arXiv research

Proves local noncollapsing estimate for mean curvature flow.

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

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

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

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.

Quantitative estimate for curvature in mean curvature flow.

Establishes 4D regularity for certain metric spaces.

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

The paper classifies noncollapsed translators in 4D space.

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

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

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

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

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

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…

Paper reconciles different Ricci flow approaches and proves weak solutions.

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.

Develops local curvature estimates for mean curvature flow.

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

Classifies ancient noncollapsed flows in 4D space.

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.

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.

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds $(M^n,g)$ with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow} (X,d)$, where $d_j$ denotes the Riemannian distance. Our main result is a solution to the codimen…

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

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…

Paper proves volume growth estimate for steady gradient Ricci solitons.

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.

Paper studies fundamental groups of certain Ricci solitons.

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…

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…

Study on biharmonic heat equation on manifolds with curvature constraints.

The paper improves estimates on singular sets in manifolds with integral curvature bounds.

Ancient solutions to Ricci flow with isotropic curvature conditions are classified.

We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.

Consider a Riemannian manifold with bounded Ricci curvature $|\Ric|\leq n-1$ and the noncollapsing lower volume bound $\Vol(B_1(p))>\rv>0$. The first main result of this paper is to prove that we have the $L^2$ curvature bound $\fint_{B_1(p)}|\Rm|^2 < C(n,\rv)$, which proves the $L^2$ conjecture. In order to prove this…

We prove that any noncompact $κ$-noncollapsed steady Ricci soliton with nonnegative curvature operator must be rotationally symmetric if it has a linear curvature decay.

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

We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any re…

In this paper we investigate the differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method that was introduced by the first two named authors in \cite{LaTi14}.

Haslhofer and Müller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly be an orbifold Ricci shrinker. In this paper we prove a compactness theorem for non…

Estimates on Einstein manifolds improve Brownian motion behavior and curvature limits.