Complete minimal surfaces in asymptotically flat spaces and localized solutions of the Einstein constraint equations
Vortrag von Prof. Dr. Alessandro Carlotto
Datum: 05.11.13 Zeit: 15.00 - 17.00 Raum: ETH
To what extent can the global theory of complete minimal surfaces in the Euclidean space can be extended to asymptotically flat 3-manifolds? Starting with the most basic question, we might ask whether complete stable minimal surfaces actually exist in presence of a positive ADM mass, thus considering a generalized Bernstein problem. The answer to this question turns out to be NO if the ambient metric has a nice expansion at infinity (namely if it is asymptotically Schwarschild, in a weak sense), while it is YES in full generality. The former rigidity result implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. The proof of this theorem is based on a characterization of finite index minimal surfaces via improved curvature estimates, on classical infinitesimal rigidity results by Fischer-Colbrie and Schoen and on the positive mass theorem by Schoen-Yau. More specifically, we also show that a minimal surface inside an asymptotically flat 3-manifold has finitely many ends and each of these is a graph of a function that has a suitable expansion at infinity, in analogy with a classical result by Schoen for Euclidean spaces. On othe other hand, the latter result (which is joint with Richard Schoen) follows from the construction of a new class of solutions of the Einstein constraint equations that have positive mass but are (exactly) Euclidean on the complement of a cone of arbitrarily small opening angle. This flexibility theorem sharply contrasts various recent scalar curvature rigidity results both in the compact and in the free-boundary setting. Our main theorems can be extended to analyze marginally outer-trapped surfaces (MOTS) in non time-symmetric initial data sets: we first prove that a non-compact stable MOTS in an initial data set (M,g,k) is conformally diffeomorphic to either the plane C or to the cylinder A and in the latter case infinitesimal rigidity holds. If the data have harmonic asymptotics, the former case is proven to be globally rigid in the sense that the presence of a stable MOTS forces an isometric embedding of (M,g,k) in the Minkowski space-time as a space-like slice.