We prove a Riemannian positive mass theorem for manifolds with a single asymptotically flat end, but otherwise arbitrary other ends, which can be incomplete and contain negative scalar curvature. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau. Contents 1. Introduction 1 2. The Dirichlet problem for µ-bubbles 6 3. Construction of the weight 9 4. Proof of Theorem 1.6 with an extra assumption 11 4.1. Construction of Σ ∞ 11 4.2. Asymptotics of Σ ∞ 14 4.3. Strong stability of Σ ∞ 23 4.4. Finishing the argument when R g > 0 far out 27 5. Proofs of the main theorems 28 Appendix A. Curvature estimate in homogeneously regular manifolds 30 References 34
We offer a mathematically rigorous basis for the widely held suspicion that full black hole evaporation is in tension with predictability. Based on conditions expressing the global causal structure of evaporating black hole spacetimes, we prove two theorems in Lorentzian geometry showing that such spacetimes either fail to be causally simple or fail to be causally continuous. These theorems, when combined with recent results [1] on the causal structure of spacetimes with timelike boundary, bear significantly on the question of whether these spacetimes permit for a predictable evolution. arXiv:1808.07303v4 [gr-qc]
Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant Λ = 0 or Λ < 0. Here, we prove a topological censorship theorem that is compatible with Λ > 0 and which can be applied to whole universes containing possibly multiple collections of black holes. The main assumption in the theorem is that distinct black hole collections eventually become isolated from one another at late times, and the conclusion is that the regions near the various black hole collections have trivial fundamental group, in spite of there possibly being nontrivial topology in the universe.
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