Analysis Seminar

Conghan DongStony Brook University
Stability of Euclidean 3-space for the positive mass theorem

Monday, October 23, 2023 - 2:55pm
Malott 406

The Positive Mass Theorem of R. Schoen and S.-T. Yau in dimension 3 states that if $(M^3, g)$ is asymptotically flat and has nonnegative scalar curvature, then its ADM mass $m(g)$ satisfies $m(g) \geq 0$, and equality holds only when $(M, g)$ is the flat Euclidean 3-space $\mathbb{R}^3$. We show that $\mathbb{R}^3$ is stable in the following sense. Let $(M^3_i, g_i)$ be a sequence of asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that $m(g_i)$ converges to 0. Then for all $i$, there is a domain $Z_i$ in $M_i$ such that the area of the boundary $\partial Z_i$ converges to zero and the sequence $(M_i \setminus Z_i , \hat{d}_{g_i} , p_i )$, with induced length metric $\hat{d}_{g_i}$ and any base point $p_i \in M_i \setminus Z_i$, converges to $\mathbb{R}^3$ in the pointed measured Gromov-Hausdorff topology. This confirms a conjecture of G. Huisken and T. Ilmanen. This talk is based on joint work with Antoine Song.