6 problems

Gromov's conjecture. There exists a constant $\Lambda$ depending only on $\sigma$ and the intrinsic geometry of $(\partial M,g|_{T(\partial M)})$ such that

Let $M$ be a compact Riemannian manifold with scalar curvature bounded below by $c\filein\mathbb{R}$. Its boundary has intrinsic geometry, and its total mean curvature is the integ…

Gromov's conjecture. There exists a constant $\Lambda$ depending only on $\sigma$ and the intrinsic geometry of $(\partial M,g|_{T(\partial M)})$ such that

Let $(B^{n+1},dx^2)$ be the unit ball in Euclidean $(n+1)$-space, and let $k=\frac{n+1}{2}$. Define … For $g=e^{2u}dx^2$, let $\mathcal{F}_k(g)$ denote the critical conformal primi…

Local stabilization conjecture. For given $r\geq 3$ there exists an $n_0 \in \mathbb{N}$, such that for all $n\geq n_0$ and for all nonnegative factorizations $M=AB$ where $M$ is o…

The simplest model is the closed half-space $\mathbb R^n_+$ with its standard flat metric $\delta$. Let $(M,g)$ be an asymptotically flat manifold with decay rate $\tau>(n-2)/2$, a…