Limit spaces & RCD

We prove that a complete noncompact gradient shrinking Ricci soliton of dimension at least two is a finite quotient of an Einstein cylinder if its scalar curvature is bounded below by the cylindrical value. The corresponding pointwise strict bound implies compactness. We also establish one-endedness criteria using curvature information at critical points of the potential, including degenerate ones, and obtain sharp one-endedness and rigidity results under global Ricci norm bounds.

We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.

We study global smooth Calabi flows on polarized Kähler surfaces $(M,L)$, assuming uniformly bounded Sobolev constants and the algebraic conditions $\chi ({\mathcal{O}}_M)=1$ and $c_1(M)\cdot c_1(L)>0$. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and $|S|\le 1$. We illustrate the hypotheses with nonstationary flows near a product cscK metric on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.

We study global Calabi flows $g(t)$ on compact Kähler surfaces with a uniform Sobolev constant. For any sequence $t_j\to \infty$, set $g_j(t):=g(t_j+t)$. After passing to a subsequence, these flows converge to an eternal Calabi flow on a compact extremal Kähler orbifold surface. The convergence holds in the spacetime ${\hat{C}}^{\infty }$-Cheeger–Gromov topology: it is uniform in the Gromov–Hausdorff sense on compact time intervals and smooth away from the orbifold singularities, using comparisons fixed in time.

We develop the weighted Sobolev and Hölder analysis needed to study perturbations of complete asymptotically conical (AC) Ricci shrinkers. Weighted elliptic estimates yield a global weighted divergence-free slice in spaces that retain the asymptotic data at infinity. As a geometric application, we prove pointed Gromov-Hausdorff rigidity of nontrivial AC shrinkers with vanishing infinitesimal solitonic deformation space.

In this paper, we establish a new estimate on the time-derivative of pointed Nash entropy and use it to give a new proof of backward pseudolocality for Ricci flow. The proof does not rely on Bamler's compactness theorem and structure theory concerning limits of Ricci flows. Our methods also work for other reasonable geometric flows, like the $G_2$-Laplacian, at a scale depending on its deviation from Ricci flow and its scalar curvature.

Let an isometric involution act on a smooth Riemannian $3$-sphere with sectional curvature at least one. We prove that every one-dimensional connected component of its fixed-point set has length at most $2\pi$, and that the bound is sharp. The proof realizes the fixed circle as the link of a complete totally geodesic surface in a positively curved expanding Ricci soliton and applies Bishop–Gromov comparison to its area growth. The analytic input is an equivariant version of Deruelle's smoothing of positively curved cones. We use Gianniotis–Schulze's equivariant local deformation and explain a marked compactness argument for arbitrary finite isotropy. In particular, the end chart retains the specified group action, rather than only an unspecified conjugate.

In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial $C^0$ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

Let $X$ be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class $[\omega ]$, Jian–Shi–Song proved that there exists a unique cscK metric in the class $2\pi c_1(K_X)+\epsilon [\omega ]$ when $\epsilon \in (0,{\epsilon }_0]$, using the properness criterion of Chen–Cheng. We prove that the cscK metrics in $2\pi c_1(K_X)+\epsilon [\omega ]$ converge, as $\epsilon \downarrow 0$, in the Gromov–Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of $X$ as conjectured by Jian–Shi–Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee–Tosatti–Zhang on the Gromov-Hausdorff convergence of the normalized Kähler–Ricci flow on $X$.

We study the $\alpha$-Ricci flow $(g(t),\alpha (t))$, which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus $M$. We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the $\alpha$-Ricci flow depends on the cohomology class $[\alpha ]\in H^1(M,\mathbb{R})$, which is invariant under the flow. If $[\alpha ]$ is rational, then as $t\to \infty$, $(M,t^{-1}g(t))$ Gromov–Hausdorff converges to $S^1$ with length $\ell ([\alpha ])/\sqrt{2}$ where $\ell ([\alpha ])$ is the primitive length of $[\alpha ]$. On the universal cover, the rescaled flow $(s^{-1}\tilde{g}(st),\tilde{\alpha }(st))$ smoothly converges to an expander $(g^{\ast }(t),{\alpha }^{\ast }(t))$ on ${\mathbb{R}}^2$ as $s\to \infty$ where ${\alpha }^{\ast }(t)$ is nonzero and constant in spacetime and $g^{\ast }(t)$ is flat and expanding in the ${\alpha }^{\ast }$ direction. If $[\alpha ]$ is irrational, then there is a sequence $s_i\to \infty$ such that $(M,s_i^{-1}g(s_{i}t))$ Gromov–Hausdorff converges to a point. Moreover, on the universal cover, $(s_i^{-1}\tilde{g}(s_{i}t),\tilde{\alpha }(s_{i}t))$ smoothly converges to $(g^{\ast }(t),{\alpha }^{\ast }(t))$.

Let compact $\mathrm{RCD}(\kappa ,N)$ spaces converge in the measured Gromov–Hausdorff sense, and suppose that they have $(\delta ,\rho )$-Reifenberg local covers for one fixed $\rho >0$, with $|\kappa |{\rho }^2\le 1$ and $\delta$ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of $O(n)$ whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with $O(n)$, with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition $A_s$ it is a closed aspherical manifold. Three appendices treat smooth manifolds with $\mathrm{Ric}\ge -(n-1)$ and small local rewinding Reifenberg error: a uniform short-time Ricci flow with $|\mathrm{Rm}|\le \alpha /t$, also for complete non-compact data, where it is unique among complete flows with curvature at most $A/t$; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov–Hausdorff approximations with small error.

We study a positive radial graph evolving by $V=H^{\alpha }+b$, with $\alpha >0$ and $b<0$, in the unit cylinder and with $u_r(1,t)=u(1,t)$. We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for $H$, $u_r/r$, and $u_{rr}$. We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.

We establish a Type I curvature bound for finite-time volume-collapsing Kähler–Ricci flows on compact Kähler surfaces. No symmetry assumption is imposed, and the initial Kähler class need not be rational. The proof combines local symplectic topology with the classification of gradient Kähler–Ricci shrinking solitons. Bamler's compactness and structure theory provides the shrinking limits used in the argument.

We prove a uniform $L^{\infty }$-estimate for the parabolic Monge–Ampère equation under uniform Hörmander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled Kähler–Ricci flows on a family of polarized Calabi–Yau manifolds over the punctured disk. An interpolation argument gives uniform Hörmander-Tian type estimates along the flows. We also establish uniform diameter bounds and Gromov–Hausdorff precompactness for this family of flows. In particular, we obtain an alternative proof of the diameter estimate by Li-Tosatti.

We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time slices of a single orbifold Ricci flow whose metric initial condition is the given polyhedral space. The proof rests on a new short-time existence and regularization theory for Ricci flow that, under two-sided volume bounds and a generalized segment inequality, replaces pointwise lower curvature control by small scale-invariant integral control of the defect from a preserved curvature cone. Although it allows arbitrarily large pointwise violations, this theory yields an existence time and positive-time curvature estimates independent of the initial upper curvature bound. As a further application, it gives rigidity consequences for manifolds with small integral curvature defect.

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.

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.