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

Shih-Kai Chiu

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.

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

Abstract

On a complete Calabi-Yau manifold 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 -forms, which follows from a new local estimate of the exterior derivative.

Paper Structure

This paper contains 5 sections, 20 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Analysis on limit Ricci-flat cones
  3. Monotonicity of the frequency function
  4. Homogeneous $1$-forms
  5. A local $L^2$ estimate for the exterior derivative

Key Result

Theorem 1.1

Let $M$ be a complete noncompact Ricci-flat manifold with maximal volume growth. Let $u$ be a harmonic $1$-form on $M$, and by this we mean that $(dd^*+d^*d)u=0$. Suppose $u$ has sublinear growth, i.e. there exist constants $C>0$ and $s<1$ such that where $r$ is the distance function from a fixed point $p \in M$. Then $u = df$, where $f$ is a subquadratic harmonic function.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • proof : Proof of Theorem \ref{['pluriharmonictheorem']}
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Lemma 2.2: Mondino-Naber MondinoNaber
  • ...and 37 more