3-manifolds

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.

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.

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 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.

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 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.

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.

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 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.

Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any $t_0>0$ and any connected component $Z_{t_0}^{\prime }$ of the time-slice $Z_{t_0}$ of the completion, we prove that $$(\overline{{\mathcal{R}}_t^{\prime }},d_{g_t})\underset{t\nearrow t_0}{\overset{\mathrm{Gromov}\text{-}\mathrm{Hausdorff}}{\to }}(Z_{t_0}^{\prime },d_{t_0}^Z),$$ where ${\mathcal{R}}_t^{\prime }$ is the corresponding connected component of the regular part and $\overline{{\mathcal{R}}_t^{\prime }}$ denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic $1$-rectifiable, and its time image has vanishing $1/2$-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most $1$. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.

We construct an explicit two-parameter family of complete, non-compact, three-dimensional, smooth steady gradient generalized Ricci solitons with $\mathrm{SO}(2)\times \mathbb{R}$ symmetry, providing a cylindrical counterpart to the spherically symmetric solitons recently found by Podestà and Raffero. The family is parametrized by a flux constant $k>0$ and a conserved quantity $\mathcal{C}\ge 0$. For $\mathcal{C}=0$, the asymptotic geometry exhibits power-law decay; for $\mathcal{C}>0$, the metric converges exponentially fast to a flat cylinder of finite radius.

Perelman's proof of the Poincare conjecture shows that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. The fundamental groups of 3-manifolds attract lots of interest from mathematicians of different fields. As it was stated in a famous survey of Allen Hatcher "The classification of 3-manifolds", one would want to know exactly which groups occur as fundamental groups of these manifolds. The Stallings-Jaco-Hempel reformulation of the Poincare conjecture inspired several connections between low-dimensional topology, equations over free groups, and combinatorial group theory. The reformulation reduces the problem to study epimorphisms from the fundamental group of a closed orientable surface onto the direct product of two free groups (they correspond to Heegaard splittings of 3-manifolds and were named splitting homomorphisms). Olshankii (1989) constructed (in non-explicit form) first non-trivial examples of such splitting epimorphisms and verified the standardness of some of them. We construct up to equivalence all the splitting coordinate-surjective homomorphisms (among them, the genuine splitting epimorphisms are exactly those for which our constructed associated group balanced presentation is trivial). We give generators and relations of the corresponding balanced presentation (so all closed orientable 3-manifold groups) that can be studied by algebraic methods. We analyse a big class of such homomorphisms/presentations (including all Olshanskii's epimorphisms) and show that splitting epimorphisms are rare, in this case the corresponding balanced presentation of the trivial group can be reduced to the standard one by Andrews-Curtis transformations and the epimorphisms are standard.

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.

In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane waves. We obtain an explicit description in global coordinates of these special homogeneous Lorentzian manifolds. We then prove that all such examples are non-gradient expanding Ricci solitons.

In this paper, we study the singular set $\mathcal{S}$ of a noncollapsed Ricci flow limit space, arising as the pointed Gromov–Hausdorff limit of a sequence of closed Ricci flows with uniformly bounded entropy. The singular set $\mathcal{S}$ admits a natural stratification: \beginequation* \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^n-2=\mathcal S, \endequation* where a point $z\in {\mathcal{S}}^k$ if and only if no tangent flow at $z$ is $(k+1)$-symmetric. In general, the Hausdorff dimension of ${\mathcal{S}}^k$ with respect to the spacetime distance is at most $k$. We show that the subset ${\mathcal{S}}_{\mathrm{qc}}^k\subset {\mathcal{S}}^k$, consisting of points where some tangent flow is given by a standard cylinder or its quotient, is parabolic $k$-rectifiable. In dimension four, we prove the stronger statement that each stratum ${\mathcal{S}}^k$ is parabolic $k$-rectifiable for $k\in \{0,1,2\}$. Furthermore, we establish a sharp uniform ${\mathcal{H}}^2$-volume bound for $\mathcal{S}$ and show that, up to a set of ${\mathcal{H}}^2$-measure zero, the tangent flow at any point in $\mathcal{S}$ is backward unique. In addition, we derive $L^1$-curvature bounds for four-dimensional closed Ricci flows. As an application, we resolve Perelman's bounded diameter conjecture for three-dimensional closed Ricci flows.

