291 problems

For $n\ge 3$, let $g$ be a smooth Riemannian metric on the torus $\mathbb{T}^n$. Suppose that every primitive class $\alpha\in H_{n-1}(\mathbb{T}^n;\mathbb{Z})$ is represented by a…

For every bounded simply-connected domain $\Omega\subset\mathbb{R}^2$, let $u$ be its torsion function, defined by $-\Delta u=1$ in $\Omega$ and $u=0$ on $\partial\Omega$, and let…

For each dimension $n\ge 2$, determine whether the following implication holds: for every domain $\Omega\subset\mathbb{R}^{n}$, every $u\in W^{2,n}_{\mathrm{loc}}(\Omega)$, and eve…

Let $m\ge 2$, let $g$ be a Riemannian metric on $\mathbb{R}\mathbb{P}^{m}$, and let $\operatorname{sys}(g)$ denote the length of the shortest loop representing the nontrivial eleme…

Conjecture 1 (Generalized Chang--Yang conjecture). For every integer $N\geq 1$ and every $\alpha\geq\frac{1}{2}$, let

For every admissible integer $n\ge 12$ and every exponent $p$ satisfying $p_{\mathrm{JL}}(n)<p<p_{\mathrm{JL}}(n-1)$, every positive smooth entire solution…

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…

In the setting of Theorem, let $\bm{l}=\bm{l}(\beta)$ be the length vector associated with the construction of $\operatorname{AF_\beta}$ gravitational instantons. Uniqueness conjec…

Let $M$ be the spatial manifold, let $[0,\tau]\times M$ be the spacetime manifold $\mathbb{M}_0$, and for each $a\geq 0$ let $S_a$ and $H_a$ denote the Sobolev and Cheeger constant…

Let $(M^d,g)$ be a complete asymptotically Euclidean manifold of real dimension $d$ with non-negative scalar curvature and an outermost minimal hypersurface $\Sigma$. Write…

Let $n\ge 4$. An ancient Ricci flow is a Ricci flow defined for all sufficiently negative times. Consider the solution on $S^n$ constructed in the paper's main existence theorem, a…

Let $K\subset\mathbb E^n$ be a smooth manifold of codimension at least $2$ with a regular embedding, and let $\mathcal W(\epsilon)$ denote the volume of the trapping locus…

Let $(X,\mathsf d,\mathcal{H}^{N})$ be a non-collapsed $\mathsf{RCD}(K,N)$ space, and let $({\mathcal R}_{N})_{\varepsilon}$ denote its $\varepsilon$-regular set. Then the interior…

Stability conjecture. If $n\leq 7$, then all level sets of $u$ are hyperplanes.

Let $\Omega$ and $k$ satisfy the assumptions in Section 1. Suppose also that $\Omega$ is convex and $\rho_->0$. An optimal-density uniqueness and convexity conjecture asserts that the…

Let $G$ be a complex Lie group whose Lie algebra is the complexification $\mathfrak g=\mathfrak g_{\mathbb R}\otimes\mathbb C$ of a simple compact real Lie algebra…

Let $M_j$ be asymptotically flat $n$-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces, with either no boundary or boundary…

Schoen's Weyl-vanishing conjecture. If $\overline{y}$ is a blow-up point, then the Weyl tensor $W_{g_0}$ of $g_0$ vanishes at $\overline{y}$ to order $\left[\frac{n-6}{2}\right]$;…

Energy conjecture. The quantity $Q(\operatorname{\stackrel{\longrightarrow}{Vol}}K^+)$ is uniquely minimized when $K$ is an ellipsoid.

Hadamard-type conjecture. (i) $\mathcal G$ is concave if and only if $G_v\geq 0$ for every $x\in\Omega$. (ii) $v$ is $-\infty$-concave, meaning that $\mathcal G$ has convex level s…

Extension conjecture. Theorem 3 should hold without the hypothesis $H\geq 0$.

Let $n\geq 3$, let $(M^n,g)$ be a smooth compact Riemannian manifold with nonempty boundary, and suppose that $\lambda_1(L_g)>0$. For a conformal metric $\widetilde g\in[g]$ with z…

Let $d\in(0,2]$, let $E\subset\mathbb C$ be compact, and let $h$ be a principal $K$-quasiconformal mapping conformal on $\mathbb C\setminus E$. Here $\mathcal M^d$ denotes $d$-dime…

Let $X$ be a compact $4$-manifold, let $\Omega$ be a symplectic form on $X$, and let $\mathcal{P}$ be a unimodular constraint manifold with negative chords that is asymptotic to a…

Volume-entropy conjecture. If