Related papers: Brunn-Minkowski inequalities for sprays on surface…

A local classification of spacelike surfaces in Minkowski 4-space, which are invariant under spacelike rotations, and with mean curvature vector either vanishing or lightlike, is obtained. Furthermore, the existence of such surfaces with…

In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent…

We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel…

We use Brunn-Minkowski inequalities for quermassintegrals to deduce a family of inequalities of Poincar\'e type on the unit sphere and on the boundary of smooth convex bodies in the $n$-dimensional Euclidean space.

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially…

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space…

Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-\kappa^2$. Using the cone total curvature $TC(\Gamma)$ of a graph $\Gamma$ which was introduced…

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, we study the flow by the mean curvature of a locally…

We first establish a family of sharp Caffarelli-Kohn-Nirenberg type inequalities on the Euclidean spaces and then extend them to the setting of Cartan-Hadamard manifolds with the same best constant. The quantitative version of these…

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in $\mathbb{R}^n$, we have $|A+B|^{1/n} \geq |A|^{1/n}+|B|^{1/n}$. Also, equality holds if and only if $A$ and $B$ are convex and homothetic sets in…

We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature…

Riemann zero mean curvature examples in the Lorentz-Minkowski space are surfaces with zero mean curvature foliated by circles contained in parallel planes. In contrast to the Euclidean case, this family of surfaces presents new and rich…

We extend to Minkowski spaces the classical result of Barbosa and do Carmo [1] that characterizes the euclidean sphere as the unique compact stable CMC hypersurface of $\mathbb R^n$. More precisely, if $K$ is a smooth convex body in…

In this paper, we establish two families of sharp geometric inequalities for closed hypersurfaces in space forms or other warped product manifolds. Both families of inequalities compare three distinct geometric quantities. The first family…

For a general family of graphs on $\mathbb{Z}^n$, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in $\mathbb{R}^n$. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality…

In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordstr\"om-anti-deSitter manifold and…

The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(\kappa)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends…

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric $\bar{g}=dr^2+{\vartheta}^2(r){\sigma}$ and non-positive radial sectional curvature. We prove, that for…

We prove a differential Harnack inequality for noncompact convex hypersurfaces flowing with normal speed equal to a symmetric function of their principal curvatures. This extends a result of Andrews for compact hypersurfaces. We assume that…

For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the…