We prove that a three dimensional compact Ricci flow that encounters a Type I singularity has uniformly bounded diameter up to the singular time, thus giving an affirmative answer - for Type I singularities - to a conjecture of Perelman. To achieve this, we introduce a concept of a neck-region for a Ricci flow, analogous to the neck-regions introduced by Jiang-Naber and Cheeger-Jiang-Naber, in the study of Ricci limit spaces. We then prove that the associated packing measure is, in a certain sense, Ahlfors regular, a result that holds in any dimension.

In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. Furthermore, we formulate a general method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [8].

This paper attempts to construct monotonic entropy functionals for four-dimensional Lorentzian spacetime under physical boundary conditions, as an extension of Perelman's monotonic entropy functionals constructed for three-dimensional compact Riemannian manifolds. The monotonicity of these entropy functionals is utilized to prove the well-posedness of applying Ricci flow to four-dimensional Lorentzian spacetime for a long flow-time, particularly for the timelike modes which would seem blow up and ill-defined. The general idea is that the Ricci flow of a Lorentzian spacetime metric and the coupled conjugate heat flow of a density on the Lorentzian spacetime as a whole turns out to be the gradient flows of the monotonic functionals for a long flow-time, so the superficial "blow-up" in the individual Ricci flow system or the conjugate heat flow system contradicts the boundedness of the monotonic functionals within finite flow interval, which gives a semi-global control to the whole coupled system. The physical significance and applications of these monotonic entropy functionals in real gravitational systems are also discussed.

This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by $-1$ and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by $-6$ and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly $C^0$-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a weighted Hölder norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].

In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking Kähler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.

In this paper, we study hypersurfaces in the product spaces ${\mathbb{Q}}_{\epsilon }^3\times \mathbb{R}$ for which the tangential component $T$ of the vector field $\frac{\partial }{\partial t}$ is a principal direction, where ${\mathbb{Q}}_{\epsilon }^3$ denotes the three-dimensional non-flat Riemannian space form with sectional curvature $\epsilon =\pm 1$, and $\frac{\partial }{\partial t}$ is the unit vector field tangent to the $\mathbb{R}$-factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field $T$. Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons.

A Ricci soliton is a natural generalization of an Einstein metric. On a pseudo-Riemannian manifold (M, g), it is defined by : $LX g + \rho = λ g, where X is a smooth vector field on M , LX denotes the Lie derivative in the direction of X, \rho is the Ricci tensor, and λ is a real constant. In this paper, we establish the existence of non-trivial Ricci solitons on a family of three-dimensional Lorentzian Walker manifolds.

In his seminal work, Rivin characterized finite ideal polyhedra in three-dimensional hyperbolic space. However, the characterization of infinite ideal polyhedra, as proposed by Rivin, has remained a long-standing open problem. In this paper, we introduce the combinatorial Ricci flow for infinite ideal circle patterns, a discrete analogue of Ricci flow on non-compact Riemannian manifolds, and prove a characterization of such circle patterns under certain combinatorial conditions. Our results provide affirmative solutions to Rivin's problem.

In this paper, we study the $t$-Gauduchon Ricci-flat condition under the Chern-Ricci flow. In this setting, we provide examples of Chern-Ricci flow on compact non-Kähler Calabi-Yau manifolds which do not preserve the $t$-Gauduchon Ricci-flat condition for $t<1$. The approach presented generalizes some previous constructions on Hopf manifolds. Also, we provide non-trivial new examples of balanced non-pluriclosed solution to the pluriclosed flow on non-Kähler manifolds. Further, we describe the limiting behavior, in the Gromov-Hausdorff sense, of geometric flows of Hermitian metrics (including the Chern-Ricci flow and the pluriclosed flow) on certain principal torus bundles over flag manifolds. In this last setting, we describe explicitly the Gromov-Hausdorff limit of the pluriclosed flow on principal $T^2$-bundles over the Fano threefold $\mathbb{P}(T_{{\mathbb{P}}^2})$.

