{P}}^1\times \mathbb{C}$. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.

For every pair of integers $3\le k<n$ with $k$ odd, we construct a compact embedded rotational torus in ${\mathbb{R}}^{n+1}$ whose homothetic dilations satisfy the unnormalised ${\sigma }_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\le p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $⟨X,\nu ⟩=-{\sigma }_k$, where $X$ is the position vector and $\nu$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.

In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.

In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete gradient shrinking Kähler-Ricci solitons in complex dimension two, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.

In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is – up to splitting Euclidean factors – either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).

In this paper, we establish several uniform estimates for Kähler–Ricci shrinkers without imposing any curvature assumptions. In particular, we prove: 1. a uniform lower bound for the entropy; 2. a uniform lower bound for the asymptotic volume ratio of Kähler–shrinkers with maximal volume growth; 3. a uniform lower bound for the scalar curvature on balls centered at a minimum point of the soliton potential for non-Gaussian shrinkers.

Let $\lambda (u,x)$ be the local Arnold multiplicity of a quasi-plurisubharmonic function $u$. Di Nezza–Guedj–Lu asked whether every maximal weak solution ${\phi }_t$ of the twisted Kähler–Ricci flow satisfies $\lambda ({\phi }_t,x)=\max \{\lambda ({\phi }_0,x)-t,0\}$. We give counterexamples on the Hirzebruch surface ${\mathbb{F}}_e={\mathbb{P}}_{{\mathbb{P}}^1}({\mathcal{O}}_{{\mathbb{P}}^1}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(-e))$, $e\ge 2$. Let $S$ be its negative section and $F_1,\dots ,F_k$ be distinct fibres. If $a,b_i>0$, ${\sum }_i{b}_i>ea$, and the initial current is $a[S]+{\sum }_i{b}_i[F_i]$, then $\lambda ({\phi }_t,x)=a-\min \{k/e,1\}t$ for $x\in S\setminus {\bigcup }_i{F}_i$ and $0<t<\min \{a,b_1,\dots ,b_k\}$. Thus the formula fails for $k<e$; the slower decay is forced by the Zariski negative part of the residual class. Under explicit SNC and positivity hypotheses, we also prove an exact surface formula for pure divisorial data. Consequences include nonlocality with fixed background data and explicit multiplier ideals. Products with smooth factors give counterexamples in every complex dimension $n\ge 2$.

We determine the long-time behavior of the canonical Ricci flow on a closed three-manifold of Thurston type ${\mathbb{R}}^3$, Nil, or Sol, starting from an arbitrary initial metric. In the Euclidean case, the metric $g(t)$ converges exponentially fast to a flat metric. The Gromov–Hausdorff limit of $(M,t^{-1}g(t))$ is a point in the Euclidean and Nil cases, and a metric circle or a compact interval of positive length in the Sol case. In all three cases, the blowdown flows lifted to the universal cover, $s^{-1}\tilde{g}(s\tau )$, converge, in the pointed Cheeger–Hamilton sense, to an explicit homogeneous expanding Ricci soliton solution.

We show that the spacelike solutions to the curvature flow for curves on the De Sitter space are in correspondence with the solutions to the inverse curvature flow on the 2-dimensional hyperbolic space, that on the De Sitter space, the timelike solutions to the curvature flow are in correspondence with the timelike solutions to the inverse curvature flow, and that the solutions curve shortening flow on the 2-dimensional hyperbolic space are in correspondence with the solutions to the spacelike solutions to the inverse curvature flow on the De Sitter space. We prove that, for spacelike curves on the De Sitter space, the curvature flow is a gradient-type flow for the arc-length functional. We observe that a spacelike or timelike curve on the De Sitter space is a soliton solution to the curvature flow (resp. inverse curvature flow) if and only if its curvature (resp. inverse of its curvature) can be written as the inner product between its tangent vector field and a fixed vector $v$ of the 3-dimensional Minkowski space. We prove that for each vector $v$, there exists a 2-parameter family of timelike (spacelike) soliton solutions to the curvature flow and to the inverse curvature flow on the De Sitter space. We show that there exists no non-trivial complete timelike soliton. There exist non-trivial complete spacelike solutions. As a consequence of curvature flow, we obtain the behavior of the soliton solutions to the inverse curvature flow on the De Sitter space and hyperbolic space.

