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This is a cross-post.

Let $M,N$ be $d$-dimensional 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 $$E_{M,N}=\inf \{ E_d(f) \, | \,\, f \in C^{\infty}(M,N) \text{ is an immersion} \}.$$

Is $E_{M,N}$ always a minimium? 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 more classical $2$-energy. This is because in the case of the $2$-energy the answer can be negative in general; it is well-known that

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

However, the identity map $\text{Id}_{M^d}$ has minimal $d$-energy among all diffeomorphisms.

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    $\begingroup$ This is the scale invariant energy, so you would not expect a minimum to always exist. A bubble can form, where a fixed part (the bubble), often with nontrivial topology, of the domain maps into a point. In particular, this was studied in the d=2 case by Sacks and Uhlenbeck, which opened the door to connecting nonlinear geometric analysis on a manifold to its topology. See math.jhu.edu/~js/Math748/sacks-uhlenbeck.pdf $\endgroup$
    – Deane Yang
    Commented Mar 26, 2018 at 14:22
  • $\begingroup$ @DeaneYang There is an explicit example due to Iwaniec and Onninen of not attaining minimum in my answer. $\endgroup$
    – Piotr Hajlasz
    Commented Apr 12, 2018 at 22:26

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I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A_*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If $$ \frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right) $$ then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is $$ h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\ \frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R, \end{cases} $$ where $\sigma$ is defined by $$ \frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right). $$ See Theorem 1.8 in T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).

It seems you get the same limit if you restrict the minimization problem to diffeomorphisms with given boundary data.

Tadeusz Iwaniec (not Henryk Iwaniec) wrote many papers regarding minimization of the n-energy so you should check his recent publications to see if you find there relevant results.

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  • $\begingroup$ Thanks, this is very interesting. This shows that the infimum of the energy over all homeomorphisms is not attained. I wonder whether this implies the infimum over immersions is also not attained. $\endgroup$
    – Asaf Shachar
    Commented Mar 29, 2018 at 9:49
  • $\begingroup$ @AsafShachar If you fix boundary conditions for a mapping between annuli (as in the result above), then an immersion has to be a diffeomorphism. Am I right? $\endgroup$
    – Piotr Hajlasz
    Commented Mar 29, 2018 at 14:03
  • $\begingroup$ A nice discussion on energy minimising maps between annuli is also in Struwe's book Variational Methods (especially about the Plateau boundary conditions) $\endgroup$
    – Pietro Majer
    Commented May 23, 2018 at 14:39

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