We prove the spacetime positive mass theorem in dimensions less than eight. This theorem
asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition,
the inequality E>=|P| holds, where (E, P) is the ADM energy-momentum vector. Previously,
this theorem was only known for spin manifolds . Our approach is a modification of the minimal
hypersurface technique that was used by the last named author and S.-T. Yau to establish the
time-symmetric case of this theorem [30, 27]. Instead of minimal hypersurfaces, we use marginally
outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first
named author . An important part of our proof is to introduce an appropriate substitute for the
area functional that is used in the time-symmetric case to single out certain minimal hypersurfaces.
We also establish a density theorem of independent interest and use it to reduce the general case of
the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics
and satisfies the strict dominant energy condition.
This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kaehler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward
second and third CR Yauís uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.
This is the very first paper to focus on the CR analogue of Yau’s uniformization conjecture in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of K¨ahler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.