13 problems

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

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\le n+1\le 7$, and let $c>0$. A closed hypersurface has constant mean curvature $c$ if its mean curvature is equal to $c$ a…

Let $M$ be a non-minimal biharmonic submanifold of the unit sphere $S^n(1)$. Constant-mean-curvature conjecture. The mean curvature of $M$ is constant. The claim is a weaker statem…

Ricci lower-bound conjecture. Then

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

Let $\mathbb{B}$ be the unit ball and let $\Delta\subset\mathbb{B}$ be the properly embedded one-manifold given in the proof of Theorem. Let $\overline{B}$ be a smooth compact Riem…

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…

Meeks–Pérez–Ros conjecture. Then

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

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 $\Sigma\subset\mathbb{R}^3$ be an immersed connected closed surface. Topping's conjecture. … This conjecture asks for the optimal lower bound for the total absolute mean curvat…

Let $K\subseteq\mathbf{R}^{n+1}$ be a convex body. For $k=1,\ldots,n$, let $H_k(K,\cdot)$ denote its pointwise $k$-th mean curvature, and write $\mathscr{H}^{n}$ and…