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

Let $\mathrm{\Omega }$ be a compact subset of a complete Riemannian manifold $(N,h)$. Suppose there exists a $C^2$-function $f:\mathrm{\Omega }⟶\mathbb{R}$ 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 $u(M)\subset \mathrm{\Omega }$ must be a constant map.

Is the inverse of this Maximum Principle true? I.e. suppose $\mathrm{\Omega }$ 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 $u(M)\subset \mathrm{\Omega }$ must be a constant map.

Natural question: Is the existence of a strictly convex function defined on $\mathrm{\Omega }$ 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”.