13 problems

Let $(M,g)$ be a closed Riemannian $3$-manifold, let $R_g$ denote its scalar curvature, and let $h(g)$ denote its volume entropy. Agol–Storm–Thurston conjecture. If … then … This c…

For $-1\leq\alpha\leq 1$, let $G_\alpha$ be the homogeneous Riemannian manifold in the interpolating family, and let $h(G_\alpha)$ denote its volume entropy, defined by … where…

Volume-entropy conjecture. If

Correction to the volume-entropy monotonicity conjecture. The volume entropy of $\mathbb{R}^{2}\rtimes_A\mathbb{R}$ was conjectured to be a monotonically decreasing function of…

Let $K\subset\mathbb{R}^n$ be a convex body, and equip its interior with the Hilbert metric. The volume growth entropy measures the exponential growth rate of metric balls in this…

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing $n\,\mathbf a$, and let $\varphi_n:\pi_1(P_n)\to\pi_1(X)$ be the induced homomorphism. Write…

Let $(P_n)$ be a sequence of quasi-optimal pseudomanifolds representing multiples of a homology class, equipped with the piecewise-flat metric in which every $m$-simplex is isometr…

Minimal entropy conjecture. This functional is minimized uniquely by the locally symmetric structure on $M$ induced by $g_0$, up to a possible homothety.

Let $g_0$ be a locally symmetric metric, and consider all Riemannian metrics whose volume equals the volume of $g_0$. The volume entropy is the exponential growth rate of the volum…

Let $\Omega\subset\mathbb{R}\mathbb{P}^n$ be a convex proper open set, and let $h_{vol}(\Omega,d_{\Omega})$ denote the volume entropy of its Hilbert geometry, defined using the Bus…

Let $X$ be a closed oriented smooth Riemannian $4$-manifold, and let $\mu(X)$ denote its volume entropy. Assume that … and that $X$ admits a quasi-non-singular solution to the norm…

Let $(\Omega,d_{\Omega})$ be a Hilbert geometry, where $\Omega$ is a convex set of dimension $n$. If the volume entropy exists, define it by … Here $B(x,r)$ is a Hilbert-metric bal…

Let $K$ be an $n$-dimensional convex set equipped with its Hilbert geometry. Fix a volume and, for a metric ball $B(o,r)$, define the upper volume entropy by … The hyperbolic space…