arXiv research

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

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…

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.

The paper solves curvature measure problem in hyperbolic space.

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

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

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

Convexity preserved in curved surfaces moving at concave speeds.

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 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 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…

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…

Study flow on de Sitter space for convex hypersurfaces.

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 $…

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} …

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

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

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.…

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

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

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

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