Subquadratic harmonic functions on Calabi-Yau manifolds with maximal volume growth

TL;DR

This paper proves that on certain Calabi-Yau manifolds, harmonic functions with subquadratic growth are essentially holomorphic, extending previous results through new local estimates and a Liouville theorem for harmonic forms.

Contribution

It generalizes Conlon-Hein's result by establishing a Liouville theorem for harmonic 1-forms on Calabi-Yau manifolds with maximal volume growth using novel local L^2 estimates.

Findings

Harmonic functions with subquadratic growth are real parts of holomorphic functions.

A new local L^2 estimate for the exterior derivative of harmonic 1-forms is developed.

The result extends the understanding of harmonic functions on Calabi-Yau manifolds.

Abstract

On a complete Calabi-Yau manifold $M$ with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville type theorem for harmonic $1$-forms, which follows from a new local $L^2$ estimate of the exterior derivative.