Geometry of three-dimensional manifolds with scalar curvature lower bound

Ovidiu Munteanu, Jiaping Wang

Published 2022-01-14, updated 2022-04-08Version 2

The paper concerns three-dimensional complete manifolds with scalar curvature bounded from below. One of the purposes is to establish a sharp comparison theorem for the bottom spectrum in the spirit of the classical result of Cheng. Another purpose is to derive volume and other geometric information in terms of the scalar curvature when the Ricci curvature is asymptotically nonnegative and the scalar curvature is positive. If the scalar curvature decays no faster than linearly, then the manifold does not admit any positive Green's function. When the scalar curvature is bounded from below by a positive constant, it is shown that the volume of unit balls must be bounded from above by the lower bound of the scalar curvature at infinity. In particular, in the case that the Ricci curvature is nonnegative, the volume of the manifold must be of linear growth. This answers a question of Gromov in the affirmative for dimension three. Volume estimates are also established for the case when scalar curvature decays polynomially.