arXiv research

New boundary condition for weak inverse mean curvature flow in bounded domains.

Inverse mean curvature flow converges to a disk in hyperbolic space.

Proves Penrose inequality for specific asymptotically flat manifolds.

Proves existence of proper solutions for inverse mean curvature flow.

Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.

Proves higher regularity for anisotropic inverse mean curvature flow.

The paper proves constant mean curvature surfaces in specific manifold types.

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean…

The paper proves existence and growth estimates for inverse mean curvature flow and related $p$-Laplacian Green kernel decay.

We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension $n\geq 3$ with boundary on the sphere.

Symmetric graphs flow without singularities on their axis.

We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and e…

Study introduces a new mean curvature flow for spherical boundaries.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

This paper extends IMCF theory to Heisenberg group, solving Penrose inequality.

Unique ancient convex flow in a ball with free boundary found.

Proves existence of mean curvature flow with surgery for free boundary surfaces.

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

The paper examines torsional rigidity bounds under geometric flows.

The paper extends flow theory with free boundaries, proving key bounds and theorems.

Paper proves mass theorems for nonnegative scalar curvature metrics.

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkow…

Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.

Study on potential behavior in special geometric spaces.

We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier $S$. Here $S$ can be any properly embedded, oriented surface in $R^{n+1}$ of bounded geometry. We also give an alternative proof that convex mean curvature flows with …

Foundations for free boundary Brakke flows established.

Study inverse curvature flows for capillary hypersurfaces in a unit ball.

Study resolves flow through cylindrical singularities, proving nonfattening.

Infinity-harmonic functions linked to IMCF clusters, revealing new properties in 2D.

Given a mean curvature flow of compact, embedded $C^2$ surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.

Proves smoothness and star-shapedness of weak IMCF solutions in hyperbolic space.

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric $\bar{g}=dr^2+{\vartheta}^2(r)σ$ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over $S^n$ the flow exists for all times and remains a graph…

The paper studies how convex surfaces shrink under mean curvature flow with a free boundary.

Unified view of monotonicity formulas for inverse mean curvature flow and $p$-capacitary potentials.

3-manifolds with positive scalar curvature and bounded geometry are contractible.

We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow …

Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.

We consider the smooth inverse mean curvature flow of strictly convex hypersurfaces with boundary embedded in $\mathbb{R}^{n+1},$ which are perpendicular to the unit sphere from the inside. We prove that the flow hypersurfaces converge to the embedding of a flat disk in the norm of $C^{1,β},$ $β<1.$

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in $\mathbb{R}^{n+1}$ to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection …

In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric $n$-dimensional hypersurface in $\R^{n+1}$ between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the m…

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of…

We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula v…

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partialΩ_0$, we show that there exists a weak solution to the null mean curvatu…

A new topological gap theorem improves the systole of 3-manifolds with positive scalar curvature.

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity $w$ in the generalized $1$-harmonic equatio…