Related papers: Two-sided non-collapsing curvature flows
In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold…
Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane…
In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's compactness theorem) for the…
Governing equations for two-dimensional inviscid free-surface flows with constant vorticity over arbitrary non-uniform bottom profile are presented in exact and compact form using conformal variables. An efficient and very accurate…
We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous…
We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the…
We consider two-dimensional Riemann boundary value problems of Euler equations for the Chaplygin gas with two piecewise constant initial data outside a convex cornered wedge. In self-similar coordinates, when the flow at the wedge corner is…
In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature and other curvature functions with suitable properties.…
We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges super-exponentially fast, then it must coincide with the cylinder itself. We also show that the result is sharp with…
We consider the flow of an { ideal} fluid in a 2D-bounded domain, admitting flows through the boundary of this domain. The flow is described by Euler equations with \textit{non-homogeneous } Navier slip boundary conditions. These conditions…
We continue the study, initiated by the first two authors in \cite{IW19}, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the…
We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space $\mathbb{H}^{n+1}$. The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain…
We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely…
An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction--diffusion process on the surface,…
In this paper, we study the $k$-Hessian curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We first prove the existence of translating solutions with given asymptotic behavior. Then, we prove that for strictly convex…
We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore…
This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. First, we introduce a class of generalized curvatures, and prove the existence and uniqueness for the…
We exhibit a concentration-collapse decomposition of singularities of fourth order curvature flows, including the $L^2$ curvature flow and Calabi flow, in dimensions $n \leq 4$. The proof requires the development of several new a priori…
We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants,…
We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We…