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Let $M,N$ be $d$-dimensional connected oriented Riemannian manifolds, possibly with boundary, $M$ compact. Let $E_d:C^{\infty}(M,N) \to \mathbb{R}$ be the $d$-energy, i.e.

$$ E_d(f)=\int_M |df|^d \text{Vol}_M.$$

Set $E_{M,N}=\inf \{ E_d(f) \, | \,\, f \in C^{\infty}(M,N) \text{ is an immersion} \}$, and suppose that $E_{M,N} >0$.

Does $E_{M,N}$ always obtained? i.e. does there exist an immersion with minimal energy? (I am assuming there exist at least one immersion from $M$ to $N$. )

I am specifically considering the $d$-energy between $d$-manifolds, and not the $2$-energy; for the $2$-energy the answer can be negative; it is known that

$$\inf_{f \in \text{Diff}(\mathbb{S}^n) } E_2(f) =0$$ when $n >2$, but there is no immersion with zero $2$-energy.

However, the identity map $\text{Id}_{M^d}$ has minimal $d$-energy among all diffeomorphisms. (So, in particular, for any simply-connected and closed $M$, we have $E_{M,M}=E_d(\text{Id}_{M})$ as any immersion is a diffeomorphism).

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  • $\begingroup$ So $|df|$ is the Hilbert-Schmidt norm $\sqrt{\operatorname{tr}(df^*df)}$? I don't think that definition is so common as to go unsaid. $\endgroup$ – Dap Mar 20 '18 at 12:40
  • $\begingroup$ Yes it is. Maybe I should add this comment. (It is rather common in the literature on harmonic maps, I think). $\endgroup$ – Asaf Shachar Mar 20 '18 at 13:04
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    $\begingroup$ I'm not especially familiar with differential geometry, so I may be misunderstanding something here, but if $M=N=[0,1]$, cannot you get arbitrarilily small positive energy by choosing immersion $x\mapsto cx$ for $c\searrow 0$? But no immersion can have zero energy. $\endgroup$ – Litho May 10 at 15:13
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    $\begingroup$ Then you would probably also want to add that $M$ is connected. Otherwise, choosing $M=N$ to be the union of $S^1$ and a closed interval gives a simple example where the minimal energy is non-zero, but still not achievable. $\endgroup$ – Litho May 10 at 16:18
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    $\begingroup$ Actually, it's better to add that $N$ is connected as well, or you could take $M=S^1$ and $N$ as the union of countably many circles with radii $1+1/n$. $\endgroup$ – Litho May 10 at 16:30
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Here is a sketch of a 4-dimensional counter-example where you allow $M$ to have boundary. The motivating 2-dimensional version of this construction is as follows:

Take $N$ which is the surface of revolution obtained by rotating the graph of $y=e^x + 0.5$ around the $x$-axis in $R^3$ (with the induced Riemannian metric). Notice that the infimum of lengths of embedded homotopically nontrivial loops on $N$ is $2\pi$ and it is not realized by any loop. Take $M$ which is the product annulus $S^1\times [0,1]$, where $S^1$ is the unit circle. Then the infimum of 2-energies of homotopically nontrivial maps $M\to N$ equals the area of $M$ but it is not achieved. As @Litho correctly noted, nevertheless $E(M,N)=0$ since you can take homotopically trivial embeddings. (Nevertheless, this is an example if you require immersions to be homotopically nontrivial. )

Similarly, if you consider connected oriented 3-manifolds and $M$ has nonempty boundary then $E(M,N)=0$ (since one can immerse every such $M$ in $R^3$).

To get the actual example, we go one dimension up. Let $M$ be a simply-connected smooth compact non-spin oriented 4-manifold with boundary. Then $M$ does not admit immersions in $R^4$.

For concreteness I will take $M$ to the complement to a 4-ball in $CP^2$. Then $M$ has structure of a disk bundle over $S^2$ (with the Euler number 1). I will equip $M$ with a Riemannian metric such that there is a fibration $M\to S^2$ over the round 2-sphere (of the unit radius) which is a submetry. I think (but I did not check this), the least 4-energy map from $M$ to such $S^2$ is given by the projection $p$, in which case, $E(M, S^2)= Vol(M)$.

Now, take a 4-manifold $N$ which is diffeomorphic to an infinite connected sum of $CP^2$'s. One needs to choose a suitable Riemannian metric on $N$ such that the $k$th $CP^1$ admits a neighborhood isometric to a rescaled (by $\lambda_k>1$) version of the above example with $\lambda_k$ converging to $1$ as $k\to \infty$.

We then have a sequence of immersions $f_k: M\to N$ whose 4-energies converge to $Vol(M)$ from above (without ever reaching it). One still needs to check that there are no immersions of smaller 4-energy. I do not see any possible candidates but proving this would require much more work that I can afford.

The real question, I think, is about maps between closed manifolds which then are necessarily finite covering maps. A paper to read is

B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), 1-17.

Or you can simply email Brian (he is at Stanford): He is a very nice guy and will help if he can.

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