arXiv research

The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…

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

Proves existence of proper solutions for inverse mean curvature flow.

Proves higher regularity for anisotropic inverse mean curvature flow.

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

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

3D metrics get scalar curvature bounds via IMCF.

Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.

New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.

The flow of a torus by inverse mean curvature keeps total curvature bounded until singularity.

Inverts operator on hyperbolic surfaces, constructing invariant distributions.

New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.

The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.

The paper proves new Minkowski inequalities for flows in warped spaces.

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

The paper studies Möbius inversion on surfaces in Minkowski 3-space.

Paper proves stability of quermassintegral inequalities using inverse curvature flow.

We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $Σ$ is strictly mean convex and star-shaped, then the flow hypersurface $Σ_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an a…

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

Geometric framework for inverse problems using foliations and dual connections.

We give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive k-th mean curvature.

We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing…

In this paper we introduce the notion of timelike surface with harmonic inverse mean curvature in 3-dimensional Lorentzian space forms, and study their fundamental properties.

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-…

We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.

In this note, we give a generalization of the inversion formulas of Pestov-Uhlmann for the geodesic ray transform of functions and vector fields on simple 2-dimensional manifolds of constant curvature. The inversion formulas given here hold for 2-dimensional simple manifolds whose curvatures close to a constant.

In this paper, we obtain a complete list of all self-similar solutions of inverse mean curvature flow in $\mathbb{R}^2$.

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

The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.

Curvature flow and inverse curvature flow solutions on 2D light cone identified.

Study on curvature flow in Minkowski space for cocompact hypersurfaces.

The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist …

It has been known in that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows by degree -1 homogeneous functions of principle curvatures in the Euclidean space. In this article, we prove that the round sphere is rigid in much stronger sens…

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

The paper classifies ruled surfaces in a specific space that move in a special way.

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 …

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Further…

Study flow on de Sitter space for convex hypersurfaces.

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 classifies shapes of translating solitons for a specific flow.

New proof shows 3-manifolds with Ricci curvature bound are either flat or grow non-Euclidean.

Proves uniqueness of geometric flow in various Riemannian manifolds.

We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.

This note revisits the inverse mean curvature flow in the 3-dimensional hyperbolic space. In particular, we show that the limiting shape is not necessarily round after scaling, thus resolving an inconsistency in the literature.

We prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form $N$ of constant sectional curvature $K_N$ with speed given by $F^{-α}$, where $α\in (0,1]$ for $K_N=0,-1$ and $α=1$ for $K_N=1$, $F$ is a smooth, symmetric homogeneous of degree one function which is inverse…

Study of curve evolution in 2D space forms converging to a circle.