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this article, we confirm this conjecture for a class of 3-manifolds. To be precise, let $M$ be an oriented compact 3-manifold with boundary, no component of which is a 2-sphere, and $\mathcal{T}$ is an ideal triangulation of $M$. If $\mathcal{T}$ satisfies properly gluing condition, and the valence is at least 6 at each ideal edge and 11 at each hyper-ideal edge, then $M$ admits an unique complete hyperbolic metric with totally geodesic boundary, so that $\mathcal{T}$ is isotopic to a geometric ideal triangulation of $M$. We use analytical tools such as combinatorial Ricci flow (CRF, abbr.) to derive the conclusions. There are intrinsic difficulties in dealing with CRF. First, the CRF may collapse in a finite time, second, most of the smooth curvature flow methods are no longer applicable since there is no local coordinates in $\mathcal{T}$, and third, the evolution of CRF is affected by certain combinatorial obstacles in addition to topology. To this end, we introduce the ideas as "extending CRF", "tetrahedral comparison principles", and "control CRF with edge valence" to solve the above difficulties. In addition, the presence of torus boundary adds substantial difficulties in this article, which we have solved by introducing the properly gluing conditions on $\mathcal{T}$ and reducing the ECRF to a flow relatively easy to handle.

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric decomposition into ideal hyperbolic tetrahedra, a result proven only for certain special 3-manifolds. This paper presents combinatorial Ricci flow as a systematic and general approach to addressing Thurston's triangulation conjecture, showing that the flow converges if and only if the triangulation is geometric. First, we prove the rigidity of the most general hyperbolic polyhedral 3-manifolds constructed by isometrically gluing partially truncated and decorated hyperbolic tetrahedra, demonstrating that the metrics are uniquely determined by cone angles modulo isometry and decoration changes. Then, we demonstrate that combinatorial Ricci flow evolves polyhedral metrics toward complete hyperbolic structures with geometric decompositions when convergent. Conversely, the existence of a geometric triangulation guarantees flow convergence.

In the present paper we study $\eta$-Ricci solitons on Kenmotsu 3-manifolds. Moreover, we consider $\eta$-Ricci solitons on Kenmotsu 3-manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Beside these, we study $\phi$-Ricci symmetric $\eta$-Ricci soliton on Kenmotsu 3-manifolds. Also Kenmotsu 3-manifolds satisfying the curvature condition $R.R=Q(S,R)$ is considered. Finally, an example is constructed to prove the existence of a proper $\eta$-Ricci soliton on a Kenmotsu 3-manifold.

We use Lott's functional and construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature. Consequently, we prove that the blowdown limit is locally an expanding Ricci soliton when the structure group is the three dimensional Heisenberg group. In addition, we classify this soliton when the base manifold is one dimensional. This, together with Lott's work in the abelian setting, yields a complete local classification of invariant Ricci flow blowdown limits on four dimensional, nilpotent principal bundles.

We explore three versions of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi–Yau 3-folds, interpreting their dimensional reductions to the Kähler geometry of the base. Precisely, we reduce Ansätze for the Laplacian coflow, modified or not by de Turck's trick, both on trivial products $C{Y}^3\times S^1$ and on contact Calabi–Yau 7-manifolds, obtaining in each case a natural modification of the Kähler–Ricci flow.

Gushel-Mukai manifolds are specific families of $n$-dimensional Fano manifolds of Picard rank $1$ and index $n-2$ where $3\le n\le 6$. A Gushel-Mukai $n$-fold is either ordinary, i.e. a hyperquadric section of a quintic Del Pezzo $(n+1)$-fold, or special, i.e. it admits a double cover over the quintic Del Pezzo $n$-fold branched along an ordinary Gushel-Mukai $(n-1)$-fold. In this paper, we prove that a general special Gushel-Mukai $n$-fold is K-stable for every $3\le n\le 6$. Furthermore, we give a description of the first and last walls of the K-moduli of the pair $(M,cQ)$, where $M$ is the quintic Del Pezzo fourfold (or fivefold) and $Q$ is an ordinary Gushel-Mukai threefold (or fourfold). Besides, we compute $\delta$-invariants of quintic Del Pezzo fourfolds and fivefolds which were shown to be K-unstable by K. Fujita, and show that they admit Kähler-Ricci solitons.

