Related papers: A sub-additive inequality for the volume spectrum

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ric$_{(M,g)}\geq (n-1)g$. If $(M,g)$ supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the…

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $g\in \mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a global smooth manifold $W\subset \mathbb{R}^m$ of dimension $p$, we…

Let $G$ be a compact connected Lie group and let $K$ be a closed subgroup of $G$. In this paper we study whether the functional $g\mapsto \lambda_1(G/K,g)\operatorname{diam}(G/K,g)^2$ is bounded among $G$-invariant metrics $g$ on $G/K$.…

The classification of solutions of the static vacuum Einstein equations, on a given closed manifold or an asymptotically flat one, is a long-standing and much-studied problem. Solutions are characterized by a complete Riemannian…

Let $M$ be a 5 dimensional Riemannian manifold with $Sec_M\in[0,1]$, $\Sigma$ be a locally conformally flat hypersphere in $M$ with mean curvature $H$. We prove that, there exists $\varepsilon_0>0$, such that $\int_\Sigma (1+H^2)^2 \ge…

Let $M$ be a compact Riemannian manifold with smooth boundary, and let $R(\lambda)$ be the Dirichlet-to-Neumann operator at frequency $\lambda$. We obtain a leading asymptotic for the spectral counting function for $\lambda^{-1}R(\lambda)$…

In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set $K\subset\mathbb{R}^{n+1}$ $(n\ge 2)$, for any embedded hypersurface $\Sigma\subset K$ with boundary $\partial\Sigma\subset \partial K$…

We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and $\gamma$-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each…

We prove that a compact, connected, and oriented 4-dimensional gradient $m$-quasi-Einstein manifold with $m\in [1, \infty]$ which is additionally a spin manifold must satisfy the Hitchin-Thorpe Inequality. We show further that the…

Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a Willmore submanifold to the following Willmore functional: $$ \int_M(S-nH^2)^{\frac{n}{2}}dv, $$ where…

We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for…

Given any divergence-free vector field of Sobolev class $W^{m,p}_0(\Omega)$ in a bounded open subset $\Omega \subset \mathbb{R}^2$, we are interested in approximating it in the $W^{m,p}$ norm with divergence-free smooth vector fields…

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \geq 2$. This paper concerns to the validity of the optimal Riemannian $L^1$-Entropy inequality \[ {\bf Ent}_{dv_g}(u) \leq n \log \left(A_{opt} \|D u\|_{BV(M)} +…

Let K be a non-Archimedean local field with the normalized absolute value $|\cdot |$. It is shown that a ``plane wave'' $f(t+\omega_1 x_1+... +\omega_nx_n)$, where f is a Bruhat-Schwartz complex-valued test function on K, $(t,x_1,...,…

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical…

Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of…

Let $\Omega$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus \Omega$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that…

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$. We will assume that the isotropy $H$-module $\mathfrak {g/h}$ has a simple spectrum, i.e. irreducible submodules are…

Let $\mathcal{M}$ be a smooth, closed and connected manifold of dimension $n\in\mathbb{N}$, endowed with a Riemannian metric $g$. Moreover, let $\mathcal{B}$ be an $(n+1)$-dimensional compact manifold with boundary equal to $\mathcal{M}$.…

We deduce various norm equivalences, and convolution estimates for the modulation space $M^{\sharp ,q}_{(\omega )}$ consisting of all $f\in M^{\infty ,q}_{(\omega )}$ such that $|V_\phi f \cdot \omega |$ satisfies a mild vanishing condition…