Let $\mathbb{M}_\kappa^n$ be the space form of sectional curvature $\kappa\in\{-1,0,1\}$, so that $\mathbb{M}_{-1}^n=\mathbb{H}^n, \mathbb{M}_{0}^n=\mathbb{R}^n, \mathbb{M}_{1}^n=\mathbb{S}^n$. Let $\Omega\subset\mathbb{M}_\kappa^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<|\Omega|<|\mathbb{S}^n|$ when $\kappa=1$. Write $0=\mu_0(\Omega)\leq\mu_1(\Omega)\leq\cdots$ for the Neumann spectrum, and let $B_R^\kappa\subset\mathbb{M}_\kappa^n$ be a geodesic ball of volume $\vert\Omega\vert/2$. We prove the sharp shifted reciprocal inequality \[ \sum_{j=2}^{n+1}\frac1{\mu_j(\Omega)} \geq \frac{n}{\mu_1(B_R^\kappa)} = \frac{n}{\mu_2(B_R^\kappa\sqcup B_R^\kappa)}. \] Equality holds if and only if $\Omega$ is the disjoint union of two equal geodesic balls. This gives an affirmative answer to the conjecture of \cite[Remark~11]{BucurMartinetNahon2025}.
Let $\mathcal H^\infty$ be the space of all Dirichlet series that admit a bounded holomorphic extension to the open right half-plane $ \{s\in \mathbb C: \operatorname{Re} s>0\}, $ and let $$ \mathcal S_N: \mathcal H^\infty \to \mathcal H^\infty; \sum_{n=1}^\infty a_n n^{-s} \mapsto \sum_{n=1}^N a_n n^{-s}. $$ be the $N...
We consider weak solutions of the second-order elliptic equation $\operatorname{div} (\mathbf{M}\nabla u) = \operatorname{div}\mathbf{F}$ in $\Omega\subset {\mathbb R}^d$ with Dirichlet boundary conditions, where $\mathbf{M}$ is a uniformly elliptic real-valued symmetric matrix $\mathbf{M}:\Omega \rightarrow \mathbb{R}...
We prove a sharp volume gap estimate: if an $n$-dimensional compact K\"ahler manifold $(X, \omega)$ satisfies $\mathrm{Ric}(\omega)\ge (n+1)\omega$ and $X\not\cong \mathbb{P}^n$, then $\mathrm{vol}(X, \omega)\le \frac{2n^n}{(n+1)^n}\mathrm{vol}(\mathbb{P}^n,\omega_{\mathrm{FS}})=\frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$...
Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-\ta...
We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator o...
We prove that the curvature equation $\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{M}})=\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{N}})$ yields uniqueness up to translation for any two closed $C^2_+$ hypersurfaces $\mathcal{M}, \mathcal{N}\hookrightarrow\mathbb{R}^{n+1}$ whenever $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\ge...
Carlos Cabezas-Moreno· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