Since Ilmanen's pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand the sharp-interface limit of systems of coupled Allen-Cahn equations. We prove - for the first time - a global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke.

We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow $(M^2,\partial M^2,g(t))$ on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.

We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE $\frac{\mathrm{d}}{\mathrm{d}t}R=Q(R)$ in dimensions $n=7,8$, thereby extending the pinching estimate established by Brendle for $n\ge 12$ and by Chen for $9\le n\le 11$. In dimension $n=8$, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at $n=8$ by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension $n=7$, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension $n=8$. The $n=8$ pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible $(n-1)$-dimensional space forms, extending a theorem of Brendle from $n\ge 12$. Together with curvature-improvement and classification results of Cho–Li and Brendle–Naff, it also yields the classification of noncompact $\kappa$-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho–Li from $n=4$ or $n\ge 12$.

Let $n\ge 3$ and $0\le \tau \le 2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint $n=3$, $\tau =2$, positive $\tau$-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.

Adapting ideas of, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in. Using this we show that low dimensional GRS are diffeomorphic to $S^3/\Gamma$ or $S^3\times S^1/\Gamma$. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion $G_2$, strong torsion $\mathrm{Spin}(7)$-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered $5$-manifolds over complex orbifolds.

We prove that every compact four-dimensional weakly Einstein Ricci soliton is Einstein. The nontrivial compact case reduces to the gradient shrinking setting, where a differential identity for weakly Einstein four-manifolds, together with the curvature identity for gradient Ricci solitons, yields the pointwise relation $|R{|}^2\nabla f=0$ for the soliton potential $f$. Consequently, no compact proper weakly Einstein four-manifold admits a Ricci soliton structure. A noncompact homogeneous example shows that the compactness assumption is essential.

We relate the generalized Ricci curvature of left-invariant generalized metrics on a Lie group with the Ricci curvature of certain metric Lie algebras. As an application, we prove subconvergence of rescaled generalized Ricci flows to expanding generalized Ricci solitons on simply connected Lie groups with positive semi-definite Killing form. Under the same hypotheses, subconvergence of pluriclosed flows to expanding pluriclosed solitons is also shown. We further prove that left-invariant Bismut-Ricci flat metrics arise as critical points for the generalized scalar curvature functional on unimodular Lie groups. In view of this, we establish dynamical stability for the generalized Ricci flow of so-called standard bi-invariant, Bismut-flat metrics on compact, simply connected, semisimple Lie groups and construct examples of dynamically unstable Bismut-flat metrics which are non-standard.

We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.

The aim of this paper is to classify some special Riemannian manifolds with cyclic parallel Ricci tensor, i.e. \beginequation D_ijk=\nabla_iR_jk+\nabla_jR_ki+\nabla_kR_ij=0\nonumber \endequation These structures include non-compact gradient shrinking Ricci soliton, compact $(m>1)$-quasi-Einstein manifolds with boundary and critical spaces. We will construct some integral identities and make use of the curvature conditions reasonably to prove that the Ricci tensor is parallel.

We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

We study Ricci solitons on the Riemannian manifold ${\mathbb{H}}^2\times \mathbb{R}$ equipped with the standard metric. A complete classification of soliton vector fields is obtained: they form a four-dimensional affine space, namely a translate of the Killing algebra $\mathfrak{isom}({\mathbb{H}}^2\times \mathbb{R})$. All corresponding solitons are expanding. In addition, gradient solitons are fully characterized and shown to form a one-parameter subfamily of the complete family of soliton fields. As a byproduct, every soliton vector field turns out to be affine, preserving the Levi-Civita connection, the curvature tensor, and the Ricci tensor.

Let $(M^n,g,f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) $Ric\ge \frac{{\nabla }_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$; (ii) $(M^n,g,f)$ smoothly converges to ${\mathbb{R}}^2\times {\mathbb{S}}^{n-2}$, we conclude that $(M^n,g,f)$ is isometric to ${\mathbb{R}}^2\times {\mathbb{S}}^{n-2}$. Notably, condition (i) is weaker than the radial flatness condition in.

