arXiv research

We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we sh…

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being sh…

We show that every complete entire self-shrinking solution on complex Euclidean space to the Kahler-Ricci flow must be generated from a quadratic potential.

We show that every entire self-shrinking solution on $\mathbb{C}^1$ to the Kähler-Ricci flow must be generated from a quadratic potential.

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the pseudo-Euclidean metric is flat if the H…

Study hexagonal network evolution under curvature flow.

We prove that every entire self-shrinking solution on $\mathbb{C}^n$ to the Kähler-Ricci flow with strictly real convex potential must be quadratic. The very same argument also gives a pointwise proof for the rigidity of entire self-shrinking solutions to Lagrangian mean curvature flow in pseudo-Euclidean space obtaine…

Paper classifies ancient solutions to 3D Ricci flow.

We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.

Study higher-dimensional Ricci flow solutions, proving uniqueness.

Paper classifies 3D breathers and generalizes Ricci soliton results.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers in arbitrary codimension, in codimension one case, our results reduce to Colding-Minicozzi's res…

In this paper, we consider a family of closed hypersurfaces which shrink self-similarly with speed of quotient curvatures. We show that the only such hypersurfaces are shrinking spheres.

As an application of his entropy formula, Perelman proved that every compact shrinking breather is a shrinking gradient Ricci soliton. We give a proof for the complete noncompact case by using Perelman's $\mathcal{L}$-geometry. Our proof follows the argument in Lu and Zheng of constructing an ancient solution, and remo…

Proof that certain complex equations have only simple solutions.

Gradient shrinking solitons from Ricci flows terminating in cones.

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere…

In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This equation reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, tr…

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

I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cas…

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show …

New examples found for a type of geometric solitons.

New classification for higher-dimensional shrinking Ricci solitons with positive isotropic curvature.

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinkin…

On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backwa…

Study on four-dimensional Ricci solitons and multiply warped Ricci flow solutions.

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

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, …

Constructs explicit solutions to Spin(7)-structures gradient flow.

We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed numbers of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable.

Study applies Huisken formula to mean curvature flow in Ricci soliton background.

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinea…

We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stab…

We use analytical and numerical methods to investigate the equations for cohomogeneity one shrinking gradient Ricci solitons. We show the existence of a winding number for this system around the subvariety of phase space corresponding to Einstein solutions and obtain some estimates for it. We prove a non-existence resu…

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which we describe. This analysis applies in particular to the ancient solution constru…

In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking Kähler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li about the non-existence…

Study extends convergence results to noncompact Ricci flows.

The study finds that only round spheres shrink self-similarly under certain curvature flows.

Solves complex equation for specific geometric solitons.

It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $κ$-solution. We prove that the every noncompact ancient $κ$-solution in dimension $3$ is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.

Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact …

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a boun…

This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The motivation for the self-similar solutions arises from natural area preserving rescaling. Further we describe the self-sim…

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable…

New stable shrinking Ricci soliton found in 4D.

Solves complex Monge-Ampère equation for Kähler-Ricci solitons.

We consider a compact, star-shaped, mean convex hypersurface $Σ^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …