Nonlinear Harmonic Forms and an Indefinite Bochner Formula

TL;DR

This paper introduces nonlinear harmonic forms that minimize energy under nonlinear constraints and presents a variant of the Bochner formula to analyze 4-manifold intersection forms, providing foundational motivations and existence results.

Contribution

It pioneers the study of nonlinear harmonic forms and develops a new Bochner formula variant for 4-manifold topology analysis.

Findings

Existence results for nonlinear harmonic forms

A new Bochner formula variant for 4-manifolds

Motivations for further research in nonlinear harmonic analysis

Abstract

We introduce the study of nonlinear harmonic forms. These are forms which minimize the $L_2$ energy in a cohomology class subject to a nonlinear constraint. In this note, we include only motivations and the most basic existence results. We also introduce a variant of the Bochner formula suitable for probing the structure of the intersection form of a 4-manifold.