256 problems

The conjecture asks for the sharp Sobolev trace inequality associated with $\sigma_2$-curvature on the round hemisphere, together with a classification of its equality cases, equiv…

Let $\mathcal{M}$ be a compact submanifold of $\mathbb{R}^n$ of dimension $m$, with codimension $k=n-m$, and let $N_{\mathcal{M}}(Q,\delta)$ denote the number of rational points of…

Does there exist a smooth embedded open annulus $S\subset\mathbb{R}^3$ with strictly negative Gaussian curvature and injective Gauss map $N:S\to\mathbb{S}^2$ that contains a simple…

The conjecture asserts a quantitative width bound for foliated positive-scalar-curvature geometry: under the appropriate completeness, foliation, boundary, index, rank, and leafwis…

For every integer $n\geq 2$, every integer $N\geq n+1$, and every immersion $f:T^n\to\mathbb{R}^N$ whose induced metric is flat, the Willmore energy…

Let $X$ be a compact connected complex manifold of complex dimension $n$, and let $f:X\to X$ be a holomorphic Anosov diffeomorphism. Thus there is an invariant splitting…

For every integer $n\geq 2$, let $F\colon B^n\to B^{2n}\subset\mathbb{C}^n$ be a smoothly immersed Lagrangian $n$-ball whose Maslov form is conformal and whose boundary is Legendri…

Let $X$ be a compact Riemann surface and let $(E,\Phi)$ be a stable $\mathrm{SL}(n,\mathbb{C})$ Higgs bundle on $X$. For each $r>0$, let $e_r(x)$ denote the energy density at…

Every connected minimal hypersurface $M^3\subset S^4$ with constant scalar curvature is isoparametric; equivalently in the local formulation, $M$ is locally congruent to an isopara…

Let $L:\mathbb{K}^2\to\mathbb{S}^3$ be Lawson's minimal Klein bottle, let $\sigma:\mathbb{S}^3\setminus\{p\}\to\mathbb{R}^3$ be a stereographic projection for a point $p$ not on…

Let $T_{\mathrm{Cl}}\subset\mathbb{C}P^2$ denote the Clifford torus, and for a closed immersed surface $\Sigma\subset\mathbb{C}P^2$ define…

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For each pair of dimensions $n,m$, there exists $\varepsilon_{n,m}>0$ such that every compact minimal submanifold $M^n\subset S^{n+m}$ satisfying $M\cong S^n$ and…

Determine whether every compact foliated band with positive leafwise scalar curvature satisfies the foliated analogue of Gromov's sharp band-width estimate, bounding its leafwise w…

For every integer $n\ge 2$, every complete simply connected Riemannian $n$-manifold $(M^n,g)$ with sectional curvature $\operatorname{Sec}_g\le \bar k\le 0$, and every volume $V>0$…

Given $n\geq 2$, a coefficient vector $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\geq 0}^n\setminus\{0\}$ that is log-concave, meaning $\alpha_j^2\geq \alpha_{j-1}\alpha_{j+1}$ for…

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…

For every smooth, strongly convex, regular Finsler metric $F$ on a smooth manifold $M$, if $F$ is a Landsberg metric, then $F$ is a Berwald metric.

For every closed generalized $m$-quasi-Einstein manifold $(M^n,g,X,\lambda)$ with $m\in\mathbb{R}\setminus\{0\}$ and constant $\lambda\leq 0$, satisfying…

Let $(M^n,g_0)$ be a closed Einstein manifold, and let $\Sigma\subset M$ be a singular set. Suppose that $g$ is a continuous (or, in the weaker formulation, $L^\infty$) Riemannian…

For each integer $n\ge 2$, let…

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…

For every integer $n\ge 3$, every compact $n$-dimensional Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature, $\operatorname{Ric}_g\ge 0$, and every principal curvature…

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