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Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided embeddings of $\Sigma$ into $M$. Let $V : \operatorname{Emb}(\Sigma, M) \to \mathbb{R}$ be the volume functional, i.e., for $\varphi \in \operatorname{Emb}(\Sigma, M)$, let

$$V(\varphi) = \int_\Sigma 1 \, \mathrm{d vol}_{\varphi^\ast g}, $$

where $\mathrm{d vol}_{\varphi^\ast g}$ denotes the volume element of the metric $\varphi^\ast g$ on $\Sigma$.

Is this functional analytic? If not, under what conditions will it be? The topology in the space of embeddings is the one induced by the analytic manifold structure in the space of all smooth mappings from $\Sigma$ to $M$.

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See this great book.

Related local question (I am more interested in this scenario): Let $U \subset C^k(\Sigma)$ be a small neighbourhood of the origin, and for a fixed embedding $\varphi : \Sigma \to M$ with unit normal $N$, define

$$E(f)(x) = \exp_{\varphi(x)}(f(x)N(x)), \quad f \in U.$$

Now consider the volume functional $V : U \to \mathbb{R}$ defined by

$$V(f) = \int_\Sigma 1 \, \mathrm{dvol}_{E(f)^\ast g}.$$

Is $V$ analytic?

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    $\begingroup$ Usually to talk about analyticity, you need to have functions equal to their Taylor expansions in suitable neighborhoods. What topology on the space of immersions do you want to use? $\endgroup$
    – Ryan Budney
    Commented May 11, 2022 at 0:07
  • $\begingroup$ In fact, I am more interested in the local picture (second part of the question). In this case, I am thinking of the $C^k$ topology for some $k \geq 2$. I’m also unsure about what is the appropriate topology in the space of immersions… $\endgroup$
    – Eduardo Longa
    Commented May 11, 2022 at 14:14
  • $\begingroup$ Since the set of embeddings is open in the set of all maps of a certain regularity from $\Sigma$ to $M$, it has a natural Banach manifold structure. $\endgroup$
    – Eduardo Longa
    Commented May 12, 2022 at 12:50
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    $\begingroup$ The answer should be "yes" even if you just consider Ck immersions. Loosely speaking the point is that charts for embeddings/immersions can be formed from sections $s$ of the normal bundle of a fixed embedding. Now you need to ask if the volume function is analytic as a function of $s$. You should convince yourself this is true (as long as $g$ is analytic). I'm sure this is explained somewhere in the green book of Leon Simon on energy minimizing maps, but there are probably other sources. $\endgroup$
    – Otis Chodosh
    Commented Aug 24, 2022 at 4:14

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