Related papers: The epsilon-regularity theorem for Brakke flows ne…

We study a class of fourth order curvature flows on a compact Riemannian manifold, which includes the gradient flows of a number of quadratic geometric functionals, as for instance the L2 norm of the curvature. Such flows can develop a…

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular…

We develop the asymptotic analysis as $\epsilon\to 0$ for the natural gradient flow of the self-dual $U(1)$-Higgs energies $$E_{\epsilon}(u,\nabla)=\int_M\left(|\nabla u|^2+\epsilon^2|F_{\nabla}|^2+\frac{(1-|u|^2)^2}{4\epsilon^2}\right)$$…

This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a…

Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes…

We use Ilmanen's elliptic regularization to prove that for an initially smooth mean convex hypersurface in Euclidean n-space moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity.…

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for…

Consider an integral Brakke flow $(\mu_t)$, $t\in [0,T]$, inside some ball in Euclidean space. If $\mu_{0}$ has small height, its measure does not deviate too much from that of a plane and if $\mu_{T}$ is non-empty, then Brakke's local…

The classical Prandtl-Batchelor theorem (Prandtl 1904; Batchelor 1956) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem…

The Triple-Deck equations are a classical boundary layer model which describes the asymptotics of a viscous flow near the separation point, and the Couette flow is an exact stationary solution to the Triple-Deck equations. In this paper we…

We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If…

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity…

This article studies special solutions to symplectic curvature flow in dimension four. Firstly, we derive a local normal form for static solutions in terms of holomorphic data and use this normal form to show that every complete static…

Bamler--Kleiner recently proved a multiplicity-one theorem for mean curvature flow in R^3 and combined it with the authors' work on generic mean curvature flows to fully resolve Huisken's genericity conjecture. In this paper we show that a…

We define a new geometric flow, which we shall call the $K$-flow, on 3-dimensional Riemannian manifolds; and study the behavior of Thurston's model geometries under this flow both analytically and numerically. As an example, we show that an…

We establish local regularity theory for parabolic systems of Uhlenbeck type with $\varphi$-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local H\"older continuity of the gradients,…

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.

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous…

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…