## 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, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, 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 convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.

The main application is a general approximation result by sections which have very restrictive local properties an open dense subsets. This shows, for instance, that given any *K*∈R every manifold of dimension at least two carries a complete *C^*1,1-metric which, on a dense open subset, is smooth with constant sectional curvature *K*. Of course this is impossible for *C^*2-metrics in general.

**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