Let $M,N$ be oriented Riemannian manifolds. Let $E$ be the Dirichlet integral, and $H(E)$ its hessian.

A harmonic map $\phi:M \to N$ is said to be weakly stable if $H(E)_{\phi}(V,V) \ge 0$ for all $V \in \Gamma(\phi^*\left( TN \right))$.

I would like to consider a stronger notion of stability;

Let's say $\phi$ is strongly stable if $H(E)_{\phi}(V,V) > 0$ for every $V$ that is not identically zero.

Somehow, I could not find a definition like this in the literature, although it seems unreasonable it does not exist.

So, is there such a standard terminology?

(Perhaps a different requirement, could be that $H(E)_{\phi}(V,V) > 0$ only for $V$ which are everywhere nonzero.).

Any reference for any of the above two definitions would be appreciated. (Even an informal statement regarding what is the common use would be helpful).


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