31 problems

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…

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 $M_j^3$ 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 $\Omega$ 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 $H_0$…

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…

Geometric inequality conjecture. One has

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

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…

Let $(M_i,g_i)$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature, and let $m(g_i)$ denote their ADM masses, with $m(g_i)\to 0$. Huisken–Ilmanen…

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…

Let $M_i$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass tending to zero. Huisken–Ilmanen's conjecture. There is a set $Zi e Mi$…

Positive mass conjecture for arbitrary asymptotically hyperbolic ends. The mass functional of the end $\mathcal{E}$ is timelike future-directed or zero.

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

Let $n\geq 3$, let $P^n$ be a Euclidean prism, $P_0\times[0,1]^{n-2}$, and let $(\Omega^n,g,k)$ be an initial data set admitting a degree-one map onto $P$. Assume that…

Positive mass conjecture. To any AF gravitational instanton $(M,\mathbf{g})$, one can assign a geometric invariant called the mass $m$, coinciding with the invariant $m$ for toric…

Let $N^n$ be a complete manifold diffeomorphic to $\mathbb{R}^n$, let $X^n$ be a complete connected non-compact manifold, and let $g$ be a smooth complete metric on … with nonnegat…

Let $\Omega$ be a compact domain with boundary data induced by a nonnegative-scalar-curvature metric, and let … where $\mathcal C_{(\gamma,H)}$ is the set of asymptotically flat ex…

Let $(M,g,\Sigma)$ be an asymptotically hyperbolic manifold with scalar curvature $R_g$ and boundary mean curvature $H_g$ satisfying … For any admissible chart $F$, let…

Let $M_j^m$ be a sequence of asymptotically hyperbolic Riemannian manifolds with scalar curvature $R\geq -m(m-1)$ and mass tending to zero. Assume that $M_j^m$ contains no closed i…

Fix $D>0$ and $r_0>0$. Let $\mathcal M$ be the class of asymptotically flat three-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal…

Let $N^n$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ and order $p>\frac{n-2}{2}$, with integrable scalar curvature $R_g$ and integrable mean curvature…

Let $N^n$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ and order $p>\frac{n-2}{2}$, with integrable scalar curvature $R_g$ and, when $\partial N$ is noncomp…

Let $\gamma \in (0,1)$, $n > 2\gamma$, and let $(X^{n+1},g^+)$ be Poincaré–Einstein. Suppose that $\Lambda^{\gamma}(M,[\hat{h}])>0$ and that either $(M^n,[\hat{h}])$ is locally con…

Positive mass conjecture. Then $m(g)\geq 0$, with equality if and only if $(M,g)=(\mathbb{R}^n,g_E)$.