This paper studies self-shrinkers and the long-time behavior of the inverse ${\sigma }_k$ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of self-shrinkers determined by their asymptotic data. More precisely, we formulate the self-shrinker equation as a fully nonlinear Dirichlet problem on hyperbolic space with prescribed data on its ideal boundary, and prove that every continuous negative boundary value determines a unique entire spacelike strictly convex self-shrinker. We further study the inverse ${\sigma }_k$ curvature flow starting from an entire spacelike strictly convex hypersurface satisfying a subsolution condition at infinity and a uniform positive lower bound for its ${\sigma }_k$ curvature. We prove global existence of the flow and show that the normalized flow converges locally smoothly to the unique self-shrinker with the same prescribed boundary value at infinity.

In this paper, we prove a gap theorem for $\mathbb{F}$-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an $\epsilon$-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).

We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of Kähler potentials. Consequently, every noncollapsed Kähler-Ricci flow through singularities in complex dimension two is modeled on a shrinker-cone-expander transition, confirming a strong form of Song's conjectural picture. We also show analogous results in higher dimensions under the Calabi ansatz, and improve known results in the compact shrinker case. These give the first compact Ricci flows through conical singularities whose small-scale behavior is fully described.

For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}{\mathbb{P}}^m\looparrowright {\overline{\mathbb{B}}}^{,N}(1)$ satisfies $\kappa (F{)}^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $\kappa (F)\le \sqrt{3/2}$ is diffeomorphic to $S^3$, $\mathbb{R}{\mathbb{P}}^3$, or $S^2\times S^1$. All three possibilities occur, while $\kappa (F)<\sqrt{3/2}$ forces $X\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh–Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar–systolic inequality $$(\underset{Y}{\min }{R}_g)\text{sys}(g{)}^2<6{\pi }^2$$ for every spherical three-space form $Y$ with $|{\pi }_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar–systolic inequality for $\mathbb{R}{\mathbb{P}}^3$ of Bray–Brendle–Eichmair–Neves.

We prove the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow. More precisely, for any compact Kähler manifold $Y$ and its blow-up $\pi :{\mathrm{Bl}}_{p}Y⟶Y$, if $[{\omega }_0]-T{c}_1(M)={\pi }^{\ast }[{\omega }_Y]$, then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker $\mathrm{Tot}({\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1))$.

This paper establishes sharp upper bounds for $p$-capacities ${\mathrm{Cap}}_{1<p<\infty }$ in the hyperbolic space ${\mathbb{H}}^n$ through hyperbolic quermassintegrals and effective curvature radii. The quermassintegral comparisons involve $W_{n-1}$, $W_{k+1}+k(n+1-k{)}^{-1}{W}_{k-1}$, and the pair $W_1\mid W_2$. For star-shaped, mean-convex or h-convex hypersurfaces, inverse mean curvature flow further produces curvature radii determined by $L^q$-averages of the normalized mean curvature and by moments of its squared hyperbolic excess. These radii convert the resulting estimates into sharp geodesic-ball comparisons for the capacity-to-area ratio. In the range $p>2m+1$, an interpolating radius combines the $2m$-th curvature-excess radius with the $L^{\infty }$ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.

We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.

We investigate a compact Einstein-type manifold whose potential vector field generates a Riemannian foliation. In particular, we prove necessary conditions for such a manifold to be taut and to have a splitting property. Additionally, some properties of taut Riemannian foliations on a compact almost Ricci solitons and a compact Einstein manifolds are provided.

We introduce and study a Fisher information metric $g_{\tau }^F$ associated to the conjugate heat kernel of a Ricci flow $(M^n,g_t)$. This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that $g_{\tau }^F$ is monotone in scale and satisfies $g_{\tau }^F\le g_t$. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect $g_t-g_{\tau }^F.$ This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities $0<g_{\tau }^F<g_t$ at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for $\phi$-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to $0$ to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's $\epsilon$-regularity theorem.

We introduce the almost isoclinic region in the oriented Lagrangian Grassmannian ${\mathrm{Lag}}^{+}(n)$ of ${\mathbb{C}}^n$, an intrinsic higher-dimensional analog of a natural convex region in ${\mathrm{Lag}}^{+}(2)\simeq {\mathbb{S}}^1\times {\mathbb{S}}^2$. A Lagrangian submanifold is called almost isoclinic if its Gauss map takes values in this region, extending the graphical condition that the characteristic angles of the tangent plane remain uniformly close. We construct a canonical positive function $\Lambda$ on this region and prove that $\log \Lambda$ is concave with respect to the invariant Grassmannian metric. This property yields subharmonicity and monotonicity formulas for minimal Lagrangians and Lagrangian mean curvature flow. As applications, we prove rigidity and Bernstein-type results, including that a complete connected almost isoclinic minimal Lagrangian with a positive lower bound for $\Lambda$ must be a Lagrangian $n$-plane.

