Title: Distance estimates in the spin setting and the positive mass theorem
Abstract: The positive mass theorem states that a complete
asymptotically Euclidean manifold of nonnegative scalar curvature has
nonnegative ADM mass. It relates quantities that are defined using
geometric information localized in the Euclidean ends (the ADM mass)
with global geometric information on the ambient manifold (the
nonnegativity of the scalar curvature). It is natural to ask whether
the positive mass theorem can be "localized", that is, whether the
nonnegativity of the ADM mass of a single asymptotically Euclidean end
can be deduced by the nonnegativity of the scalar curvature in a
suitable neighborhood of E. I will present the following localized
version of the positive mass theorem in the spin setting. Let E be an
asymptotically Euclidean end in a connected Riemannian spin manifold
(M,g). 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, for
spin manifolds, to Schoen and Yau's question on the positive mass
theorem with arbitrary ends. 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. I will
also present explicit quantitative distance estimates in case the
scalar curvature is uniformly positive in some region of the chosen
end E. The bounds obtained are reminiscent of Gromov's metric
inequalities with scalar curvature. This is joint work with Rudolf
Zeidler.