Related papers: A general regularity theory for weak mean curvatur…
We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn…
In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity…
We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling…
The quantum field theory of two-dimensional sigma models with bulk and boundary couplings provides a natural framework to realize and unite different species of geometric flows that are of current interest in mathematics. In particular, the…
Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a…
We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by…
We discuss a monotone quantity related to Huisken's monotonicity formula and some technical consequences for mean curvature flow.
By carrying out a point-wise estimate for the second fundamental form, we prove a rigidity theorem of complete noncompact ancient solutions to the mean curvature flow in codimension one. Moreover, we derive an optimal growth condition.
The regularity theory of the Campanato space $\mathcal{L}^{(q,\lambda)}_k(\Omega)$ has found many applications within the regularity theory of solutions to various geometric variational problems. Here we extend this theory from…
We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is $C^\infty$ in space…
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…
In this paper we study the blow up sequence of mean curvature flow of surfaces in $\mathbb R^3$ with additional forces. We prove that the blow up limit of a mean curvature flow of smoothly embedded surfaces with additional forces with…
Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$",…
We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is…
We prove a low-regularity version of Hawking's singularity theorem for Lorentzian metrics in $W^{1,p}$ with Riemann curvature in $L^p$, where $p>2n$ and $n$ the dimension of spacetime. This extends previous results beyond the Lipschitz…
We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schr\"odinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness…
We prove existence and uniqueness of weak solutions to anisotropic and crystalline mean curvature flows, obtained as limit of the viscosity solutions to flows with smooth anisotropies.
We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become…
We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex.…
We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in $\mathbb{R}^3$. With the aim of…