We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Ampère equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.

For a closed connected Riemannian manifold $(M,g)$, the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding $x\mapsto p_t(x,\cdot )d{\mathrm{vol}}_g$. The resulting family ${\tilde{g}}_t$ agrees with Ricci flow to first order in $t$, but in general not to second order. We prove that $${\tilde{g}}_t=g-2t{\mathrm{Ric}}_g+t^2\left(-\Delta {\mathrm{Ric}}_g+2{\mathrm{Ric}}_g^2-\frac{2}{3}{\mathcal{Q}}_g\right)+O_{C^0}(t^3),$$ where ${\mathcal{Q}}_g$ is quadratic in the full curvature tensor. The term $-\Delta {\mathrm{Ric}}_g$ also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while ${\tilde{g}}_t$ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is $O(t^2)$, and round spheres show that this estimate is sharp.

We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.

We construct the local-in-time solution of the generalised anisotropic direction-dependent curvature flow in dimension $d\ge 2$ forced by a white-in-time and smooth-in-space Gaussian noise. This seems to be the first construction with a white-in-time noise which also allows spatial dependence, even in the simpler case of isotropic stochastic mean curvature flow. The main difficulty is that the stochastic PDE describing the flow has a multiplicative noise depending nonlinearly on both the solution and its gradient. The key technique is a transform developed in based on rough characteristics that removes this rough multiplicative term. We also illustrate the relationship of this transform with previously known special situations. As an application of the construction, we show that in a short time interval, the sharp interface limit of the stochastic Allen-Cahn equation with nonlinear diffusion and the same noise (slightly smoothened in time) is given by the above direction-dependent stochastic curvature flow.

The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n+m\ge 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\text{singular minimal cone as the asymptotic model}$.

In this paper, we study compactness, rigidity, and related geometric properties of complete Kähler Ricci shrinkers through the polarized Fano fibration structure.

We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential $C^{\infty }$ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.

In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant $\alpha =-1$ (or $+1$), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.

We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in $W^{1,\infty }$ by smooth profiles compatible with the boundary condition. Within this class, if the initial Lipschitz seminorm is sufficiently small, the corresponding solution is global; on a bounded domain, it also converges exponentially to the flat graph. Positive-time regularization is quantified by time-weighted Hölder estimates whose weighted quantities remain bounded as $t\downarrow 0$. The main analytic ingredient is a boundary Schauder theory for variable-coefficient parabolic systems. On the half-space, it combines coefficient freezing with parity extensions and boundary identities intrinsic to the graphical system. On curved domains, localization and boundary flattening lead to anisotropic estimates, from which normal derivatives are recovered recursively.

Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-Einstein manifold in dimension $5$ and $7$. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with $S^1$.

We prove that the Feldman–Ilmanen–Knopf (FIK) blowdown singularity has a nonempty open nonlinear formation basin in the full space of smooth Riemannian metrics and, on a smaller neighborhood, local marked first-order asymptotic moduli. Given a closed connected oriented Riemannian four-manifold, an implantation point, and a positive time bound, its oriented blow-up admits a relatively $C^{2,\alpha }$-open set of smooth metrics whose Ricci flows form, before that time, a localized FIK singularity with global Type-I curvature control. No symmetry, Kähler condition, or finite-dimensional tuning is imposed. Curvature stays uniformly bounded outside the implantation region; fixed-convention marked rescalings converge along the full singular-time sequence to the ancient FIK flow; and the exceptional sphere has sharp FIK asymptotics for area, intrinsic diameter, and curvature. To our knowledge, this is the first open full-metric formation theorem for a finite-time singularity modeled on a noncompact, noncylindrical shrinker. A self-contained spectral certification identifies the weighted FIK operator's nine-dimensional nonnegative space with the scaling and diffeomorphism directions, and exact modulation removes this block. On a smaller little-Hölder $h^{2,\alpha }$ neighborhood, a fixed positive-time restart yields the marked first-profile coordinate ${\mathfrak{A}}_1={\lambda }_{\infty }^{-{\gamma }_1}{V}_{\infty }\in E_1$. This amplitude is a split $C^1$ submersion and locally the projection onto $E_1$. Equality of amplitudes characterizes marked first-order agreement; a transverse disk uniquely realizes each sufficiently small amplitude; normalized two-flow differences converge to the corresponding Jacobi field; and the amplitude determines the first quadratic scale and phase response. This gives a complete local marked first-order profile coordinate for FIK blowdown.

