arXiv research

Establishes 4D regularity for certain metric spaces.

We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound are topologically spheres. As an application we show that for any finite dimensional Alexandrov space $X^n$ with $n\ge 5$ there exists an Alexandrov space $Y$ homeomorphic to $X$ which can not be o…

The paper shows that certain Einstein orbifolds cannot be limits of smooth Einstein metrics.

Proves curvature tensor convergence for smoothable spaces.

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.

We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence $X_i$ of Alexandrov spaces with curvature bounded below Gromov-Hausdorff converging to a compact Alexandrov space $X$, $X_i$ is homeomorphic to $X$ for all large $i$.

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$δ$ inner regions" which avoid the boundary by a distance $δ$. We prove Gromov-Hausdorff compactness theorems for sequences of the…

In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncoll…

The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.

Proves local noncollapsing estimate for mean curvature flow.

Given a sequence of complete(compact or noncompact) Kähler manifolds $M^n_i$ with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of complex structure of …

Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.

Study noncollapsed F-limit metric solitons, proving properties similar to smooth Ricci shrinkers.

Study confirms conjectures on Ricci limit spaces and their topological properties.

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an ap…

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$ with positive mean curvature is $κ$-noncollapsing, and a blow-up sequence conve…

New proof classifies ancient flows in 3D space.

In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the implication in the other direction, that for an ancient solution with bounded no…

In this paper, we analyze the asymptotic behavior of $κ$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $κ$-noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.

Ancient solutions to Kähler Ricci flow classified completely.

Paper classifies singularity models for 3D hypersurfaces in R^4.

Paper disproves potential singularity models for 3D hypersurfaces in R^4.

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative c…

Study on ancient Ricci flows with positive curvature, proving noncollapsedness.

Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the actio…

In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.

We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.

The study characterizes and rules out collapsing in convex ancient mean curvature flow.

Enhanced estimates for ancient ovals and translators in 3D and 4D.

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in …

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph $(y,u(|y|))$ over $\mathbb{R}^n$, $n\geq 2$, satisfying there exist positive constants $ε$, $c$ and $N$ such that $u'(r)\geq c r^ε$ for $r\geq N$, the longtime solution to mean curvature flow with initial data $(y,…

The paper studies how certain surfaces evolve in space without collapsing.

Quantitative estimate for curvature in mean curvature flow.

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $κ$-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the rotationally symm…

Sharp bound on singular set dimension for specific geometric problems.

The paper classifies noncollapsed translators in 4D space.

Paper studies fundamental groups of certain Ricci solitons.

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

The paper shows that ancient noncollapsed mean curvature flows have a blowdown of at most n-2 dimensions.

In this paper, we study $κ$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$. By making use of the properties of…

In this paper, we prove that any $κ$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $ε$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $κ$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature oper…

The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.

The paper studies singular sets in Ricci flow limits, proving rectifiability and curvature bounds.

Classifies ancient noncollapsed flows in 4D space.

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k…