Critical points of green’s functions on complete manifolds

Alberto Enciso Consejo Superior de Investigaciones Cient´ıfica Daniel Peralta-Salas Consejo Superior de Investigaciones Cient´ıfica

Differential Geometry mathscidoc:1609.10278

Journal of Differential Geometry, 92, (1), 1-29, 2012
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 Rn (n > 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 R^n.
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@inproceedings{alberto2012critical,
  title={CRITICAL POINTS OF GREEN’S FUNCTIONS ON COMPLETE MANIFOLDS},
  author={Alberto Enciso, and Daniel Peralta-Salas},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231111211198940},
  booktitle={Journal of Differential Geometry},
  volume={92},
  number={1},
  pages={1-29},
  year={2012},
}
Alberto Enciso, and Daniel Peralta-Salas. CRITICAL POINTS OF GREEN’S FUNCTIONS ON COMPLETE MANIFOLDS. 2012. Vol. 92. In Journal of Differential Geometry. pp.1-29. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913231111211198940.
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