Special Metrics in Spin Geometry

• Applicant: Dr. Nikolai Nowaczyk
• Subject Area: Mathematics
• Term: from 2014 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 271963318

We propose two research projects that deal with special metrics in spin geometry.Project A: Prescribing Dirac Eigenvalues of higher MultiplicitiesThe Dirac equation $D \psi = \lambda \psi$ is a fundamental equation that has its origin in Dirac's formulation of quantum mechanics. For physical as well as for mathematical reasons, one is interested in those $\lambda$ for which the Dirac equation has a solution $\psi \neq 0$, i.e. in the eigenvalues of the Dirac operator $D$. Formulated on a compact Riemannian spin manifold, the Dirac operator always has a discrete real spectrum that depends on the metric. Dahl conjectured in 2005 that one can prescribe a finite part of the spectrum arbitrarily by deforming the metric (as long as one respects some side constraints). He proved his conjecture for simple eigenvalues. In my PhD thesis I was able to show that Dirac eigenvalues of higher multiplicities always exist in dimensions $m \equiv 0,6,7 mod 8$. In this project, I want to improve this result further by showing that one can prescribe simple and double eigenvalues within a given spectral interval.Project B: Critical Points of the spinorial Enery Functional and the Willmore FunctionalTo any compact oriented surface immersed in $R^3$ one can compute the associated Willmore energy. This is a measure of how strongly the surface is bended. The famous Willmore conjecture states that any torus has Willmore energy greater or equal to $2 pi^2$, which was proven recently by Neves and Marques. The proof relies on a clever construction of a 5-parameter family of surfaces, called canonical family.On the other hand, Ammann, Weiss and Witt defined a spinorial energy functional on the bundle of universal unit spinor fields. This functional can be regarded as an extension of the classical Willmore functional. We want to find new critical points of the spinorial energy functional by ``catching'' these with a suitable k-parameter family of maps analogous to the canonical family. Conversely, we want to study which Willmore tori are critical points of the spinorial energy functional.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Host: Professor Dr. André Neves