Related papers: Mean value formulas on surfaces in Grushin spaces
We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\Phi$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\Phi$-convex…
We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of…
We prove that if $\Sigma$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation…
We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $\Sigma$ with nonpositive Yamabe invariant that are properly immersed in a Riemannian $n$-dimensional manifold $M$ with scalar curvature $R^M$ and mean curvature of…
The paper presents the bosonic and fermionic supersymmetric extensions of the structural equations describing conformally parametrized surfaces immersed in a Grasmann superspace, based on the authors' earlier results. A detailed analysis of…
We derive some estimates for stable minimal hypersurfaces in $R^{n+1}$. The estimates are related to recent proofs of Bernstein theorems for complete stable minimal hypersurfaces in $R^{n+1}$ for $3\le n\le 5$ by Chodosh-Li,…
The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, but displays on the right-hand side an…
Let $\mathbb C$ be the complex plane, $E$ be a measurable subset in a segment $[0, R]$ of the positive semiaxis $\mathbb R^+$, $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$. The main result of this article is an upper…
In this paper, we study the asymptotic Plateau problem in hyperbolic space for constant sum Hessian curvature. More precisely, given a asymptotic boundary $\Gamma$, one seeks a complete hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ satisfying…
In this paper, we will give an upper bound and a lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which are uniform and explicit. These two bounds have the optimal dominant terms. As an application, we use…
In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…
We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1<p<\infty$, following the scheme described in Manfredi et al. (2009) for the Euclidean case. The new tool that allows us…
We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical…
The notions of (metric) hypersurface data were introduced in [Mars,2013] as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudo-riemannian manifolds. In this paper, general geometric properties of…
We supplement the result of the first part of the work with estimates of the integrals of the difference of subharmonic functions in measure with some deterioration of the absolute constants, but these estimates have the form of a…
We investigate integral conditions involving the mean curvature vector $\vec{H}$ or mixed higher-order mean curvatures, to determine when a codimension-two submanifold $\Sigma$ lies on a shear-free (umbilical) null hypersurface in a…
It is shown that for a subset $A\subset\R^n$ that has the global Gabrielov property, a Vitushkin-type estimate holds. Concrete examples are given for sub-level sets of certain classes of functions.
We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.
We obtain general analytical forms for the solutions of the one-loop renormalization group equations in the top/bottom/$\tau$ sector of the MSSM. These solutions are valid for any value of $\tan \beta$ as well as any non-universal initial…
Formulae for the value of a harmonic function at the center of a rectangle are found that involve boundary integrals. The central value of a harmonic function is shown to be well approximated by the mean value of the function on the…