Vortrag: "Nonlinear Dirac equations and surfaces of constant mean curvature"

Abstract:

The talk will start with an introduction to the spinorial version of the Weierstrass representation for surfaces in $R^3$. For constant mean curvature surfaces this leads to a non-linear eigenvalue equation for the Dirac operator, namely $D\phi=\lamba \|\phi\|^2 \phi$. The $n$-dimensional generalization of this equation, namely the equation $D\phi=\lambda  \|\phi\|^{2/(n-1)}\phi$, defined on a closed $n$-dimensional spin manifold, also plays an important role when we study the infimum of the first positive eigenvalue of the Dirac operator among all volume-$1$-metrics in a given conformal class. We discuss when the infimum is attained, adopting a strategy similar to the solution of the Yamabe problem. We will describe a related obstruction for the mean curvature of conformal embeddings $S^2\to R^3$. If time admits we might comment on some related surgery theorems, we might discuss some associated conjectures for non-linear Dirac equations or or its non-spinorial brother, the Yamabe equation.

We intend to start the talk in a way suitable for mathematicians not-specialised in spin geometry or surfaces theory, later the talk will touch the research interests of several scientists in Gießen and Marburg. Parts of the talk are collaborations with Emmanuel Humbert, Mattias Dahl, Nadine Große and Mohameden Ould Ahmedou. My interest in this subject was essentially stimulated by discussions with Thomas Friedrich who will unfortunately no longer be able to listen to the talk.

Tee um 16:30 Uhr im runden Zimmer, Vortrag um 17:00 Uhr