382 problems

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain equipped with the conformal metric $g=\omega\lvert dz\rvert^2$, where…

For every admissible Bartnik boundary datum $(\Sigma,\gamma,H)$, the Bartnik mass…

For each $n\ge 4$ and each $\sigma>0$, there should exist a constant $C(n,\sigma)<\infty$ such that every smooth closed $n$-manifold with isotropic curvature bounded below by…

Does every complete noncompact Riemannian manifold $(M^n,g)$ with $\operatorname{Ric}_g\ge 0$ satisfy, for some (equivalently any) $p\in M$,…

For every integer $n\ge 3$, every complete $n$-dimensional Riemannian manifold $(M^n,g)$ with $\operatorname{Ric}_g\ge 0$, and every point $q\in M$, is the normalized scalar-curvat…

Let $M$ be a complete open $5$-manifold with nonnegative Ricci curvature. If its universal cover $\widetilde M$ has Euclidean volume growth, meaning that for some…

Let $M$ be a compact connected orientable Riemannian surface with boundary, and suppose that its boundary is isometric, with respect to the induced distance in $M$, to the circle o…

Let $n\ge 3$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold with positive scalar curvature. If, for some (equivalently any) $x_0\in M$,…

Let $M$ be a complete Riemannian manifold satisfying the geometric hypotheses in Holopainen’s conjecture. Is every nonnegative $p$-harmonic function $v$ on $M$ such that…

For every closed Riemannian $n$-manifold $(M,g)$ and every real number $\lambda$, there exists a constant $C(M,g,\lambda)$ such that, for every compact Riemannian $(n+1)$-manifold…

Given a complete weighted Riemannian manifold $(M,g,e^{-f})$ with bounded potential $f$ and nonnegative Bakry–Émery Ricci curvature, namely…

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For every integer $n\ge 2$, if $(M^{n+1},g)$ is a compact Riemannian manifold with boundary satisfying $\operatorname{Ric}_g\ge n g$ and $\operatorname{II}_g\ge 0$ on $\partial M$,…

For every integer $n\ge 3$, there exists a constant $\varepsilon_n\in(0,1)$ such that, for every connected, closed, smooth Riemannian manifold $(M^n,g)$, if…

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Let $(M,g_1)$ and $(M,g_2)$ be compact Riemannian manifolds with smooth boundary, with $\dim(M)\geq 3$. For each metric, let $C_{g_i}$ denote the Cauchy data set of harmonic functi…

Conjecture 1.3 (Brin--Karcher, 1984). Let $M$ be a compact connected manifold with a $C^3$ Riemannian metric of negative variable sectional curvature $K$, satisfying…

Let $g$ be a smooth complete metric on $\mathbb R^n$, where $n\geq 3$, and let $g_{\mathrm{euc}}$ denote the Euclidean metric. Assume that … and … as $x\to\infty$. Gromov's Euclide…

Let $M$ be a closed smooth manifold, and let $g$ be an $L^\infty$-metric on $M$, meaning that for some smooth reference Riemannian metric $g_0$ and constant $\Lambda>0$,…

Let $g$) be a smooth Riemannian metric on the closed hemisphere $\mathbb{S}^3_+$ satisfying the following conditions: its scalar curvature obeys ${\mathrm R}_g\geq 6$; its induced…

Let $(M^{n},g)$ be a closed, simply connected Riemannian manifold of dimension $n\ge 4$. Define the minimum sectional curvature and normalized scalar curvature at $x\in M$ by … Her…

Let $(M,g)$ be a compact Riemannian manifold without boundary, and let $\varphi$ be an eigenfunction satisfying … Write $\mathcal{L}_{\varphi}(M)$ for its nodal volume. Yau's conje…

Let $(M^n,g,f)$ be a CPE metric, meaning that $(M^n,g)$ is a closed, oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature and $f:M^n\to\mathbb{R}$ is…

Gromov's upper bound conjecture. There exists a constant $C(M,\sigma)$ such that for every such $X$,

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…