arXiv research

Study on shrinking solitons of generalized Ricci flow.

Study Bernstein results for self-shrinking solutions in Lagrangian flow.

Paper proves genus of surfaces decreases in mean curvature flow.

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…

Gradient shrinking solitons from Ricci flows terminating in cones.

Researchers create nonconvex, non-soliton ancient flows in various dimensions.

Study hexagonal network evolution under curvature flow.

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 use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.

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…

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that …

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

Uniqueness proven for specific types of geometric structures.

Ancient flows of elliptic functionals classified in various dimensions.

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.

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

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

Paper classifies 3D breathers and generalizes Ricci soliton results.

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…

Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructe…

Rigidity shown for spherical product Ricci solitons.

The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.

Study classifies ancient convex solutions to curve shortening flow.

Space curves with convex projections evolve smoothly until shrinking to a point.

Estimates heat equation on shrinking Ricci solitons with uniform bounds.

In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of $\mathcal{F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…

We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an ap…

Paper shows perturbed Taub-Bolt metric becomes singularity under Ricci flow.

Study pinched solutions of mean curvature flow in higher dimensions, proving singularities are specific types.

Paper classifies ancient solutions to 3D Ricci flow.

The $H^1(ds)$-gradient flow shrinks circles with radius $r_0$ to a point.

Study higher-dimensional Ricci flow solutions, proving uniqueness.

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…

Study ancient Ricci flow solutions, proving unique asymptotic behavior.

Study extends convergence results to noncompact Ricci flows.

The paper studies how convex surfaces shrink under mean curvature flow with a free boundary.

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…

The study explores $S^1$-invariant Laplacian flow on 6-manifolds.

The study finds only spheres shrink self-similarly with quotient curvature speeds.

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional…

Curve shortening flow shrinks curves to points.

New examples found for a type of geometric solitons.

To every Ricci flow on a manifold M over a time interval I, we associate a shrinking Ricci soliton on the space-time M x I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by considera…

The paper proves a unique solution to a shrinking flow equation converging to a soliton.

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

Backward propagation rules for warped products under Ricci flow.

In [Cheeger-Tian 2005], Cheeger-Tian proved an $ε$-regularity theorem for $4$-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensional manifolds and higher dim…

Suppose $M$ is a compact n-dimensional manifold, $n\ge 2$, with a metric $g_{ij}(x,t)$ that evolves by the Ricci flow $\partial_tg_{ij}=-2R_{ij}$ in $M\times (0,T)$. We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality.