Critical points of Green’s functions on complete manifolds. (English) Zbl 1278.53042

Enciso, Alberto; Peralta-Salas, Daniel

J. Differ. Geom. 92, No. 1, 1-29 (2012).

The authors analyze critical points of a Li-Tam Green’s function \(G\) on a complete open Riemannian surface. They show (Theorem 1.1) that in the case of a surface of finite type the number of critical points of \(G\) is bounded from above by the first Betti number. Moreover, if this upper bound is attained, then \(G\) is Morse.
Next they show that in higher dimension this is not the case. Let \(n\geq 3\) and \(N\) be any positive integer. Let \(\Sigma\) be a closed submanifold of \(\mathbb R^n\). The authors construct real analytic Riemannian manifolds \(M_j\), \(j=1,2,3\), such that (Theorem 1.2):

(i)

The minimal Green’s function of \(M_1\) has at least \(N\) nondegenerate critical points.

(ii)

The minimal Green’s function of \(M_2\) has a level set diffeomorphic to \(\Sigma\).

(iii)

The critical set of the minimal Green’s function of \(M_3\) has codimension \(\leq 3\).

Moreover, in all cases, the Green’s functions tend to zero at infinity.

Reviewer: Wojciech Kozlowski (Łódź)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
35J08	Green’s functions for elliptic equations
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

Green’s functions; Morse function; complete open Riemannian surface; Betti number