arXiv research

Study shows Reeb orbits on starshaped hypersurfaces grow logarithmically with period.

The study finds starshaped compact hypersurfaces in warped products with curvature estimates.

Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.

We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.

We consider the evolution of starshaped hypersurfaces in the Euclidean space by general curvature functions. Under appropriate conditions on the curvature function, we prove the global existence and convergence of the flow to a hypersurface of prescribed curvature.

New findings on hypersurfaces with specific curvature properties in space forms.

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Bbb R$. For a positive function $ψ$ defined on $\bar M$, we generalized the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $σ_k(κ)=ψ(V,ν(…

The paper studies a modified scalar curvature flow and proves convergence to a sphere.

Let $\mathcal{W}^{n}$ be the class of $C^{\infty }$ complete simply connected $n-$dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let $$ and let $A$ be a subset of $W$. This article aims at characterization and bu…

Let $ψ$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $ψ$, when evaluated on $M$, coincides with the $m-$th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness…

The paper proves Michael-Simon inequalities in hyperbolic space using novel curvature flows.

In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $σ_2(κ(X))=f(X, ν(X)).$ In this note, we give a simpler proof of this result, and extend it to space forms.

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.

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\varphi(r)$. If $\varphi'(r)>0$ and $\varphi''(r)\geq 0$, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of $\varphi''(r)$ and …

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(κ)=σ\in(0, 1)$ with a prescribed asymptotic boundary $Γ$ at infinity has at least one smooth solution with uniformly bounded hyperbol…

New flow for capillary surfaces converges to spherical caps.

Proves Riemannian starshape of capacitary potential levels.

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…

The study examines hypersurfaces in warped products and their properties.

Proves a generalized Minkowski inequality for starshaped domains.

We give a simple proof of the insoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a reslut of Gerhardt \cite{G} and Urbas \cite{U} on an expanding geometric curvature flow.

The paper proves new inequalities and flow properties for hypersurfaces.

In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $φ$ into $§^{n+1}$. Let $q>n$ and $A>0$ be some real numbers satisfying $|M|^\frac{1}{n…

In this note, we observe that if $B$ is a ball in a Euclidean space with dimension $n$, $n\geq3$, then a stable CMC hypersurface $Σ$ with free boundary in $B$ satisfies \[ nA\leq L\leq nA\left( \frac{1+\sqrt{1+4(n+1)H^2}}{2} \right)\,, \] where $L$, $A$ and $H$ denote the length of $\partial Σ$, the area of $Σ$ and the…

In this paper, we construct a Rabinowitz-Floer type homology for a class of non-linear problems having a \emph{starshaped} potential; we consider some equivariant cases as well. We give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model…

Find conditions for starshapedness of level sets in Heisenberg group.

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $\varphi(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut v…

We study the evolution of strictly mean-convex entire graphs over $R^n$ by Inverse Mean Curvature flow. First we establish the global existence of starshaped entire graphs with superlinear growth at infinity. The main result in this work concerns the critical case of asymptotically conical entire convex graphs. In this…

The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.

Classification of hypersurfaces in homogeneous spaces with specific properties.

Study on biconservative hypersurfaces with constant scalar curvature in space forms.

The paper classifies various types of hypersurfaces in a product space.

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

Classifies hypersurfaces with constant isotropic curvature in space forms.

We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…

New global section found for geodesic flows on convex hypersurfaces.

New tensors capture intrinsic embedding data of conformal hypersurfaces.

The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.

Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.

Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. The study of these hypersurfaces is motivated by the fact that for indefinite almost Hermitian manifolds this class of lightlike hypersurfaces does not exist. We also establish that radical transversal…

In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…

New compact mean convex hypersurfaces found for positive λ.

In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $δ(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of G…

Study on minimal hypersurfaces in a special normed space.

The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.

Classifies and describes hypersurfaces in Siklos spacetimes.