9 problems

Converse finite-volume conjecture. Then

Let $(M^{n},g)$ be a complete non-compact $n$-dimensional Riemannian manifold with $Ric_{g}\ge0$. For a point $p\in M$, let $B^{g}_{l}(p)$ denote the geodesic ball of radius $l$, l…

Let $M$ be an open manifold with nonnegative Ricci curvature, and consider the vector space of biharmonic functions on $M$, where biharmonic means satisfying … For a fixed growth r…

Let $M$ be an open manifold with nonnegative Ricci curvature, and let $u$ be a bounded biharmonic function on $M$, meaning that $abla^4 u=0$ in the source's terminology via the bi…

Let $M$ be a complete noncompact (open) Riemannian manifold with nonnegative Ricci curvature. For a fixed growth rate, consider the vector space of harmonic functions on $M$ with p…

Let $M$ be the surface setting of the main theorem, so $n=2$, and let the theorem's other hypotheses and notation remain in force. Surface-case conjecture. Theorem 1.1 holds withou…

Topping's Ricci-flow existence conjecture. The Ricci flow equation

Let $M^n$ be a compact Riemannian manifold with nonempty boundary $\partial M$. Assume that $M$ has nonnegative Ricci curvature and that $\partial M$ is strictly convex with respec…

Let $N^n$ be an $n$-dimensional complete open manifold with Ricci curvature $\operatorname{Ric}_N\geq 0$, and let … where $B(p,r)$ is the geodesic ball of radius $r$ and $w_n$ is t…