Related papers: Some existence results for the modified binormal c…

For certain non linear evolution equations, existence of global in time flows for large data is a fundamental and difficult question. In general, for dispersive and wave equations high regularity of the data does not automatically guarantee…

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal…

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set…

We provide a curvature flow approach to the regular Christoffel-Minkowski problem. The speed of our curvature flow is of an entropy preserving type and contains a global term.

In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable…

In this paper, we prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves Baker's convergence theorem. In particular, we obtain a new differentiable sphere theorem for submanifolds in…

We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results.

In this paper, for a given compact 3-manifold with an initial Riemannian metric and a symmetric tensor, we establish the short-time existence and uniqueness theorem for extension of cross curvature flow. We give an example of this flow on…

We give a new proof of Brakke's partial regularity theorem up to C^{1,\varsigma} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The…

The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schr\"odinger…

We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open…

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or…

In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's…

We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in $\mathbb R^3$, which converges to a plane with multiplicity $2$ as time approaches infinity.

We investigate the gradient flow of the $L^2$ norm of the Riemannian curvature on surfaces. We show long time existence with arbitrary initial data, and exponential convergence of the volume normalized flow to a constant scalar curvature…

We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.

The level-set curvature G-equation, a well-known model in turbulent combustion, has the following form $G_t + \left(1-d\, \mathrm{dvi}\left({\frac{DG}{|DG|}}\right)\right)_+|DG|+V(X)\cdot DG=0.$ Here the cutoff correction $()_+$ is imposed…

Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution…

We consider the forced mean curvature flow in 2-d, finite range of dependence and positive random forcing. We prove flatness and existence of effective speed for initially flat propagating fronts. This is the analogue, in random media, of a…

There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations.…