arXiv research

In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in $\mathbf{R}^{n+1}$ in arbitrary …

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…

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we p…

The paper proves the existence of a hyperbolic inverse mean curvature flow under specific conditions.

Study introduces a new mean curvature flow for spherical boundaries.

Proves higher regularity for anisotropic inverse mean curvature flow.

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.

The paper finds at least four prime closed characteristics on star-shaped hypersurfaces in 8D space.

The paper proves inequalities for star-shaped and $F$-mean convex hypersurfaces in $\mathbb{R}^{n+1}$.

In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.

The study examines evolving star-shaped hypersurfaces in hyperbolic spaces, influenced by ambient geometry.

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

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

Study eigenvalues for special curvature equations on star-shaped surfaces.

We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in $C^\infty$ to…

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.

We consider the inverse curvature flows $\dot x=F^{-p}ν$ of closed star-shaped hypersurfaces in Euclidean space in case $0<p\not=1$ and prove that the flow exists for all time and converges to infinity, if $0<p<1$, while in case $p>1$, the flow blows up in finite time, and where we assume the initial hypersurface to be…

Study anisotropic flow for capillary hypersurfaces, proving new inequalities.

Proves optimal isoperimetric inequality in de Sitter space.

The study proves the existence of $k$-convex hypersurfaces for specific curvature equations.

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We p…

Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.

Flow turns star-shaped curves into circles.

Inverse Mean Curvature Flow bounds star-shaped surfaces' lifetime.

In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.

The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of com…

Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.

We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to…

In this paper we consider a star-shaped hypersurface flow by mean curvature. Without any assumption on the convexity, we give a new proof of gradient estimate for a short time. As an application, we also give a lower bound for the blowing up time.

Sharp bounds for Steklov eigenvalues on star-shaped domains and paraboloids.

The paper studies curvature flows in hyperbolic space and proves convergence to spheres under certain conditions.

The paper studies curvature flows of star-shaped hypersurfaces and proves convergence to spheres.

The study uses symplectic capacities to bound the systole on the sphere.

In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic $3$-space.

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypers…

The paper studies curvature flows in Euclidean and hyperbolic spaces, proving smooth convergence to spheres.

Paper estimates curvature of convex hypersurfaces with prescribed curvature.

The paper proves new inequalities in hyperbolic space using Euclidean methods.

Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.

The paper studies a curvature flow on hypersurfaces in R^(n+1).

We consider a compact, star-shaped, mean convex hypersurface $Σ^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which …

Paper studies inverse curvature flows and solves related geometric problems.

We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique cons…

Given a convex cone in the \emph{prescribed} warped product, we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve along the inverse mean curvature flow, then, by using the convexity of …

In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.

Proves inequalities for convex hypersurfaces in Euclidean space.