Eigenfunctions for quasi-laplacian

2018/07/03 by Min Chen, Chen, Min
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1807.01108

openalex publication_date 2018/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To study the regularity of heat flow, Lin-Wang[1] introduced the quasi-harmonic sphere, which is a harmonic map from M=(ℝ^m,e^{-(|x|2)/(2(m-2))}ds₀²) to N with finite energy. Here ds₀² is Euclidean metric in ℝ^m. Ding-Zhao [2] showed that if the target is a sphere, any equivariant quasi-harmonic spheres is discontinuous at infinity. The metric g=e^{-(|x|2)/(2(m-2))}ds₀² is quite singular at infinity and it is not complete. In this paper , we mainly study the eigenfunction of Quasi-Laplacian Δ_g=e^{(|x|2)/(2(m-2))} ( Δ_g0 - ∇_g0h⋅ ∇_g0) =e^{(|x|2)/(2(m-2))} Δ_h for h=(|x|²)/(4). In particular, we show that non-constant eigenfunctions of Δ_g must be discontinuous at infinity and non-constant eigenfunctions of drifted Laplacian Δ_h=Δ_g0 - ∇_g0 h⋅ ∇_g0 is also discontinuous at infinity.

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