Séance Séminaire

Séminaire Gaston Darboux

The positive mass theorem and distance estimates in the spin setting

Let E be an asymptotically Euclidean end in a connected Riemannian spin manifold (M,g). We show that if E has negative ADM-mass, then there exists a constant R > 0, depending only on the geometry of E, such that M must either become incomplete or have a point of negative scalar curvature in the R-neighborhood around E in M. This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions <8 assuming Schwarzschild asymptotics on the end E. We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E. Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature. This is joint work with Rudolf Zeidler.