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401-4491-75L 4 Credits DR , MSC D-MATH

Geometric Heat Flows

Lecturers & Examiners: Prof. Dr. Tom Ilmanen
VVZ CR n/a

Last Updated: 2026-06-01 11:30:59

Abstract

Geometric heat equations: the Mean Curvature Flow and the Ricci flow.These equations make geometric objects evolve in time -- submanifolds and Riemannian manifolds respectively. At first, the object smooths out. Later, singularities and even topological changes may occur. With luck, as t -> infty, the object settles down to a static or self-replicating configuration with an optimal geometry.

Objective

The course is a survey (a glimpse) of the field. Many results will be described intuitively and geometrically.

Content

The equations are dx/dt = H and dg/dt = -2 Rc(g) respectively. The first governs a surface (or more generally, a submanifold) evolving in space by its mean curvature vector. The second describes the evolution of the metric of a Riemannian manifold by its Ricci curvature. Topics: Evolution equations for derived quantities. Smoothing properties. The maximum principle. Monotone quantities. Elementary estimates. Examples of singularity formation. Self-similar solutions (solitons). Weak solutions. Topological changes. Topological monotonicity. Curves evolving in the plane. Surfaces evolving in space. Gradient structure of Ricci flow (Perelman). Uniformization of surfaces. Thurston's geometrization theorem for three-manifolds. Analogies between the two evolution equations will be stressed.

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 20 minutes

Course Components

Type Title Time & Place Hours
lecture Geometric Heat Flows
no class on 29 September as of 6 October, the class will take place in HG E 33.5
  • Mon 16:15-18:00 (HG E 33.5)
  • Mon 16:15-18:00 (ML F 39)
2 h weekly

Offered In