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Bi 2 Sr 2-x La x CuO 6+δ and Bi 2-y Pb y Sr 2-x La x CuO 6+δ high-T c superconductors in a wide doping range from overdoped to heavily underdoped were studied by x-ray absorption and photoemission spectroscopy. The hole concentration p was determined by an analysis of the Cu L 3 -absorption edge. Besides the occupied density of states derived from photoemission, the unoccupied density of states was determined from the prepeak of the O K-absorption edge. Both, the occupied as well as the unoccupied density of states reveal the same dependence on hole doping, i.e. a continuous increase with increasing doping in the hole underdoped region and a constant density in the hole overdoped region. By comparing these results of single-layer BSLCO with previous results on single-layer LSCO it could be argued that besides the localized holes on Cu sites the CuO 2 -planes consist of two types of doped holes, from which the so-called mobile holes determine the intensity of the prepeak of the O 1s absorption edge

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

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For manifolds with a distinguished asymptotically flat end, we prove a density theorem which produces harmonic asymptotics on the distinguished end, while allowing for points of incompleteness (or negative scalar curvature) away from this end. We use this to improve the "quantitative" version of the positive mass theorem (in dimensions 3 ≤ n ≤ 7), obtained by the last two named authors with S.-T. Yau [LUY21], where stronger decay was assumed on the distinguished end. We also give an alternative proof of this theorem based on a relationship between MOTS and µ-bubbles and our recent work on the spacetime positive mass theorem with boundary [LLU21].