The mixed Yamabe problem for harmonic foliations

TL;DR

This paper investigates a Yamabe-type problem for harmonic foliations, aiming to prescribe constant mixed scalar curvature through conformal changes, and employs spectral analysis of Schrödinger operators to find solutions.

Contribution

It formulates a new Yamabe problem for harmonic foliations and develops a method using spectral parameters to solve the associated nonlinear elliptic equation.

Findings

Derived leafwise elliptic equation for harmonic foliations

Established existence of solutions using spectral analysis

Identified solutions as attractors of a nonlinear heat equation

Abstract

The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a harmonic foliation, we derive the leafwise elliptic equation and explore the corresponding nonlinear heat type equation. We assume that the leaves are compact submanifolds and the manifold is fibered instead of being foliated, and use spectral parameters of certain Schr\"odinger operator to find solution, which is attractor of the equation.