In this paper, we compute the canonical and Kobayashi-Nomizu connections, together with their curvature, on Lorentzian four-dimensional nilpotent Lie groups endowed with a product structure. We also classify the algebraic Ricci solitons associated with these connections.

Let (M,g) be a compact n-dimensional Riemannian manifold, n>2. We introduce a restricted variational principle for the Einstein-Hilbert functional by requiring the admissible metric variations to satisfy the harmonic gauge condition. We derive the corresponding Euler-Lagrange equation and show that a metric is critical with respect to all volume-preserving harmonic variations if and only if its Einstein tensor differs from a multiple of the metric by an element of the image of the adjoint Bianchi operator. We prove that every harmonic critical metric determines a compact Ricci soliton whose soliton constant is given by the normalized Einstein-Hilbert functional. By Perelman's theorem, every such metric is in fact the metric of a compact gradient Ricci soliton. Conversely, every compact gradient Ricci soliton satisfies the restricted Euler-Lagrange equation. Thus, a compact Riemannian metric is harmonic critical if and only if it is the metric of a compact gradient Ricci soliton. We further show that the gauge one-form differs from the negative differential of a soliton potential by a Killing one-form. In particular, if the Ricci tensor is negative definite, then the gauge one-form vanishes and the metric is Einstein. Moreover, every non-Einstein harmonic critical metric is necessarily shrinking.

We present a numerical method that constructs geometrically defined coordinates on black-hole horizons, from which the multipole moments are computed without assuming axisymmetry. This method, which we denote the conformal-mapping method (CMM), provides a numerical realization of the conformal construction proposed by Ashtekar et al. in 2022, combining discrete Ricci flow, spectral embedding onto the unit sphere, and Möbius gauge fixing by the vanishing-area-dipole condition. We first test the CMM against analytic Kerr benchmarks, and then apply it to an equal-mass, non-spinning binary black-hole merger. We also compare it with an approximate-symmetry-based method. The CMM allows the multipole moments to be expressed in a fixed reference frame, whereas the symmetry-adapted frame can reorient abruptly when the preferred approximate axis changes. In a frame aligned with the orbital angular momentum, the amplitude of the quadrupole mode grows during inspiral and decays after merger, displaying a qualitative ringdown behavior. These results show that the CMM is a useful tool for studying horizon geometry in dynamical situations where no stable symmetry axis is available.

In this note, we prove that, on weighted graphs, the Lin–Lu–Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\ge 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier–Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin–Lu–Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.

Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on ${\mathbb{S}}^2$ and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.

We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.

We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.

We prove the Penrose inequality with charge under the 2-convexity condition recently introduced by Dong. More precisely, given a complete, connected and asymptotically flat Einstein-Maxwell initial data set $(M,g,k;E,B)$ satisfying the charged dominant energy and the 2-convexity conditions, with divergence-free electromagnetic vector fields $(E,B)$ and a connected outermost past apparent horizon $\Sigma$ that satisfies $|\Sigma |\ge 4\pi q^2$ - where $q$ is the total charge - we show that the following inequality for the ADM mass $m$ holds: $m\ge \sqrt{\frac{|\Sigma |}{16\pi }}+q^2\sqrt{\frac{\pi }{|\Sigma |}}$, with equality if and only if $k\equiv 0$ and $(M,g;E,B)$ is isometric to a canonical slice of sub-extremal Reissner-Nordström spacetime. Building on Dong's proof of the uncharged case, we use his $\mathbf{P}$-inverse mean curvature flow and its weak formulation, which only depends on $(g,\mathbf{P})$ and hence applies to the charged setting unchanged. The novelty of our work is the modification of the monotonicity formula to account for the additional charge term. For time-symmetric data ($k\equiv 0$), the flow reduces to the classical inverse mean curvature flow and our monotonicity formula to Jang's monotonicity of the charged Hawking mass, recovering the charged Riemannian Penrose inequality.

