A direct approach to prescribing scalar curvature on bundles

TL;DR

This paper explores conditions under which scalar curvature functions can be realized on bundle total spaces, using Kazdan-Warner results and variational methods, with applications to bundles over Calabi-Yau manifolds.

Contribution

It introduces a direct approach to prescribing scalar curvature on bundles, combining classical results with variational techniques, and applies these to complex geometric settings.

Findings

Characterization of scalar curvature functions on bundle total spaces

Conditions for realizability of scalar curvature functions via variational methods

Application to bundles over Calabi-Yau manifolds

Abstract

This note intends to demonstrate how to discuss scalar curvature functions' admissibility on bundles by directly applying some of the Kazdan--Warner results. Proofs of the concept include determining which functions are realizable as scalar curvature functions on the total space of several bundles over exotic manifolds. In addition, we employ traditional variational methods to provide reasonable conditions to smooth functions on the total spaces of some fiber bundles to be realized as scalar curvature functions for some Riemannian submersions metrics. We apply these last results to bundles over Calabi--Yau manifolds.