Talk

On the (non-)Inversibility of the Maximum Principle for harmonic maps (on manifolds thick at infinity).

  • Renan Assimos (Leibniz Universität Hannover)
A3 02 (Lab)

Abstract

Let Ω be a compact subset of a complete Riemannian manifold (N,h). Suppose there exists a C2-function f:ΩR which is strictly convex in the geodesic sense. It is a classical consequence of the maximum principle then that every harmonic map u:(M,g)(N,h), where (M,g) is complete compact without boundary, with u(M)Ω must be a constant map.

Is the inverse of this Maximum Principle true? I.e. suppose Ω is a subset of (N,h) now with the following property:

•) every harmonic map u:(M,g)(N,h), where (M,g) is complete compact without boundary, with u(M)Ω must be a constant map.

Natural question: Is the existence of a strictly convex function defined on Ω what is preventing non-constant harmonic maps to exist inside this subset?

In this talk I will present a counter example to this inverse maximum principle and comment on a result by M. Gromov where he proves that a certain weaker version of this question is true for the case of minimal hypersurfaces on a class of manifolds called “thick at infinity”.