29 problems

Ricci lower-bound conjecture. Then

Let $\Sigma$ be the boundary of a compact Riemannian manifold $\Omega$. Suppose that the scalar curvature $R$ of $\Omega$ satisfies … for a constant $\sigma$. Gromov's conjecture.…

Discreteness conjecture. Under either of these alternatives, the spectrum of the Laplace–Beltrami operator on $M$ is discrete.

Symmetry conjecture under the Main Assumption and Condition S. Any smooth compact connected embedded hypersurface $M$ in $\mathbb R^{n+1}$ satisfying the Main Assumption and Condit…

Let $(M,g)$ be a closed and oriented Riemannian surface, let $H$ be a function on $(M,g)$, and let ${\cal F}$ be a regular homotopy class of immersions of $M$ into $\mathbb{R}^3$.…

Let $(M,g)$ be a closed oriented Riemannian surface, and let $H:M\to\mathbb{R}$ be an arbitrary non-constant function. Two isometric immersions are congruent if they differ by a ri…

Quadratic decay conjecture. On each boundary component $\Sigma$, there exists a constant $C'$ such that

Let $X$ be an infinite mean-convex domain in $\mathbb{R}^{n}$, with boundary $\partial X$. A connected component of $\partial X$ has its mean curvature separated away from zero if…

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

Twin-Bubble Conjecture. There are at least two almost-embedded hypersurfaces in $M$ with mean curvature prescribed by $h$.

Twin-Bubble Conjecture. For every $c>0$, there are at least two almost-embedded hypersurfaces in $M$ with constant mean curvature $c$.

Weighted mean-curvature inequality conjecture. Inequality $$ holds in a $C^1$ neighborhood of the ball without any curvature assumption and without replacing $H$ with $H^{+}$.

Let a bounded, smooth surface have its mean curvature ordered in a particular direction. Nirenberg–Li symmetry conjecture. The surface must be symmetric with respect to some hyperp…

Let $\mathcal{F}$ be a codimension one foliation of a complete Riemannian manifold $\overline{M}^{n+1}$. Assume that there is $K_0 \geq 0$ such that $\overline{M}$ has Ricci curvat…

Let $\Sigma\subset\mathbb{R}^3$ be a $C^{1,1}$-regular axisymmetric surface, with mean curvature $H$ and induced volume form $\mathrm{d}\operatorname{vol}_g$. Dalphin's conjecture.…

Let $M$ be a compact Riemannian manifold with scalar curvature bounded below by $c\filein\mathbb{R}$. Its boundary has intrinsic geometry, and its total mean curvature is the integ…

Let $\Omega\subset\mathbb{R}^3$ be a compact set with smooth boundary. Call $\Omega$ outer-minimizing if it is outer-minimizing, and call it mean-convex when its boundary has posit…

The setting is that of Theorem 1: a compactly supported and asymptotically mean convex sequence of sets in Euclidean space satisfying the theorem's finite-total-curvature and almos…

Let $\Sigma\subset\mathbb{R}^3$ be a closed surface with non-negative mean curvature. The Minkowski inequality conjecture. There holds … with equality if and only if $\Sigma$ is a…

Let $M$ be a compact hypersurface in hyperbolic space $\mathbb{H}^{n+1}$, possibly with boundary $\partial M$, and let $f$ be a positive smooth function on $M$. Write…

Higher-mean-curvature hyperbolic Michael–Simon conjecture. For $1\le k\le n$, the inequality

Let $(\Sigma,g_\Sigma)$ be a closed Riemannian $n$-manifold associated with a smooth function $h:\Sigma\to \mathbf R$. A fill-in is a compact Riemannian $(n+1)$-manifold…

Let $\Omega$ be a smooth domain, and let $P_{\delta}$ denote the concentration points associated with the optimizers as $\delta\to0$. Maximal mean-curvature concentration conjectur…

Gromov's conjecture. There should exist a continuous self-map $R:X\to X$ such that $R(X)$ has topological dimension $n-2$ and

Let $M\hookrightarrow\mathbf{R}^n$ be an immersed connected closed surface, with $n=3$. Define its extrinsic diameter by … Let $H$ be the mean curvature vector, normalized so that…