7 problems

A Cartan-Hadamard manifold $M^n$ is a complete simply connected Riemannian manifold of nonpositive sectional curvature. For a bounded region $a9\subset M^n$ of finite perimeter, wr…

Greene–Wu conjecture. There exists an entire bounded nonconstant harmonic function on $M$. This conjecture concerns the existence of nonconstant bounded harmonic functions on Carta…

Let $(M^n,g)$ be a Cartan–Hadamard space, meaning a complete simply connected Riemannian $n$-manifold with non-positive sectional curvature. For a smooth open subset…

Let $\Gamma$ be a closed surface immersed in a $\operatorname{CAT}^3(0)$ manifold, and let $\widetilde{\mathcal{G}}(\Gamma)$ denote its total absolute Gaussian curvature. If $g$ is…

Let $\Omega$ be a domain in a surface $\mathcal{H}^{2}$ with curvature bounded above by a constant $\mathcal{K}\leq 0$. Let $A$ be its area and $L$ the length of its perimeter. Two…

Let $\mathcal{H}^{n}$ be a Cartan–Hadamard manifold whose sectional curvature is bounded above by a negative constant $\mathcal{K}$. Let $\Omega$ be a domain in $\mathcal{H}^{n}$.…

Let $(M^n,g)$ be a Cartan–Hadamard manifold, meaning a simply connected manifold with sectional curvature less than or equal to $0$. Let $\Omega\subseteq M$ be a finite perimeter s…