19 problems

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

Let $M$ be an asymptotically flat manifold of low dimension, and let $c$ be a prescribed mean curvature. A closed hypersurface in $M$ has constant mean curvature $c$ when its mean…

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…

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 $Zi e Mi$…

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

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…

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 $n$ and $q$ be integers with $n\geq 3$ and $1\leq q<n/2$. Let $(M^n,g)$ be asymptotically flat of order $\tau>(n-2q)/(q+1)$, with integrable nonnegative $q$-th Gauss–Bonnet cur…

Let $n$ and $q$ be integers with $n\geq 3$ and $1\leq q<n/2$. Let $(M^n,g)$ be asymptotically flat of order $\tau>(n-2q)/(q+1)$, with integrable nonnegative $q$-th Gauss–Bonnet cur…

Positive mass conjecture. The ADM mass of $(M,g)$ is nonnegative. Moreover, if the mass is zero, then $(M,g)$ is isometric to Euclidean space $(\mathbb{R}^n,\delta)$.

The simplest model is the closed half-space $\mathbb R^n_+$ with its standard flat metric $\delta$. Let $(M,g)$ be an asymptotically flat manifold with decay rate $\tau>(n-2)/2$, a…

Let $(M^n,g)$, $n \ge 3$, be a complete, asymptotically flat, smooth $n$-manifold with nonempty compact boundary $\Sigma=\partial M$ smooth with respect to both the smooth structur…

Limiting Riemannian Penrose inequality conjecture. Then

Conformal Conjecture. The function $u$ is harmonic with respect to $h$ and tends to 1 at infinity; in the metric $h_0$, the areas of $\widetilde \Sigma$ and $\Sigma$ differ by at m…