arXiv research

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

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

The paper classifies noncollapsed translators in 4D space.

Classifies ancient noncollapsed flows in 4D space.

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

We prove that any translating soliton for the mean curvature flow which is noncollapsed and uniformly 2-convex must be the rotationally symmetric bowl soliton. In particular, this proves a conjecture of White and Wang, in the 2-convex case in arbitrary dimension.

In this paper, we consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a rotationally symmetric translating soliton.

Proves local noncollapsing estimate for mean curvature flow.

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.

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,…

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

Classifies ancient solutions to curvature flows, finding two main types.

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.

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.

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.

Study ancient flows in 4D, classifying based on bubble-sheet eigenvalues.

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

We consider ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq γ_1 |H|^2, |\nabla^2 A| \leq γ_2 |H|^3$. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, …

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 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.

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…

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.

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.

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…

We found a 6D manifold with specific geometric properties.

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

This research shows that steady solitons in higher dimensions always reduce at infinity.

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…