42 problems

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 $(M^3,g)$ be an asymptotically flat Riemannian 3-manifold in the asymptotically Schwarzschild setting. Consider the volume-preserving stable constant mean curvature spheres con…

Let $(M,g)$ be a Riemannian manifold of dimension $3\leq n\leq 7$ that is asymptotically flat of rate $\tau=n-2$, with non-negative scalar curvature. Suppose that $M$ contains a no…

Let $(M,g)$ be an asymptotically flat manifold with non-negative scalar curvature, and let $m_{\text{ADM}}$ denote its ADM mass. Positive mass conjecture. The ADM mass of $M$ shoul…

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

Nonexistence conjecture. There is no such manifold for which

Let $(M,g)$ be a complete asymptotically flat Riemannian manifold with nonnegative scalar curvature, and let its mass be defined in the asymptotically flat sense. Positive mass con…

Mass-systole conjecture. If $(M,g,E)$ has nonnegative scalar curvature, then

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…

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

Let $(M,g)$ be an asymptotically flat complete Riemannian $3$-manifold with nonnegative scalar curvature, and let $m(g)$ be its ADM mass. Ilmanen's conjecture. There exists a surfa…

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 $Zi e Mi$…

Let $(M,g)$ be the asymptotically flat manifold considered in Theorem, and let $\Sigma$ be an area-constrained Willmore sphere with associated quantities $\rho(\Sigma)$ and…

Let $(\mathbb R^n,g)$ be an asymptotically flat initial data set with ADM mass $m_{\mathrm{ADM}}(g)$ and energy and momentum densities satisfying the dominant condition…

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 $M=(\mathbb R_{+}^{3},g)$ be an asymptotically flat three-dimensional Riemannian manifold with boundary, and let $p\in\partial M$. A free boundary minimal plane is a complete,…

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass, and assume that $partial M$ is empty or minimal. For $V>0$ sufficiently la…

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass. A Bernoulli surface is a surface $S=partial A$ for a compact set $A$ such…

Dimension-two extension conjecture. The assertion of the main result also holds in dimension $n=2$.

An asymptotically flat manifold models an isolated gravitational system, with local mass density and total mass measured at spatial infinity. Positive mass conjecture. If the syste…

Let $(N,g)$ be an asymptotically flat Riemannian $n$-manifold with non-compact boundary, of order $p>0$, and let $m(g)$ be its mass, assuming the defining limit exists. Positive ma…

Let $M'$ be an outermost region of a geometrostatic manifold, written as … where each $\Omega_i$ is diffeomorphic to a three-ball, has stable minimal boundary…

Let $(M'_k,g_k,x_k)$ be a sequence of pointed asymptotically flat three-manifolds with nonnegative scalar curvature and outermost minimal-surface boundaries. Suppose that … The man…