arXiv research

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

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 …

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 …

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

Introduces Star-Shaped deviation measures for risk analysis.

New star-shaped acceptability indexes generalize existing methods.

New set-valued star-shaped risk measures introduced for better risk assessment.

Flow turns star-shaped curves into circles.

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

The paper studies dynamic star-shaped risk measures and their representation.

The paper characterizes dynamic return and star-shaped risk measures via BSDEs.

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.

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 $\mathbb{R}^{n+1}$.

The paper characterizes law-invariant star-shaped risk measures.

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…

Introduces Star-Shaped DDPMs for non-Gaussian distributions.

The paper generalizes convex and star-shaped concepts to symplectic spaces and studies variational problems.

Paper characterizes star-shaped risk measures and their properties.

This paper connects monetary and star-shaped risk measures by showing their equivalence under certain conditions.

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

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.

Proves optimal isoperimetric inequality in de Sitter space.

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

Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.

Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.

New Minkowski inequality for capillary surfaces in half-space.

We consider the warped product manifold, $\mathbb{R}_+ \times_{\bf{Id}} M^n$, with Riemannian metric $γ\equiv \mathrm{d} r^2 \oplus r^2 σ$, where $(M^n, σ)$ is a smooth closed Riemannian $n$-manifold. We investigate what sufficient curvature condition is required of $σ$ to ensure that a solution to the inverse mean cur…

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.

This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $Ω$ be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenv…

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …

The paper explores risk measures and arbitrage in financial markets.

The flow of symmetric spheres converges to a round sphere.

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

Proves higher regularity for anisotropic inverse mean curvature flow.

The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.

In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounde…

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…

We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex polygons, for instance their inscriptibility in a circle and their regularity i…

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…

Study on closed curves on negatively curved surfaces, linking number formula, and restrictions.

The manifold $\mathcal{M}$ of star-shaped curves in $\mathbb{R}^n$ is considered via the theory of connections on vector bundles, and cyclic $\mathcal{D}$-modules. The appropriate notion of an "integral curve" (i.e. certain admissible deformations) on $\mathcal{M}$ is defined, and the resulting space of admissible defo…

The study constructs symplectic 4-manifolds with exotic structures.

Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.

Paper develops a new method for solving IBVPs on star-shaped domains.

The radius of the star-shaped set need not have been continuous.

New algorithm for robust density estimation in corrupted data.