5 problems

Let $N$ be the dimension, let $\mathbb H^N$ be hyperbolic $N$-space, and let $Omegasubsetmathbb H^N$ be a horoconvex domain. Write $\Gamma(\Omega)$ for its fundamental gap, namely th…

Let $T$ be a triangle with maximal interior angle $\omega$, and let $C_{\rm st,2}$ denote the stability constant associated with the hybrid high-order method. For any…

Let $(M^n,g)$ be a compact Riemannian manifold satisfying $\operatorname{Ric}(M)>K$, and let $v$ be a one-form with $\|v\|_\infty<C$. Suppose that there exists $u\in W^{2,p}(M)$ sa…

Let $n\geq 2$, and let $\epsilon>0$ and $K>0$. For an $n$-dimensional closed Riemannian manifold $(M,g)$, write $\lambda_{n+1}(\bar{\Delta}^{E},M)$ for the $(n+1)$-st eigenvalue of…

Funano and Shioya's conjecture. For every natural number $k$, there exists a positive constant $C_k$ depending only on $k$ such that