Nonlinear funtionals of smooth maps between Riemannian manifolds

Question

Let two smooth Riemannian manifolds $M$ and $N$, and let $C^{\infty}(M, N)$ be the family of smooth maps between them.

I would like to study functionals of the form $$E[\cdot]: C^{\infty}(M, N) \to \mathbb{R},$$ where for each $F \in C^{\infty}(M,N)$ i.e., a smooth map $$F:M\to N,$$ $E[F]$ represents strain or energy of the map $F$ (but the energy can be negative etc.). The following three properties define what I mean by strain:

1. Continuity, i.e., $E[F]$ is continuous in $F$ (in some norm on $C^{\infty}(M, N)$, see definition below for elaboration)
2. $E[F]$ is a valuation on $M$ (please see definition below in elaborations)
3. $E[F]$ is invariant to rigid deformation of $F$ (please see definition below in elaborations)

My goal is to better understand this family of functionals and how they can be represented. To get handle of the subject, I would like to understand if one can represent them using integrals on $M$.

So that given arbitrary $E[\cdot]$ we can always write

$$ E[F] = \int_M D_F(x) dV_n, \tag{$\ast$} $$

where $dV_n$ is volume form on $M$ and $D_F(x), x\in M$ is some kind of density on $M$ that depends on $F$ in a reasonable fashion.

To make progress with this question (and really understand if it makes sense to pose it) here are few things I struggle with:

1. Find a good reference for functional analysis in this general case, where the functional is not linear.

2. Is there some hope to prove my claim $(*)$, if yes what one can try to say about the density $D_F(x)$, how does it depend on $x$ or on $F$, e.g., is it for example necessarily continuous in $F$?

3. Is there any principal way to generate such functionals, if not by using integral, perhaps one can approximate any such functional by some simpler nice functionals (just like continuous functions can be approximated by smooth bump functions)?

Below are elaborations and definitions of the terms used above with an example of such a functional: Dirichlet energy.

Any comments, answers or thoughts on the subject would be greatly appreciated.

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Elaborations:

I have a smooth map $F$ from a smooth Riemannian n-manifold $M$ to $N$

$$ F: M \to N. $$

Suppose I want to compute Dirichlet energy of $F$ in this very general setting and better understand what kind of functional this energy is.

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To this end I define a functional, $E_D[F]:C^{(\infty)}(M,N) \to \mathbb{R}$, given by:

$$ E_D[F] := \int_{M} \mathrm{Tr}\left\{\mathrm{det}(DF)^\top\mathrm{det}(DF)\right\} dV. \tag{$\ast\ast$} $$

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The above can be interpreted using the following inner product on $C^{\infty}(M, N)$ (smooth maps between smooth manifolds $M$ and $N$):

let $F,G \in C^{\infty}(M, N)$ define: $$ \langle F, G \rangle_D := \int_{M} \mathrm{Tr}\left\{\mathrm{det}(DF)^\top\mathrm{det}(DG)\right\} dV, $$ $$ \|F\|_D^2 := \langle F, F \rangle_D. $$ So that $E[F] = \|F\|_D^2.$

What properties does $E_D[F]$ have as a functional on $C^{\infty}(M, N)$?:

(Edited: As pointed out in the comments, this definition of inner product is a sloppy one and one needs to introduce connection on manifold $N$ to make the product of Jacobians meaningful)

• Continuity. The functional $E_D[F]$ is continuous (in the above defined norm)

i.e., for small enough $$ \|F- G\|_D <\delta, $$ we have $$ \left|E_D[F] - E_D[G]\right| < \varepsilon. $$

• Valuation. In addition, the functional $E_D[F]$ is a valuation on $M$.

By this I mean that if we cover $M$ by submanifolds $U, V \subset M$, i.e., $M = U \cup V$, such that $U \cap V$ is also submanifold of $M$. Then we can compute $E_D[F]$ by considering the following maps: $$ F|_U:U \to N, $$ $$ F|_V:V \to N, $$ $$ F|_{V\cap U}:V\cap U \to N. $$ Then $$ E_D[F] = E_D[F|_U] + E_D[F|_V] - E_D[F|_{V\cap U}]. $$

• Rigid deformation in-variance Let $V$ be a smooth vector field on $N$, we can use $V$ to define, for each point $p \in N$, an integral curve $\gamma:[0, b] \to N$, which is the curve on $N$ characterized by the property $$ \gamma(0) = p, $$ $$ \gamma'(t) = V_{\gamma(t)},~~t \in [0, a]. $$ Roughly, all of the integral curves of $V$ define a flow $\Theta_V: N \times [0, a] \to N$ of $V$ on $N$. With the help of integral curves, we can now make sense of what we mean by applying $V$ to $N$ ($V$ acts on $N$). This can be interpreted as mapping each point $p\in N$ by the flow of $V$ . Which in turn means that we are moving each point along the integral curve of $V$ for some time $t \in [0,a]$. So that for a fixed $t \in [0,a]$ we have

$$ \Theta_V^{(t)}: N \to N, $$

we can now make sense of composition $\Theta_V^{(t)} \circ F: M \to N$, and compute Dirichlet energy of this composition:

$$ E_D[\Theta_V^{(t)} \circ F]. $$

Dirichlet energy has this property, that if the Vector field $V_K$ is Killing vector field (this is specific field that preserves the Riemannian metric $g$, i.e., for Lie derivative we have $\mathcal{L}_V g = 0$ and therefore Killing vector field defines isometric transforms of $N$) we have:

$$ E_D[F] = E_D[\Theta_{V_K}^{(t)} \circ F]. $$

Reiterating my question, I want now to consider the family of all functionals that have the listed property and understand if they all have the form $(**)$, more generally if one can write them as an integral with those other density.

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References - I found this question on mathoverflow that seems to be somewhat
related. - Another paper that looks at a similar setting

Answer 1

The Dirichlet energy example is well understood, and you can find a wealth of literature under the name minimizing harmonic maps (this is what its minimizers are called). Partially answering your question no. 1, you can expect some unexpected features in the analysis of similar functionals:

• The function space is not a linear space, let alone an inner product space (as pointed out in a comment, there's a flaw in your description of the inner product).
• It makes a difference whether your function space consists of smooth functions or Sobolev functions. That is, it's possible that smooth functions are not dense in the corresponding Sobolev space $W^{1,2}$.
• Related to the previous point, there's something called the Lavrentiev phenomenon: the infimum of the Dirichlet energy can be strictly smaller on $W^{1,2}$ than on $C^\infty$ (say, if you prescribe some boundary data).
• The Euler-Lagrange system for this functional is nonlinear (if you write it in coordinates or via an embedding) and smoothness doesn't follow from it automatically (like it does for similar functionals in the flat setting). And indeed, Sobolev minimizers can have singularities.

Ad 2, I cannot say anything for certain, but it does look like your "valuation" axiom forces the functional to look like a measure. If it was replaced it with a stronger axiom $$ E[F|_M] = \sum_i E[F|_{U_i}] \quad \text{whenever $U_i$ are disjoint and sum up to $M$,} $$ then we would say that $\mu_F(U) := E(F|_U)$ is a measure and thus $E(F|_M) = \int_M d \mu_F(x)$ - which means that $E$ is an integral of a measure, although a priori this measure doesn't have to be absolutely continuous. In larger generality, there might be some pathological examples (similarly to how not every finitely additive measure is also countably additive etc.), but I'm not sure if this is what you're looking for.