Related papers: A Poincar\'{e}-type inequality and a related eigen…

We prove the following operator-coefficients free Poincar\'{e} inequality: $$|f(X)-E[f(X)]|_{2}\leq2|X|_{2}\|\widehat{\partial}_{X:B}[f(X)]\|_{\pi}, \quad f(X) \in \mathrm{dom}(\widehat{\partial}_{X:B}),$$ where $\|\cdot\|_{\pi}$ is the…

In this paper, we carry out in-depth research centering around the Harnack inequality for positive solutions to nonlinear heat equation on Finsler metric measure manifolds with weighted Ricci curvature ${\rm Ric}_{\infty}$ bounded below.…

"V - E + F = 2", the famous Euler's polyhedral formula, has a natural generalization to convex polytopes in every finite dimension, also known as the Euler-Poincar\'e Formula. We provide another short inductive proof of the general formula.…

We prove the following generalization of the classical Lichnerowicz vanishing theorem: if $F$ is an oriented flat vector bundle over a closed spin manifold $M$ such that $TM$ carries a metric of positive scalar curvature, then $<\widehat…

We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincar\'e…

We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the…

The sphere formula states that in an arbitrary finite abstract simplicial complex, the sum of the Euler characteristic of unit spheres centered at even-dimensional simplices is equal to the sum of the Euler characteristic of unit spheres…

Let $\mathcal{F}$ be a smooth Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and $E \subset M$ is a closed set. Fix a hermitian metric $g$ on $M \setminus E$ and assume that all leaves of $\mathcal{F}$ are…

In this paper, we consider the non-linear general $p$-Laplacian equation $\Delta_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is…

We present an argument which leads from the Brunn-Minkowski inequality to a Poincare' type inequality on the boundary of convex bodies with smooth boundary and positive Gauss curvature

We consider a solution f of a certain Dirichlet Problem on a domain in $S^{(n+1)}$ whose boundary is a minimal hypersurface and we prove a Poincare type inequality for f. One have equality iff Yau's conjecture about the first non-zero…

Our main result is an abstract good-$\lambda$ inequality that allows us to consider three self-improving properties related to oscillation estimates in a very general context. The novelty of our approach is that there is one principle…

We generalize the Fenchel theorem for strong spacelike closed curves of index $1$ in the 3-dimensional Minkowski space, showing that the total curvature must be less than or equal to $2\pi$. Here strong spacelike means that the tangent…

Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except…

Let $\Omega$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality…

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown…

Let $\Omega \subset \mathbb{R}^n$ be a convex. If $u: \Omega \rightarrow \mathbb{R}$ has mean 0, then we have the classical Poincar\'{e} inequality $$ \|u \|_{L^p} \leq c_p \mbox{diam}(\Omega) \| \nabla u \|_{L^p}$$ with sharp constants…

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq…

We use the Bakry-\'{E}mery curvature-dimension criterion and $\Gamma$-calculus to establish the Poincar\'{e} inequality with monomial Gaussian measure, and then apply the duality approach to study its improvements and its gradient…

We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincar\'e inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This…