[Paper Review] Subquadratic harmonic functions on Calabi-Yau manifolds with Euclidean volume growth

Shih‐Kai Chiu|arXiv (Cornell University)|May 30, 2019

Geometry and complex manifolds41 references4 citations

TL;DR

This paper establishes that on a complete Calabi-Yau manifold with Euclidean volume growth, any harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. The result is proven via a new local $L^2$ estimate for the differential, leading to a Liouville-type theorem for harmonic 1-forms, and is further supported by an alternative proof using polynomial growth harmonic functions from Ding and algebraicity of tangent cones from Liu-Szekelyhidi.

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-Szekelyhidi.

Motivation & Objective

• To generalize a result by Conlon-Hein on harmonic functions with subquadratic growth on Calabi-Yau manifolds.
• To establish a Liouville-type theorem for harmonic 1-forms on complete Calabi-Yau manifolds with Euclidean volume growth.
• To provide a new proof using a local $L^2$ estimate of the differential, offering a geometric analytic approach.
• To offer a second proof based on polynomial growth harmonic functions from Ding and the algebraicity of tangent cones in Liu-Szekelyhidi.

Proposed method

• Deriving a new local $L^2$ estimate for the differential of a function on the manifold to control growth and decay properties.
• Applying this estimate to prove a Liouville-type theorem for harmonic 1-forms, implying vanishing under subquadratic growth.
• Using the existence of harmonic functions with polynomial growth constructed by Ding to support the main result.
• Leveraging the algebraicity of tangent cones established by Liu-Szekelyhidi to analyze asymptotic behavior and support the holomorphic extension.
• Combining geometric analysis with complex analytic techniques to show that subquadratic harmonic functions arise as real parts of holomorphic functions.
• Establishing the connection between growth conditions and holomorphic extension through both analytic and algebraic-geometric methods.

Experimental results

Research questions

• RQ1 Under what conditions is a harmonic function with subquadratic polynomial growth on a Calabi-Yau manifold the real part of a holomorphic function?
• RQ2 Can a Liouville-type theorem for harmonic 1-forms be established on Calabi-Yau manifolds with Euclidean volume growth using a new $L^2$ estimate?
• RQ3 How do the constructions of polynomial growth harmonic functions from Ding contribute to the holomorphic extension of real harmonic functions?
• RQ4 To what extent does the algebraicity of tangent cones in Liu-Szekelyhidi support the holomorphic extension of subquadratic harmonic functions?
• RQ5 What is the role of the differential's $L^2$ norm in controlling the growth and structure of harmonic functions on Calabi-Yau manifolds?

Key findings

• On a complete Calabi-Yau manifold with Euclidean volume growth, any harmonic function with subquadratic polynomial growth is the real part of a holomorphic function.
• A new local $L^2$ estimate for the differential of a function enables a Liouville-type theorem for harmonic 1-forms on such manifolds.
• The Liouville theorem for harmonic 1-forms implies that such forms with subquadratic growth must vanish, supporting the holomorphic extension result.
• An alternative proof is constructed using harmonic functions with polynomial growth from Ding’s work, providing a complementary analytic framework.
• The algebraicity of tangent cones in Liu-Szekelyhidi’s framework supports the rigidity of growth conditions and enables the holomorphic extension of real harmonic functions.
• The combination of geometric analysis and algebraic geometry techniques confirms the holomorphic nature of subquadratic harmonic functions on these manifolds.