THE HYPERBOLIC POSITIVE MASS THEOREM AND VOLUME COMPARISON INVOLVING SCALAR CURVATURE

Abstract

In this thesis, we study how the scalar curvature relates to the ADM mass and volume of the manifold. The positive mass theorem, first proven by Schoen and Yau in 1979, states that nonnegative local energy density implies nonnegative total mass. The harmonic level set technique pioneered by D. Stern [Ste19] has been used to prove a series of positive mass theorems, such as the Riemannian case [BKKS19], the spacetime case [HKK20] and the charged case. Using this novel technique, we prove the hyperbolic positive mass theorem in the spacetime setting, as well as some rigidity cases. A new interpretation of mass is introduced in this context. Then we solve the spacetime harmonic equation in the hyperbolic setting. We not only prove the positive mass theorem, but we also give a lower bound for the total mass without assuming the nonnegativity of the local energy density.Additionally, We prove a scalar curvature volume comparison theorem, assuming some boundedness for Ricci curvature. The proof relies on the perturbation of the scalar curvature [BM11].