Related papers: Curvature estimates for graphs in warped products
A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic $\lambda$-hypersurfaces (\cite{…
The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold…
In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic…
We study a half-space problem related to graphs in $\mathbb{H}^2\times\mathbb{R}$, where $\mathbb{H}^2$ is the hyperbolic plane, having constant mean curvature $H$ defined over unbounded domains in $\mathbb{H}^2$.
In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such…
The CD inequalities and CDE inequalities are useful in the estimate of curvature on graphs. This article is based on the ufinite graph with large girth, and finally concludes some curvature estimate in CD and CDE.
Real-world graphs naturally exhibit hierarchical or cyclical structures that are unfit for the typical Euclidean space. While there exist graph neural networks that leverage hyperbolic or spherical spaces to learn representations that embed…
This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical…
On the ambient space of a Lie group with a left invariant metric that is isometric and isomorphic to a semidirect product $\mathbb{R}^2\rtimes_A\mathbb{R}$, we consider a domain $\Omega\subseteq \mathbb{R}^2\rtimes_A\{0\}$ and vertical…
In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N\times\mathbb{R}$, submanifolds with bounded $\varphi$-mean curvature in the hyperbolic space, and successfully give…
We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we…
We obtain area growth estimates for constant mean curvature graphs in $\mathbb{E}(\kappa,\tau)$-spaces with $\kappa\leq 0$, by finding sharp upper bounds for the volume of geodesic balls in $\mathbb{E}(\kappa,\tau)$. We focus on complete…
We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of…
We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…
In this paper we find a tight estimate for Gromov's waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature ($\mathrm{CAT}(\kappa)$-spaces), and…
We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various…
We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for…
We study interior curvature estimates for convex graphs which satisfy the quotient equation $\frac{\sigma_{n}}{\sigma_{n-2}}(\lambda)=f(X)>0$ in this paper.
In this article, we consider compact surfaces $\Sigma$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $\Gamma=\partial\Sigma\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the…
We study the Gromov--Sormani MinA scalar curvature compactness conjecture for warped product metrics on $\mathbb{S}^2\times\mathbb{S}^1$ of the form introduced by Kazaras-Xu in \cite{KazarasXu2023} as follows: \[…