Related papers: Existence and regularity theorems of one-dimension…

Let $X$ be a compact K\"ahler manifold. We prove that the K\"ahler-Ricci flow starting from arbitrary closed positive $(1,1)$-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has…

We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing…

We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some…

In this paper we prove an analogue of the Brakke's $\varepsilon$-regularity theorem for the parabolic Allen-Cahn equation. In particular, we show uniform $C^{2,\alpha}$ regularity for the transition layers converging to smooth mean…

In this paper we aim to study the consistency of the mean curvature flow via discretization. We will use discretizations by volumetric varifolds, and derive a Brakke approximate equality involving the masses of the volumetric varifolds and…

We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called $k$-dimensional $(\varepsilon,R)$ Reifenberg flat sets in $\mathbb{R}^n$. Our results generalize the ones…

We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of…

This paper studies singularities of mean curvature flows with integral mean curvature bounds $H \in L^\infty L^p_{loc}$ for some $p \in ( n, \infty]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$.…

We show that on any Riemannian surface for each $0<c<\infty$ there exists an immersed $C^{1,1}$ curve that is smooth and with curvature equal to $\pm c$ away from a point. We give examples showing that, in general, the regularity of the…

In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex…

Given two curves bounding a region of area $A$ that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time $A/\pi$. We prove…

We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial…

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our…

We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$…

We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of…

We solve the problem of topological classification for smooth structurally stable flows on closed four-dimensional manifolds, the non-wandering set of which contains exactly two saddle equilibria, and the wandering set contains isolated…

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…

In 1978 Brakke introduced the mean curvature flow in the setting of geometric measure theory. There exist multiple variants of the original definition. Here we prove that most of them are indeed equal. One central point is to correct the…

This paper focuses on the analysis of stratified steady periodic water waves that contain stagnation points. The initial step involves transforming the free-boundary problem into a quasilinear pseudodifferential equation through a conformal…

We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and…