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Geometric pdes on manifolds (Yamabe problem) SemesterWiSe 2022 / 23 Lecturer Bernd Ammann Type of course (Veranstaltungsart) Vorlesung German title Geometrische PDGl auf Mannigfaltigkeiten (Yamabe-Problem) Contents The main theme of the lecture is the solution of a specific geometric problem, the so-called Yamabe problem. More precisely we will show: If M is a compact manifold without boundary and with a Riemannian metric g, then there is a metric g1=f2g with constant scalar curvature. Its solution requires to solve an interesting non-linear partial differential equation. On the way to solve the Yamabe problem we will learn many analytic concepts that we will need to solve such partial differential equations, i.e. semi-linear elliptic partial differential equation with a non-linearity of critical type. In the first part of the lecture I will both give an introduction into the differential geometric preliminaries (such as e.g. scalar curvature) and into the required tools for solving the associated partial differential equations. Then we will solve the Yamabe problem, by using the positive mass theorem (PMT). The PMT is an important theorem in general relativity, which will be proved for a large class of manifolds. I see the interest in this lecture in the interplay between analysis in geometry. For example we will study equations of the type Δ u + c u = λ up on an n-dimensional domain. For 1 ≤ p < (n+2)/(n-2) the solution is similar to the linear case (p=1), and therefore relatively easy. However, when p converges to (n+2)/(n-2) "bubbles" may develop in solution, and this explains why the limiting case p = (n+2)/(n-2) both a particular interesting and challenging case. The lecture is designed in a self-contained way. This means: in principle only the basic lectures such as Analysis 1-4 and Linear Algebra 1 and 2 will be required. However, having heard lecture such as functional analysis, pde 1 and differential geometry makes it easier to follow the lecture. The lecture thus can be combined with visiting the lecture differential geometry I, which gives deeper insight into the curvature aspects of this lecture. Literature
Recommended previous knowledge Analysis 1-4 and Linear Algebra 1-2. If you want to know whether you can follow and if you speak German, start reading our skript in the list of literature Time/Date Tuesday 8-10 and Friday 10-12 Location M101 (Tuesday), M102 (Friday) Course homepage https://ammann.app.uni-regensburg.de/lehre/2022w_yamabe (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
There will be a weekly exercise class Modules BV, MV, MGAGeo, MAngAn ECTS 9 |