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

Abstract

We prove that on a complete Calabi-Yau manifold $M$ with Euclidean 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 differential. We also give another proof based on the construction of harmonic functions with polynomial growth in Ding, and the algebraicity of tangent cones in Liu-Sz\'ekelyhidi.Comment: 30 pages. Comments are welcom