# Members & Guests

## Dr. Diego Corro

Universität zu Köln

E-mail: dcorro(at)math.uni-koeln.de
Homepage: www.diegocorro.com

## Project

43Singular Riemannian foliations and collapse

## Publications within SPP2026

We expand upon the notion of a pre-section for a singular Riemannian foliation $$(M,\mathcal{F})$$, i.e. a proper submanifold $$N\subset M$$ retaining all the transverse geometry of the foliation. This generalization of a polar foliation provides a similar reduction, allowing one to recognize certain geometric or topological properties of $$(M,\mathcal{F})$$  and the leaf space $$M/\mathcal{F}$$. In particular, we show that if a foliated manifold  $$M$$ has positive sectional curvature and contains a non-trivial pre-section, then the leaf space $$M/\mathcal{F}$$ has nonempty boundary. We recover as corollaries the known result for the special case of polar foliations as well as the well-known analogue for isometric group actions.

 Journal Ann. Global Anal. Geom. Link to preprint version

Related project(s):
43Singular Riemannian foliations and collapse

 Journal Journal of Geometric Analysis Volume 32 Link to preprint version Link to published version

Related project(s):
43Singular Riemannian foliations and collapse

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.