15 problems

Let $(\boldsymbol{\rm X},\boldsymbol{\rm g})$ be an asymptotically flat, stationary vacuum Einstein spacetime. Black hole uniqueness conjecture. The spacetime is the Kerr spacetime…

Let $(M,g)$ be a non-flat $\operatorname{AF}_0$ gravitational instanton. Schwarzschild uniqueness conjecture. The metric $g$ is, up to isometry, the Schwarzschild metric. Evidence…

Consider a rotationally symmetric, asymptotically flat Ricci flow with no minimal hypersphere present initially. Immortality and convergence conjecture. The flow is immortal and co…

Let an asymptotically flat manifold evolve by Ricci flow, and let its mass be the ADM mass of the asymptotically flat metric. Mass-decrease conjecture. The mass should decrease und…

Let $\beta\in(0,1)$ and $n\geq 1$, and let $X_n$ be the manifold in the source. Let $g_{\beta,n}$ denote the metric constructed in Theorem. AF$\beta$ uniqueness conjecture. The met…

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…

Let $X$ be a conical $\mathcal{AF}$ 4-manifold with a simply connected end, and let its asymptotic cone be the metric cone at infinity. Petrunin–Tuschmann conjecture. There is no c…

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…

Riemannian no-hair conjecture. The euclidean Kerr solution is the only AF gravitational instanton.

Let $\Sigma$ be a surface, and let its full (spacetime) Bartnik mass be defined using admissible asymptotically flat initial-data extensions with the prescribed boundary data. A st…

Let $(M,g)$ be a non-compact Riemannian manifold with asymptotically flat metric $g$, and let $R_g$ denote its scalar curvature. The ADM mass is the invariant…

Weakening conjecture. The nonexistence conclusion for sequences of connected closed stable constant mean curvature surfaces with

Let $(M,g)$ be a noncompact asymptotically flat Riemannian manifold, let $\mathfrak m_{(M,g)}$ denote its ADM mass, and let $R_g$ be its scalar curvature. Positive mass conjecture.…

Global area-minimization conjecture. These constant-mean-curvature spheres also globally minimize area among surfaces in the same homology class enclosing the same volume.

ADM mass limit conjecture. As $V$ tends to infinity,