Two perspectives on the geometry of Ricci-flat manifolds

23.11.2022, 14:00  –  Golm, Haus 9, Zimmer 2.22

Bernd Ammann, Uni Regensburg und Klaus Kröncke, KTH, Stockholm

14:00 Bernd Ammann (Uni Regensburg): From Pythagorean triples to constant mean curvature surfaces

14:45 Tee und Kaffee Pause

15:15 Klaus Kröncke (KTH, Stockholm): From the heat flow to the Ricci flow: Stability and singularity models


Bernd Ammann, Uni Regensburg: From Pythagorean triples to constant mean curvature surfaces
Finding Pythagorean triples essentially amounts to determining the rational points on a unit circle; the stereographic projection defines a bijection from these points to the field of rational numbers. One can parametrize all (irreducible) Pythagorean triples (modulo a factor i) by means of a formula used to parametrize the quadric X^2+Y^2+Z^2=0 in the three-dimensional complex space. Around 1860 Weierstraß used this parametrization in order to describe conformally parametrized surfaces in the three dimensional real space in terms of two complex-valued functions. The functions are holomorphic if and only if the parametrized surface is minimal. It is convenient to turn this pair of complex-valued functions into a section of the spinor bundle, and in the 1980's, it was realized that this spinor satisfies a non-linear Dirac equation involving the mean curvature.   This is also the Euler-Lagrange equation to a variational problem that can be solved similarly to the solution of the Yamabe problem.
I will also report on joint work with the brothers Sergiu and Andrei Moroianu, where I generalized the 3-dimensional hypersurfaces in four dimensional Ricci-flat manifolds.

Klaus Kröncke, KTH, Stockholm: From the heat flow to the Ricci flow: Stability and singularity models
The Ricci flow is an evolution equation for Riemannian metrics on a manifold which has been a powerful tool for proving several breakthrough results obtained in the 21st century, including the celebrated resolution of the Poincaré and Thurston geometrization conjectures on the classification of three-dimensional manifolds. We will first recall some facts about this evolution equation, which has a similar mathematical structure as the heat equation and shares many features with it. However, in contrast to solutions to the heat equation, Ricci flows will in general develop singularities.
The structure of singularities is deeply related to the geometry and stability of self-similar solutions of the Ricci flow, called Ricci solitons. We will discuss Ricci solitons, which form an interesting class of Riemannian manifolds which contain Einstein manifolds and in particular, Ricci-flat manifolds. In this talk I will mainly focus on stability properties of these manifolds and discuss tools which are used to prove stability or instability of a given Ricci soliton.

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