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Let $M,N$ be compact Riemannian manifolds, and let $\phi:M \to N$ be harmonic.

The index of $\phi$ is defined to be the dimension of the maximal subspaces of $\Gamma(\phi^*TN)$ on which the hessian of the Dirichlet energy is negative definite.

Why is it true that the index is always finite?

Does it follow from the ellipticity of the Jacobi operator?

I know that the hessian of the Dirichlet energy is given by $$ H(E)_{\phi}(V,V)= \int_{M} \langle J_{\phi}(V),V \rangle \text{Vol}_g $$ where $J:\Gamma(\phi^*TN) \to \Gamma(\phi^*TN))$ is the Jacobi operator, i.e.

$$ J_{\phi}(V)=-\text{trace}_{g} R^{TN}(V,d\phi(\cdot))d\phi(\cdot)+\nabla^*\nabla V. $$

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