Related papers: Convexity of $\lambda$-hypersurfaces

For the Laplace operator with Dirichlet boundary conditions on convex domains in $\mathbb H^n$, $n\geq 2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any…

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 $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…

We prove that a connected mean convex region in $\mathbb{R}^{n+1}$ with at least two components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature…

We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss…

The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(n\geq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal…

A variant of the Gauss curvature flow for closed and convex hypersurfaces is considered. We reveal that if the initial hypersurface is pinched enough, then this property is preserved. Furthermore, based on some structure assumptions on the…

In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $\sigma_2(\kappa(X))=f(X, \nu(X)).$ In this note, we give a simpler proof of this result, and…

In this paper, we prove there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct closed characteristics on every compact convex hypersurface $\Sg$ in $\R^{2n}$. Moreover, there exist at least $[\frac{n}{2}]+1$ geometrically distinct…

Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\le\omega_1$. It is also proved that…

Suppose that $\{T_{a}:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\left\{T_{a}x:a\in G\right\}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt{2}$, then the set of common fixed points…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

Following the overall strategy of the paper ``Convex hypersurfaces in contact topology" by Ko Honda and Yang Huang on contact convexity in high dimensions, we present a simplified proof of their main result.

In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and…

In $\mathbb{R}^d$, a closed, convex set has zero Lebesgue measure if and only its interior is empty. More generally, in separable, reflexive Banach spaces, closed and convex sets are Haar null if and only if their interior is empty. We…

In this paper we prove some general results on constant mean curvature lamination limits of certain sequences of compact surfaces $M_n$ embedded in $\mathbb R^3$ with constant mean curvature $H_n$ and fixed finite genus, when the boundaries…

We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups…

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with…

We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod\limits_{i = 1}^n (H - \kappa_i) = \big( (n - 1) \sigma \big)^n$ for $n = 4$ and has prescribed asymptotic boundary…

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $\delta(n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb{S}^{n+1}$,…

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex.