# Positive mass theorem and applications to the Yamabe problem<

## Language/Sprache

The lecture is English on demand. If there is at least one participant who is not able to follow a lecture in German, then the lecture will be held in English. The formal part of the annoucement is kept in German.

## Content of the lecture

The content of the lecture is the positive mass theorem and its application to the Yamabe problem. The lecture thus has two goals
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, there might be at least two reasons, why students can be interested in this lecture: understand the positive mass theorem, its significance in general relativity and its proofs. Or to see the last missing step to solve the Yamabe problem. The precise structure of the lecture, in particular the weight of these two parts, will thus depend on the audience.

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

siehe Skript

### 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

Tuesday, 10-12 in M101.

## Registrierung

Bitte auf der GRIPS-Seite (link siehe unten) registrieren, da wir insbesondere Nachrichten via GRIPS versenden.

## Prüfung

Die Prüfung ist eine 30-minütige mündliche Prüfung.

## Kriterien für Leistungsnachweise

Siehe kommentiertes Vorlesungsverzeichnis.
Bernd Ammann, 31.01.2023
Impressum und Datenschutzerklärung