## Prof. Dr. Bernhard Hanke

### Speaker, Member of Programme committee, Project leader

**Professor für Differentialgeometrie**

Universität Augsburg

E-mail: hanke(at)math.uni-augsburg.de

Telephone: +49 821 598 - 22 38

Homepage: https://www.math.uni-augsburg.de/prof/di...

## Publications within SPP2026

We show that local deformations of solutions to open partial differential relations near suitable subsets can be extended to global solutions, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with ordinary differential inequalities, convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry. The main application concerns Riemannian metrics. We prove an approximation result which implies, for instance, that every compact surface carries a *C^{*1,1}-metric with Gauss curvature ≥1 a.e. on a dense open subset. For *C*^2-metrics this is, of course, impossible if the genus of the surface is positive.

**Related project(s):****5**Index theory on Lorentzian manifolds**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

The canonical map from the \(\mathbb{Z}/2\)-equivariant Lazard ring to the \(\mathbb{Z}/2\)-equivariant complex bordism ring is an isomorphism.

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds