I am looking for an example of a harmonic mapping between compact Riemannian manifolds, which is stable but not a local minimizer of the Dirichlet energy.
A harmonic map $f:M \to N$ is said to be stable if the hessian of the energy at $f$ is non-negative.
(This is mentioned in "Two reports on harmonic maps", by Eells and Lemaire, without a proof or a reference).
Take the surface of revolution $M\subset R^3$ obtained by rotating the curve $$ x^{2n} + y^{2n}=1 $$ ($n\ge 2$) around the $x$-axis. On the surface $M$ take the closed curve $C$ $$ y^2+z^2=1 $$ parameterized by its arclength. This curve will give you a stable minimal map $S^1\to M$ which is not a local minimum of the energy functional.