# Do there exist energy-minimizing immersions?

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).

• 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. – Dap Mar 20 '18 at 12:40
• Yes it is. Maybe I should add this comment. (It is rather common in the literature on harmonic maps, I think). – Asaf Shachar Mar 20 '18 at 13:04
• 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. – Litho May 10 at 15:13
• 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. – Litho May 10 at 16:18
• 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 $1+1/n$. – 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.