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

$$ E(f)=\int_M |df|^2 \text{Vol}_M.$$

For any given homotopy class $\alpha$ of smooth maps $M \to N$, define $$E_{\alpha}=\inf \{ E(f) \, | \,\, f \in \alpha \}.$$

(In some cases, we know $E_{\alpha}$ is realized, e.g. when $N$ is closed and of negative curvature.)

Now define $$E:=\inf_{\alpha \text{ is not trivial }} E_{\alpha}$$

Is $E$ always a minimium? i.e. does there exist a homotopy class $\alpha$ such that $E=E_{\alpha}$?

Does the answer change if we replace the Dirichlet energy with the $d$-energy? (i.e. integrate $|df|^d$ instead of $|df|^2$?)

In the case $M=N=\mathbb{S}^1$, one can prove directly, via Holder's inequality, that the answer is positive:

$$ \big(\deg f \cdot \text{Vol}(\mathbb{S}^1)\big)^2=\big(\int_{\mathbb{S}^1} |f'|\big)^2 \le \int_{\mathbb{S}^1} |f'|^2 \int_{\mathbb{S}^1} 1^2=\text{Vol}(\mathbb{S}^1)E(f),$$ so we get

$$ E(f) \ge (\deg f)^2 \cdot \text{Vol}(\mathbb{S}^1),$$

and for a map of degree $n$, the minimal energy is obtained exactly for $f(e^{i\theta})=e^{in\theta}$.

So, the non-trivial homotopy class with minimal infimal energy is the class of degree-one maps.


More generally, if we consider the $d$-energy, and assume $M,N$ are closed, then

$$ E_d(f)=\int_M |df|^d \text{Vol}_M \ge d^{\frac{d}{2}} \int_M \det df \text{Vol}_M=d^{\frac{d}{2}} \int_{M} f^*\text{Vol}_N=d^{\frac{d}{2}} \deg f \int_{N} \text{Vol}_N,$$ so $$ E_d(f) \ge d^{\frac{d}{2}} \deg f \cdot \text{Vol}(N),$$ with equality if and only if $f$ is conformal. I am not sure this helps to settle the question though.

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  • $\begingroup$ I think the case $M=S^1$ gives $E=\ell^2/2\pi$ where $\ell$ is the length of the shortest noncontractible loop i.e. homotopy $1$-systole. In this case the minimizer won't exist if $N$ is for example $\mathbb R^2$ with a disc of radius $\tfrac 1n$ removed from around $(n,0)$ for each $n.$ Even the case $M=S^1$ and $\pi_1(N)$ finitely generated could be interesting perhaps $\endgroup$ – Dap Mar 20 '18 at 13:58
  • $\begingroup$ Oh, to get it $d$-dimensional: $M=S^1\times [0,1]^{d-1}$ should work, I think. $\endgroup$ – Dap Mar 20 '18 at 14:26
  • $\begingroup$ Thanks, this is interesting. Do you mean removing open disks from $\mathbb{R}^2$, so the closed non-contractible geodesics you are referring to, are "boundary geodesics" (just the circles, i.e. the boundaries of the removed sets)?. $\endgroup$ – Asaf Shachar Mar 21 '18 at 6:26
  • $\begingroup$ Since there are no restrictions on $N$ and since there doesn't need to be a minimizing $f,$ you could remove either closed or open discs. You could even get a closed $N$ by gluing something into the holes, like a gluing of the manifold to a copy of itself (using collars). $\endgroup$ – Dap Mar 21 '18 at 7:10
  • $\begingroup$ Ah, OK you are right. So your example shows the minimizer won't necessarily exist if $N$ is non-compact. Interesting. I wonder what can be said when $N$ is compact. I will think about it... $\endgroup$ – Asaf Shachar Mar 21 '18 at 12:03

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