11 problems

Let $G=(V,E)$ be a graph with degree bound $d_*$, and let $B(x,r)$ denote the open ball of radius $r$ about $x$. Assume the degree and curvature hypotheses referred to as Assumptio…

Volume growth and asymptotic cone dimension conjecture. (1) If

Let $(M^n,g)$ be a complete non-compact Riemannian manifold, let $B_R(p)$ denote the geodesic ball of radius $R$ centered at $p\in M$, and define … Write $\operatorname{Ric}$ and…

Asymptotic volume growth conjecture. There is a universal constant $C(n)>0$ such that, if $R\geq 2$, then

Given $n$ and $L\in(0,1]$, let $M$ be an open $n$-manifold with $\operatorname{Ric}\ge 0$, and suppose that the Riemannian universal cover of $M$ has Euclidean volume growth of con…

Let $T_p$ be the infinite $p$-regular tree, and let $T_p^q$ be its $q$-fold Cartesian product, which is $pq$-regular. For a graph $G$ and vertex $x$, write…

Let $d_{\max}\in{\mathbb N}$. A graph $G=(V,E)$ has vertex degrees $d_x\leq d_{\max}$ for all $x\in V$ and satisfies $CD(0,\infty)$. Bishop comparison conjecture. There are constan…

Let $(M^n,g)$ be a complete Riemannian manifold of positive sectional curvature. A manifold has maximum volume growth when its volume growth is maximal in the relevant comparison s…

Let $X$ be a compact Kähler manifold and let $\omega(t)$ be a solution of the Kähler-Ricci flow defined on $[0,\infty)$. Volume-growth conjecture. The Kodaira dimension satisfies…

The conjecture on uniform continuity and weakly Euclidean points. Under these assumptions,

Let $G$ be an infinite graph with the doubling property: there exists a universal constant $C>0$ such that for every $r>0$ and every vertex $x$, … where $B(x,r)$ is the ball of rad…