70 problems

Geroch's conjecture. Any Riemannian metric on a torus with non-negative scalar curvature must be flat.

Let $g$ be a smooth complete metric on $\mathbb R^n$, where $n\geq 3$, and let $g_{\mathrm{euc}}$ denote the Euclidean metric. Assume that … and … as $x\to\infty$. Gromov's Euclide…

Let $M_i$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass tending to zero. Let $M_{i,\mathrm{ext}}$ denote the exterior region, le…

Schoen–Yau's conjecture. The ADM mass of $\mathcal{E}$ is non-negative.

Let $P$ be a convex polyhedron in Euclidean space $(\mathbb R^n,\bar g)$, and let $g$ be a Riemannian metric on $P$. Assume that $g$ has nonnegative scalar curvature, each face of…

Let $(M_j,g_j)$ be a sequence of complete asymptotically flat $n$-dimensional manifolds with nonnegative scalar curvature and ADM masses converging to zero. Convergence can be cons…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$, and let $m_{\frac{n-2}{4(n-1)}\operatorname{scal}_g}$ denote the mass associated with the conformal Laplacian. P…

Positive mass theorem. If $R_g,H_g\geq 0$, then $m(g)\geq 0$, and equality holds if and only if $M$ is isometric to $\mathbb{R}^n_+$. This is the expected positive mass statement n…

Let $(M,g,v^m d\operatorname{vol}_g)$ be an asymptotically Schwarzschild smooth metric measure space, with weighted scalar curvature and modified weighted mean curvature satisfying…

Let $M_j^3$ be a sequence of three-dimensional manifolds in the class $\mathcal{M}$, and let $m_{ADM}(M_j^3)$ denote their ADM masses. Let ${\mathbb E}^3$ denote Euclidean three-sp…

Let $M$ be a strictly pseudoconvex CR manifold of real dimension greater than three, and consider its CR mass under the relevant hypotheses for defining that mass. Higher-dimension…

Positive mass conjecture. The ADM mass of an asymptotically flat initial data set satisfying the dominant condition is positive unless it is a Cauchy hypersurface of Minkowski spac…

Positive mass conjecture. The mass of an asymptotically hyperbolic vacuum initial data set is positive unless it is a Cauchy hypersurface of Minkowski space.

Let $(M^n,g)$ be a complete non-compact asymptotically Euclidean Riemannian manifold, with ADM mass $m_\text{ADM}(g)$ and scalar curvature $\operatorname{Scal}^g$. Positive mass co…

Reference-mass conjecture. The mass of the reference should appear in the inequality of the positive mass theorem.

Let $(M^n,g,k)$ be an asymptotically hyperboloidal initial data set, where $M$ is an $n$-dimensional smooth manifold, $g$ is a Riemannian metric asymptotic to the hyperbolic metric…

Let $\Omega$ be a convex smooth domain in $\mathbb R^n$, with $n\geq 3$, and let $(\mathbb R^n\setminus\Omega,g)$ be an asymptotically flat Riemannian manifold. Let $H_g$ and $H_0$…

Lee's positive mass conjecture. The positive mass theorem holds for $g$; in particular, its ADM mass is nonnegative, with equality only in the appropriate flat rigidity case.

Let $(M,g)$ be an asymptotically flat Riemannian manifold with non-compact boundary, scalar curvature $R$ and boundary mean curvature $H$. Almaraz–Barbosa–de Lima conjecture. If…

Let $(M,g)$ be a continuous Riemannian $3$-manifold, and interpret nonnegative scalar curvature, denoted $R_g\geq 0$, in an appropriate weak sense. The Huisken's weak isoperimetric…

Geometric inequality conjecture. One has

Non-spin extension conjecture. Theorem 3.1 holds even if $M$ is not spin; equivalently,

Schoen–Yau–Lesourd–Unger–Yau conjecture. The ADM mass of $\mathcal{E}$ is non-negative.

Discrete positive mass conjecture. For an asymptotically flat graph $G$ with non-negative scalar curvature, one has $E(G)\geq 0$. Moreover, $E(G)=0$ if and only if $G$ is a standar…

Let $(M_i,g_i)$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass $m(g_i)$ tending to $0$. Suppose $Z_i\subset M_i$ are the subsets…