Members & Guests

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

As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive
scalar curvature metrics on closed simply connected manifolds in dimensions at least 
five appears on spin manifolds, and is given by the non-vanishing of the \(\alpha\)-genus 
of Hitchin. 
When unobstructed we will in this paper realise  a positive scalar curvature metric by an 
immersion into Euclidean space whose dimension is uniformly close to the classical Whitney 
upper-bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure 
for constructing positive scalar curvature metrics.

 

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

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products.

 

Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimensions at least five which have odd order abelian fundamental groups, are non-spin and  atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fundamental groups.

 

 

 

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

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):
5Index theory on Lorentzian manifolds15Spaces 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):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

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