27 problems

Let $(M^n,g)$ be a complete Riemannian manifold with infinite filling radius. Let $\omega_n$ be the volume of the unit $n$-ball, and let $V(R)$ denote the volume of a radius-$R$ ba…

Gromov's conjecture. There exists a dimensional constant $C(n)$ such that, for every $p\in M$,

Let $(\mathcal{T},d_{\mathcal{T}},\mu)$ be the continuum random tree, and for $\sigma\in\mathcal{T}$ and $r>0$ define … The cited global volume estimate gives, almost surely, rando…

Let $G$ be the random graph on vertex set ${\mathbb Z}$ in which each pair of vertices $u,v$ is joined independently with probability $p_{|u-v|}$, where $p_1=1$ and…

Let $M$ be a complete Kähler manifold with nonnegative bisectional curvature that admits a nonconstant holomorphic function of polynomial growth, and suppose its bisectional curvat…

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…

Chen–Xu–Zhang's conjecture. Under Euclidean volume growth, equivalently $\mathrm{AVR}(g)>0$, one has

Let $(M^n,g,p)$ be a non-compact pointed Riemannian manifold of dimension $n$ with non-negative Ricci curvature and scalar curvature bounded below by a strictly positive constant.…

Ledoux's conjecture. If this inequality holds with ${\sf C}={\sf K}_{\rm opt}$, without any curvature assumption on $(M,g)$, then

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…

Let $(M^n,g)$ be a Riemannian manifold, let $B_R(p)$ denote the geodesic ball of radius $R$ centered at $p\that is in M$, and write $\operatorname{Ric}$ and $\operatorname{Sc}$ for…

Let $(M^n,g)$ be an $n$-dimensional complete non-compact Riemannian manifold with nonnegative Ricci curvature and scalar curvature satisfying $\Sc_g\geq1$. Gromov's volume growth c…

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

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

Yau's and Gromov's conjectures. Yau conjectured that if

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 $G$ be a planar graph with uniform polynomial volume growth. In the preceding context, recurrence means that simple random walk on $G$ returns to its starting vertex almost sur…

Let $M$ be an arbitrary complete connected Riemannian manifold, let $B(o,r)$ denote the geodesic ball of radius $r$ centered at $o$, let $cmu$ be the Riemannian measure, let $q$ be…

Yomdin's local volume-growth conjecture. A similar estimate to the known surface estimate

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 $G$ be a connected, infinite, locally finite planar graph. For $x \in V(G)$ and $r \geqslant 0$, write … Assume that, for sufficiently large $r$, uniformly for all $x\in V(G)$,…

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,