This is a survey on the Strominger system and a geometric flow known as the anomaly flow. We will discuss various aspects of non-Kähler geometry on Calabi-Yau threefolds. Along the way, we discuss balanced metrics and balanced classes, the Aeppli cohomology class associated to a solution to the Strominger system, the equations of motion of heterotic supergravity, and a version of Ricci flow in this special geometry.

This article carries out the investigation of a three-dimensional Riemannian manifold $N^3$ endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on $N^3$. It is established that a $N^3$ with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of $N^3$ with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either $N^3$ is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold $N^3$ with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and $m$-quasi Einstein solitons of gradient type, respectively.

We explicitly determine the optimal degenerations of Fano threefolds $X$ in family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration $(\mathcal{X},{\xi }_0)$ of $X$ such that $({\mathcal{X}}_0,{\xi }_0)$ is weighted K-polystable, which is equivalent to $({\mathcal{X}}_0,{\xi }_0)$ admitting a Kähler-Ricci soliton (KRS) by and. Furthermore, we study the moduli spaces of $({\mathcal{X}}_0,{\xi }_0)$. The $\mathbf{H}$-invariant of $X$ divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves $C\subseteq {\mathbb{P}}^1\times {\mathbb{P}}^1$, and the other one is a single point.

We prove the existence of a one-parameter family of pairwise non-isometric, complete, positively curved, steady generalized Ricci solitons of gradient type on ${\mathbb{R}}^3$ that are invariant under the natural cohomogeneity one action of SO(3). In the context of generalized Ricci flow, this result represents the analogue of Bryant's construction of the complete rotationally invariant steady soliton for the Ricci flow.

We study the behavior of a three-dimensional dynamical system with respect to some set $S$ given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter $a\in (0,1/2)$, as for $S$ it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that $S$ is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and $S$ for all $a\in (0,1/2)$. These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases $a\in \{1/9,1/8,1/6\}$ of generalized Wallach spaces.

In this paper we study 4d gradient steady Ricci solitons, which are weak $\kappa$-solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those for 3d compact $\kappa$-solutions found in [ABDS22]. This is the first step towards the classification of 4d gradient steady Ricci solitons and more general ancient Ricci flows.

We develop a general theory of flows in the space of Riemannian metrics induced by neural network gradient descent. This is motivated in part by recent advances in approximating Calabi-Yau metrics with neural networks and is enabled by recent advances in understanding flows in the space of neural networks. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel, a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman's formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi-Yau metrics, as is expected since the associated neural networks do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of neural network metric flows therefore explains why neural networks are better at learning Calabi-Yau metrics than fixed kernel methods, such as the Ricci flow.

We find Fano threefolds $X$ admitting Kähler-Ricci solitons (KRS) with non-trivial moduli, which are $\mathbb{T}$-varieties of complexity two. More precisely, we show that the weighted K-stability of $(X,{\xi }_0)$ (where ${\xi }_0$ is the soliton candidate) is equivalent to certain GIT-stability. In particular, this provides the first examples of strictly weighted K-semistable Fano varieties. On the other hand, we generalize Koiso's theorem to the log Fano setting. Indeed, we show that the K-stability of a log Fano pair $(V,{\Delta }_V)$ is equivalent to the weighted K-stability of a cone $(Y,{\Delta }_Y,{\xi }_0)$ over it. This also leads to new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups. To achieve these, we establish the weighted Abban-Zhuang estimate generalizing the work of, which gives a lower bound of the weighted stability threshold ${\delta }_{\mathbb{T}}^g(X,\Delta )$. This is an effective way to check the weighted K-semistablity of a log Fano triple $(X,\Delta ,{\xi }_0)$. This estimate is also useful in testing (weighted) K-polystability based on the work of.

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.

We introduce a classification conjecture for $\kappa$-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of ${\mathbb{Z}}_2^2\times {\mathrm{O}}_3$-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow $g(t)$ emerging from an arbitrary 3D complete noncompact Riemannian manifold $(M^3,g_0)$ which has nonnegative Ricci curvature. We show $g(t)$ is complete for positive times provided $g_0$ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show $g(t)$ is complete for positive times provided $g_0$ is a compactly supported perturbation of a nonnegative sectional curvature metric on ${\mathbb{R}}^3$.