Morse theory of harmonic forms

TL;DR

This paper investigates whether harmonicity assumptions can improve Novikov inequalities for closed 1-forms, concluding that under certain conditions, such improvements are impossible, supported by theoretical analysis and examples.

Contribution

It demonstrates that harmonicity does not enhance Novikov inequalities under specific assumptions, using Calabi's theorem and illustrative examples.

Findings

Harmonicity does not improve Novikov inequalities under certain conditions

Calabi's theorem characterizes harmonic 1-forms with respect to some metric

Examples illustrate the limitations of harmonic forms in inequality improvements

Abstract

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use, in an essential way, a theorem of E.Calabi characterizing 1-forms which are harmonic with respect to some metric. We also study some interesting examples illustrating our results.