Related papers: Self-Expanders to Inverse Curvature Flows by Homog…
We construct nonradial, self-similar solutions to the two-dimensional incompressible Euler equations without assuming rotational symmetry. These solutions extend the study of self-similar algebraic spiral flows, initiated by Elling and…
The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct…
We prove a complete family of `cylindrical estimates' for solutions of a class of fully non-linear curvature flows, generalising the cylindrical estimate of Huisken-Sinestrari for the mean curvature flow. More precisely, we show that, for…
We address the asymptotic behavior of the $\alpha$-Gauss curvature flow, for $\alpha >1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the 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…
We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that…
We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a…
In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space $\mathbb{R}^{n+1}$, \begin{align*}…
In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the…
In this paper, we study the $\sigma_k$ curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show…
We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact…
In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under…
Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along…
In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $\Lambda_{n-2}$. In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity…
This expository paper presents the current knowledge of particular fully nonlinear curvature flows with local forcing term, so-called locally constrained curvature flows. We focus on the spherical ambient space. The flows are designed to…
Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean…
We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by…
In this paper, inspired by the work of Spruck-Xiao [27] and based partly on a result of Derdzi\'nski [11], we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in $\mathbb{R}^{n+1}$. More…
In the paper, we construct, for $\lambda>0$, complete embedded and non-convex $\lambda$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $\lambda$-hypersurfaces share a common conclusion on the planar…
Given a smooth, symmetric, homogeneous of degree one function $f\left(\lambda_{1},\cdots,\,\lambda_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\,n$, and a rotationally symmetric cone $\mathcal{C}$ in $\mathbb{R}^{n+1}$, we…