We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton $(M,g,J,f)$ and a real-valued pluriharmonic function $u$, we investigate conditions under which $u$ must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that $u$ is constant whenever $${\int }_M|\nabla u{|}^p\mathrm{d}v<\infty$$ for some $0<p<\infty$. In the shrinking case, we prove the same conclusion for $0<p\le 2$. Finally, we construct a complete Kähler example showing that the extension to the range $0<p<1$ relies essentially on the soliton structure and does not hold on general complete Kähler manifolds.

We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions $n\ge 3$ show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every $n\ge 2$, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere.

We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the $\alpha$-Gauss curvature flow ${\partial }_{t}X=-K^{\alpha }\nu$, $\alpha >0$. We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If $\alpha >\frac{1}{n+2}$, we apply a Cayley-type conformal map that sends the extinction point to the origin of a Euclidean half-space and then normalize the enclosed half-space volume. The resulting normalized hypersurfaces converge smoothly to the unit hemisphere. The proof combines boundary identities for the spherical free boundary, a boundary-adapted Tso estimate, an almost-monotonicity formula for a half-space entropy, and uniform curvature estimates for the normalized flow.

In this paper, we study Ricci solitons on the three-dimensional solvable Lie group ${\mathrm{Sol}}_{m,n}^3$ equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical ${\mathrm{Sol}}^3$ geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties of the integral curves of the Ricci soliton vector field. We also characterize harmonic linear maps from ${\mathrm{Sol}}_{m,n}^3$ into Euclidean spaces.

We show that any complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient Kähler-Ricci soliton on such a resolution. This confirms a special case of the uniqueness part of a conjecture of Song-Zhang. Some other consequences are also discussed.

We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an $\epsilon$-uniformly nondegenerate initial metric, we establish the short-time existence and uniqueness of smooth solutions to the combinatorial Yamabe flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow.Furthermore, under an integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a local well-posedness theory for the hyperbolic combinatorial Yamabe flow on infinitely triangulated surfaces.

We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an $S^1$-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.

We establish the following Miyaoka-Yau inequality for any $n$-dimensional klt Fano variety $X$, possibly K-unstable, in terms of its delta invariant: $$\left(2(n+1){\hat{c}}_2(X)-n{c}_1(X{)}^2\right)\cdot c_1(X{)}^{n-2}\ge -n{\left(1-\min \{1,\delta (X)\}\right)}^2\cdot c_1(X{)}^n.$$ Furthermore, inspired by recent work of Inoue and Hallam-Lahdili, we formulate and prove an equivariant version of this inequality in the soliton setting, in which the Chern classes and the delta invariant are replaced by the equivariant Chern classes and the weighted delta invariant, respectively. As a consequence, we obtain the equivariant Miyaoka-Yau inequality for every smooth Fano manifold admitting a Kähler-Ricci soliton.

We first derive a formula for the mean curvature and the squared norm of the shape operator of equifocal hypersurfaces in simply-connected irreducible symmetric spaces of compact type. The formulas are given explicitly in terms of the tangential focal data of the equifocal hypersurfaces. Third, we study the backward mean curvature flow for equifocal hypersurfaces. The long-time existence of this flow for an equifocal hypersurface was established by Liu and Radeschi. We analyze the time evolution of the mean curvature and the squared norm of the shape operator along the long-time solution, thereby we generalize the result of Liu and Terng for isoparametric hypersurfaces in the sphere. Our analysis also gives an extension of Liu-Terng conjecture on the backward mean curvature flows in the sphere to the simply-connected irreducible symmetric space of compact type.

Given an embedded shrinker $\Sigma$ in ${\mathbb{R}}^{n+1}$ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with ${\mathbb{R}}^k$, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on $\Sigma$. We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.

In this paper, we study three-dimensional combinatorial Yamabe flows on locally finite infinite triangulations in Euclidean and hyperbolic background geometries. Under suitable non-degeneracy and bounded-degree assumptions, we establish the short-time existence and uniqueness of the original flows. We further introduce the extended flows by using the continuous extension of solid angles, and prove long-time existence for both extended flows under suitable initial assumptions.

We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we obtain rigidity and non-existence results for Ricci solitons whose potential field is the Reeb vector field of almost contact metric and almost $\alpha$-cosymplectic manifolds. In dimension three, we derive consequences for almost $\alpha$-cosymplectic and contact metric manifolds, and we compare our results with the classification of homogeneous almost $\alpha$-cosymplectic Ricci solitons due to Li and Liu. Several examples and non-examples are included to illustrate the necessity of the finite-volume and sign assumptions.

We study the spectrum of the $f$-Laplacian on complete gradient Ricci shrinkers. Upper and lower bounds for the $k$-th eigenvalue are established in terms of the volume growth rate. Both bounds are shown to be sharp in the exponent. The method extends to $f$-Laplace-type operators on vector bundles; as an application we obtain explicit upper bounds for the Betti numbers.

Let $(M^3,g,\mathbf{k})$ be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon $\Sigma$. We prove the Penrose conjecture, namely that $m_{\mathrm{ADM}}(g)\ge \sqrt{\frac{|\Sigma |}{16\pi }}$, under the assumptions of the dominant energy condition and the $2$-convexity condition that the sum of the two smallest eigenvalues of $\mathbf{k}$ is nonnegative. The main tool is the $\sigma$-inverse mean curvature flow, together with a monotonicity formula developed in.

In this paper, we show that the modified mean curvature flow starting from an arbitrary graph in a Fuchsian manifold exists for all time and converges smoothly to an equidistant surface of constant mean curvature as $t\to \infty$. This result generalizes earlier work to the modified mean curvature flow setting and removes the restrictive global gradient bound initially required for the standard mean curvature flow by Huang, Zhou, and the second author.

Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\ge 4$, and $\ell =\lfloor \frac{n}{2}\rfloor$. We prove the sharp pointwise estimate $$q_2(E)\ge -\frac{2(\ell -1)}{3\ell }\mathrm{Scal}(E){\mathrm{Id}}_{{\Lambda }^2{V}^{\ast }}$$ for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature. Applying this estimate to the decomposition ${\mathrm{Rm}}_g=K_{\min }I+E$, we obtain the vanishing of $H^2(M;\mathbb{R})$ under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields $b_2(M)=0$ in odd dimensions and $b_2(M)\le 1$ in even dimensions. At even-dimensional endpoint, $b_2(M)>0$ forces $(M,g)$ to be isometric, up to scaling, to ${\mathbb{CP}}^{\ell }$ with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion $$K_{\min }\ge \frac{n(n-1)}{n^2-n+12}{S}_0⟹PIC2.$$ The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.

We develop a weak formulation of spacelike mean curvature flow in pseudo-Euclidean space. The framework is based on spacelike integer rectifiable varifolds and a pseudo-Euclidean version of first variation. Particular attention is paid to the presence of a fixed spacelike boundary. We introduce a generalized mean curvature vector and a weak spacelike conormal along the boundary. Under natural uniform spacelikeness and curvature assumptions, we prove a closure theorem for spacelike varifolds with boundary. We then define spacelike Brakke flows by means of the corresponding pseudo-Euclidean Brakke inequality, and prove a compactness theorem for sequences of spacelike Brakke flows with fixed boundary. Basic monotonicity properties are established, reflecting the sign structure of the ambient indefinite metric. Finally, we adapt Ilmanen's elliptic regularization procedure to prove existence of spacelike Brakke flows, and the local regularity theorem of White to the pseudo-Euclidean setting. The results provide a varifold setting for studying weak spacelike mean curvature flow beyond singularities.

In this paper, we establish a geometric correspondence between the Lorentzian Raychaudhuri equation and Perelman's non-collapsing theorem for the Ricci flow. By interpreting the Raychaudhuri equation as a Lorentzian analogue of Ricci flow, we connect geodesic focusing in general relativity to the monotonicity and entropy functionals of geometric analysis. Using this correspondence, we derive a Lorentzian non-collapsing theorem and introduce a covariant entropy functional governing causal volume evolution. Finally, we propose the concept of geodesic entropy capacity, a curvature-bounded limit on the information that can be stored in spacetime regions, providing a unified geometric framework linking gravitation, thermodynamics, and information.

