arXiv research

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

We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is $σ_k$-convex.

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

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

We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type…

Convexity preserved in curved surfaces moving at concave speeds.

Study flow on de Sitter space for convex hypersurfaces.

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the c…

We study relations of some classes of $k$-convex, $k$-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{$k$-circular convex} and \textrm{$k$-circular visible} ones. Investigati…

Proves existence and uniqueness of hypersurfaces with prescribed curvature in Minkowski space.

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

Let $\cM$ be a Brakke flow of $n$-dimensional surfaces in $R^N$. The singular set $\cS\subset\cM$ has a stratification $\cS^0\subset\cS^1\subset...\cS$, where $X\in \cS^j$ if no tangent flow at $X$ has more than $j$ symmetries. Here, we define quantitative singular strata $\cS^j_{η,r}$ satisfying $\cup_{η>0}\cap_{0<r} …

The paper solves curvature measure problem in hyperbolic space.

In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow.…

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weig…

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

The paper establishes Pogorelov type estimates for solutions to Hessian quotient equations in hyperbolic space.

In this paper, we consider smooth, properly immersed hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second fundamental form of these hypersurfaces in some space-time region $B_R(y)\times (0, t_0)$ …

The paper derives new inequalities for non-convex domains and flows.

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

This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a do…

We prove Obata's rigidity theorem for metric measure spaces that satisfy a Riemannian curvature-dimension condition. Additionally, we show that a lower bound $K$ for the generalized Hessian of a sufficiently regular function $u$ holds if and only if $u$ is $K$-convex. A corollary is also a rigidity result for higher or…

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable…

In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of $σ_k^α$-flow must be a round sphere. We also obtain a similar result for the solutions of $F=-\langle X, e_{n+1}\rangle \, (*)$ with a non-homogeneous function $…

We prove that in metric measure spaces where the entropy functional is K-convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one opti…

Paper solves overdetermined $k$-Hessian equation in exterior domains.

We combine Gromov's amenable localization technique with the Poincaré duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the trajectories that are tangent to the boundary in a particular canonical fashion. Sp…

The paper shows that ancient noncollapsed mean curvature flows have a blowdown of at most n-2 dimensions.

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.

Proves convexity of certain hypersurfaces with negative λ.