369 problems

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…

…

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 $(M^n,g)$ be a complete noncompact Riemannian manifold with $n\ge5$. Write $R_g$ for its scalar curvature, and let $\mathcal S^2$ denote the $L^2$-scalar curvature functional.…

Let $(M,g)$ be a fixed compact smooth two-dimensional Riemannian manifold with smooth boundary, and let $\mu^{\text{loop}}$ denote the Brownian loop measure on $(M,g)$. Let $K$ be…

Let $M$ be a simply connected surface with a complete Riemannian metric with no conjugate points. Suppose that $\mathcal F$ is a foliation on $M$ whose leaves are all geodesic line…

Let $(\mathcal{M},g)$ be a Riemannian manifold, let $A$ be a smooth vector field, and let $\mu$ be the loop measure determined by the diffusion with generator … Assume that the dif…

Let $M$ be the closed manifold and let $\Gamma_\infty(M)$ denote the space of smooth Riemannian metrics on $M$. Let $\mathcal{M}_{ei}$ be the set of metrics for which the set of cr…

Convex geodesic Swiss cheese conjecture. There exists such a finite disjoint union whose removal from $N$ delivers a metric $g_M$ of gradient type on $M$.

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…

Let $M$ be a compact, locally irreducible Riemannian manifold that is rigidly scalar-flat, meaning that it admits a scalar-flat metric but no metric of positive scalar curvature. I…

Intrinsic characterization conjecture. $(\mathbb S^n,g)$ admits a smooth global isometric embedding

Let $(X,\tilde g)$ be the universal cover of a $d$-dimensional smooth Riemannian manifold $(M,g)$. A mediatrix is the locus equidistant from the reference point and one of its nont…

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…

Universal-cover conjecture. The universal covering space $\widetilde{M}$ is isometric to $G/\operatorname{\textsl{SU}}(2)$ equipped with its unique $G_2$ structure with parallel to…

Entropy-or-flatness conjecture. Either $M$ is flat or its geodesic flow has positive topological entropy.

Compact nearly Kähler conjecture. Every compact nearly Kähler manifold is a 3-symmetric space.

Let $(M^n,hyp)$ be a closed hyperbolic manifold, and let $g$ be another metric on $M$ with $Vol(M,g)<Vol(M,hyp)$. Hyperbolic volume comparison conjecture. … for all $R$. This stren…

Let $(M^n,g)$ be a complete Riemannian manifold, and let $(S^n,g_0)$ be a round sphere. Choose the radius of the round sphere so that the filling radius of $(M^n,g)$ equals that of…

Let $\Sigma$ be a closed surface, and let $I(\Sigma)$ denote its Friedlander–Nadirashvili invariant, defined as the infimum over conformal classes $C$ on $\Sigma$ of the supremum…

Minimal immersion conjecture. The immersion $f$ must be totally geodesic.

Let $M$ be a closed Riemannian manifold, let $\widetilde{M}$ be its universal cover, and let $\operatorname{Isom}(\widetilde{M})$ be its isometry group containing $\pi_1(M)$. Quant…

Let $(M,g)$ be a real analytic Riemannian manifold with ergodic geodesic flow, and let $\{\varphi_j\}$ be a density-one sequence of ergodic eigenfunctions. For a test function $f$,…

Let $M$ be complete, let $O$ be a fixed origin, and suppose that for some $r_0>0$ the Ricci curvature is bounded below by $(n-1)/(4r^2)$ whenever $r\geq r_0$. Higher-dimensional Ri…

Let $(M,g)$ satisfy the hypotheses of the preceding one-endedness conjecture, and suppose that its Ricci curvature agrees with that of a paraboloid for $r\leq r_0$. Paraboloid rigi…