A Note on Harmonic Functions on surfaces
Jean C. Cortissoz
TL;DR
This paper reviews Liouville type properties of harmonic functions on surfaces with complete Riemannian metrics, providing elementary proofs and establishing a gap theorem for harmonic function growth under nonnegative Gaussian curvature.
Contribution
It offers simplified proofs of known properties and introduces a new gap theorem for harmonic functions on surfaces with nonnegative Gaussian curvature.
Findings
Elementary proofs of Liouville type properties
A gap theorem for harmonic function growth
Characterization of harmonic functions on curved surfaces
Abstract
We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this metric has nonnegative Gaussian curvature.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Advanced Mathematical Modeling in Engineering