We determine the asymptotics of a game inspired by classic vector balancing problems in combinatorial discrepancy theory. In this game, which we call the online Komlós game, two players, Paul and Carol, update the state vector $y$ in ${\mathbb{R}}^m$, initially placed at $0$. At each round, Paul chooses freely a set of $n$ vectors in the Euclidean unit ball, and Carol chooses, for each such vector, whether to leave it unchanged or reverse its sign. The resulting vectors are all added to $y$, and the game proceeds to a new round. After $T$ rounds, the game ends, and the ${\ell }_{\infty }$ norm of the state vector $y$ is determined. Paul's objective throughout the game is to maximize this norm, and Carol's objective is to minimize it. As $T$ gets large, we establish that the leading order term of the value of this game is $\sqrt{T/2\tau }$, where $\tau$ is the extinction time of the unit cube in ${\mathbb{R}}^m$ under a curvature-based flow characterized by the values of $m$ and $n$. When $n\ge m-1$, this flow is the mean curvature flow, and we show that $1/\sqrt{2\tau }=\Theta (\sqrt{\log m})$. Our results build upon the work of Kohn and Serfaty on deterministic games and mean curvature flow, combined with Banaszczyk's ${\ell }^2$ analogue of the Beck-Fiala theorem. As the large $T$ limit of the online Komlós game amounts to a localization of the classic Komlós problem, we hope this work can shed light on this and other vector balancing problems. Our results generalize to the version of the online Komlós game with the final value given by an arbitrary norm in ${\mathbb{R}}^m$.

We study the geometric regularization of a positive closed current by the (twisted) Kähler-Ricci flow on a compact Kähler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete Kähler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.

In geometry, it is important to study whether a given Lie group admits a special left-invariant geometric structure. In our previous work, we constructed nilpotent Lie algebras from finite quivers without cycles by utilizing paths within these quivers. Also, we proved that the simply-connected Lie groups corresponding to these nilpotent Lie algebras always admit left-invariant Ricci solitons. In this paper, we extend our approach by constructing solvable Lie algebras from finite quivers without cycles, adding vertices as paths of length zero. We demonstrate that the simply-connected Lie groups corresponding to these solvable Lie algebras always admit left-invariant Ricci solitons. Moreover, we prove that when the quivers are oriented multi-trees, these Ricci soliton Lie groups are rigid, that is the direct product manifold of a flat manifold and an Einstein manifold.

An $\infty$-harmonic function is a viscosity solution of ${\nabla }^{2}u(\nabla u,\nabla u)=0$, or equivalently, an absolute minimizer of $\Vert \nabla u{\Vert }_{L^{\infty }}$. We prove a variety of new structural and regularity results in two dimensions, including: 1. $\infty$-harmonic functions in domains of ${\mathbb{R}}^2$ are $C^{1,1/3}$. 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating $\infty$-harmonic functions to inverse mean curvature flow (IMCF) clusters – which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the $p\to \infty$ limit of the classical duality between $p$-harmonic and $q$-harmonic functions in ${\mathbb{R}}^2$, where $\frac{1}{p}+\frac{1}{q}=1$.

We investigate the stochastic Ricci flow of spherically symmetric perturbations of the Schwarzschild–Anti de Sitter black-hole metric. Elaborating on the Ricci-flow analysis of Headrick and Wiseman, we include a negative cosmological constant through a Ricci-target term and study how the flow is correlated with the thermodynamic heat capacity of the black hole. Numerical simulations show that, in the positive heat-capacity regime, perturbations of the angular sector of the metric relax toward the Schwarzschild–Anti de Sitter fixed point, while in the negative heat-capacity regime they grow under the deterministic Ricci flow. We then introduce a multiplicative stochastic noise and find that sufficiently strong stochasticity can suppress the growth of these perturbations, effectively stabilizing configurations that would otherwise be thermodynamically unstable. Finally, we reformulate the dynamics in terms of an entropy variable evolving on a thermodynamic free-energy landscape, and support the metric-flow results through Monte Carlo simulations and the associated Fokker–Planck equation. These results suggest that stochastic fluctuations can modify the relation between geometric stability under Ricci flow and thermodynamic stability in asymptotically Anti de Sitter black-hole spacetimes.