Singular Riemannian Foliations and the prescribing scalar curvature problem

TL;DR

This paper investigates singular Riemannian foliations, establishing foundational theorems and applying them to the Yamabe problem and scalar curvature on exotic spheres, advancing geometric analysis in foliated manifolds.

Contribution

It proves a Kondrakov embedding theorem and a symmetric criticality principle for orbit-like foliations, and applies these to scalar curvature problems.

Findings

Established a version of Kondrakov Embedding Theorem for orbit-like foliations.

Proved an analogue of Palais' Principle of Symmetric Criticality in this setting.

Demonstrated existence results for metrics with constant scalar curvature on exotic spheres.

Abstract

An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of $M$ into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of orbit-like foliations. As proof of concepts, we study not only the corresponding Yamabe problem in the setting, but also to the case of fiber bundles with homogeneous fibers, seeking for the existence of metrics with constant scalar curvature that respect the respective Riemannian Foliation decomposition. An application to the existence of positive constant scalar curvature on exotic spheres is presented. In an upcoming version we shall extend the results to the corresponding Kazdan--Warner problem.