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).

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.