Biharmonic functions and bi-eigenfunctions on some model spaces

TL;DR

This paper provides a detailed analysis of biharmonic and bi-eigenfunctions on model spaces, including explicit formulas, classifications, and examples, advancing understanding of biharmonic functions on curved spaces.

Contribution

It introduces a convenient formula for the bi-Laplacian on spheres, describes eigenvalues and eigenfunctions, and classifies proper biharmonic functions on various model spaces.

Findings

Explicit formulas for bi-Laplacian on spheres

Complete eigenvalue and eigenfunction descriptions

Classification of proper biharmonic functions on space forms

Abstract

In this paper, we first give a convenient formula for bi-Laplacian on a sphere and the complete description of its eigenvalues, buckling eigenvalues, and their corresponding eigenfunctions. We then show that the radial (or rotationally symmetric) solutions for biharmonic equation on the model space $( \mathbb{R}^+\times S^{m-1}, dr^2 + \sigma^2(r)\, g^{S^{m-1}})$ can be given by an integral formula. We also prove that the model space always admits proper biharmonic functions as the products of any eigenfunctions of the factor sphere with certain radial functions. Many explicit examples of proper biharmonic functions on space forms are given. Finally, we give a complete classification of proper biharmonic functions with positive Laplacian on the punctured Euclidean space.