Let $S$ be a closed manifold and $(M,g)$ be a Riemannian manifold. Minimal submanifolds are by definition the critical points of the volume functional $$F: \mathcal{Imm}(S,M) \to \mathbb{R} \qquad \iota \mapsto \int_S \mathrm{vol}_{\,\iota^* g}$$ defined on the space $\mathcal{Imm}(S,M)$ of immersions of $S$ into $M$. There exists a well-known expression of the Hessian of $F$ in terms of the Jacobi operator $J$. The Jacobi operator is Fredholm of index $0$ but has a non-trivial kernel in general. Vector fields annihilated by $J$ are called Jacobi fields.

Since the critical points of $F$ are degenerate (in general), we cannot apply the Morse lemma to get a quadratic normal form of $F$. However, the splitting lemma of Gromoll & Meyer applies and entails that $F$ is locally equivalent to $$\frac{1}{2}\langle J \cdot, \cdot \rangle + G$$ where $G$ is a function defined on the finite-dimensional space of Jacobi vector fields (and the quadratic part is the Hessian of $F$).

Question: What is known about the singular part $G$? If the concrete form of $G$ is not known, what can be said about its zero set $G^{-1}(0)$?


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