Open Access
September 2012 Critical points of Green's functions on complete manifolds
Alberto Enciso, Daniel Peralta-Salas
J. Differential Geom. 92(1): 1-29 (September 2012). DOI: 10.4310/jdg/1352211221


We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by constructing, for each nonnegative integer $N$, a Riemannian manifold diffeomorphic to $\mathbb{R}^n\: (n \ge 3)$ whose minimal Green’s function has at least $N$ non-degenerate critical points. Variations on the method of proof of the latter result yield contractible $n$-manifolds whose minimal Green’s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of $\mathbb{R}^n$.


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Alberto Enciso. Daniel Peralta-Salas. "Critical points of Green's functions on complete manifolds." J. Differential Geom. 92 (1) 1 - 29, September 2012.


Published: September 2012
First available in Project Euclid: 6 November 2012

zbMATH: 1278.53042
MathSciNet: MR2998897
Digital Object Identifier: 10.4310/jdg/1352211221

Rights: Copyright © 2012 Lehigh University

Vol.92 • No. 1 • September 2012
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