Positive mass theorem and applications to the Yamabe problem

Summer term 2023

Prof. Bernd Ammann (Room M119)

Language/Sprache

Content of the lecture

1. Explain the role of the ADM mass in general relativity. Roughly speaking the ADM mass describes the mass of a system of some stars and black holes in a universe that is asymptotically flat at infinity in general relativity. It may be extended to the ADM energy-momentum vector. The positive mass theorem (PMT) says: if the mass density of the space-time is non-negative, then its total mass is non-negative as well.
2. Apply this to the Yamabe problem: in the winter term 2022/23 I gave a lecture "Geometric partial differential equations on manifolds (Yamabe Problem)" in which the Yamabe problem was solved, assuming the Aubin-Schoen inequality. This inequality follows easily from the positive mass theorem.

Thus, I do not require that the audience has followed the lecture about the Yamabe problem, but if you have, you will have an additional motivation. The lecture will be adjusted accordingly.

Prerequisites

• Linear Algebra I and II
• Analysis I, II, IV
• One lecture (or equivalent knowledge from books) out of the following topics: Differential geometry I, Geometric partial differential equations on manifolds or some lectures about general relativity.

Literature

Important References

• Das Skript zur Vorlesung in der aktuellen Version (wird laufend angepasst)
• Das Skript einer ähnlichen Vorlesung vor gut 4 Jahren.
• Lee, Parker, The Yamabe problem, Bull. AMS 17 (1987), Seite 37 ff.

Further References

Quellen zu den Grundlagen

• Skripte zur Analysis IV: Ammann (SS 2020), Friedl (SS 2016) und Garcke (zB SS 2017 auf GRIPS, evtl. auch später)
• Garcke, Funktionalanalysis (zB WS 2017/18 auf GRIPS, evtl. auch später)
• Christian Bär, Elementare Differentialgeometrie, de Gruyter
• Diverse Skripte von Christian Bär und Mitarbeitern, insbesondere das Skript zur Differentialgeometrie
• M. do Carmo, Riemannian Geometry, Birkhäuser

Weiterführende Literatur über Riemannsche Geometrie

• Cheeger, Ebin, Comparison theorems in Riemannian Geometry
• B. O'Neill, Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press
• F. Warner, Foundations of differentiable manifolds and Lie groups, Springer
• T. Sakai, Riemannian Geometry, Transl. Math. Monogr., AMS
• W. Kühnel, Differentialgeometrie, Vieweg
• J. Lee, Introduction to topological manifolds, Springer
• J. Lee, Introduction to smooth manifolds, Springer
• J. Lee, Riemannian manifolds, Springer

Time and location

Registrierung

Prüfung

Verbundene Webseiten

• Link zur G.R.I.P.S.-Seite (zusammen mit der Vorlesung im WS2022/23
• Angaben im Kommentierten Vorlesungsverzeichnis

Kriterien für Leistungsnachweise