Optimal, conformal diffeomorphisms between two surfaces in 3D

Question

Let $S_1$ and $S_2$ be two smooth, closed surfaces embedded in $\mathbb{R}^3$.

Q. Is there a natural definition of the optimal, conformal diffeomorphism between $S_1$ and $S_2$?

I am imagining $S_1$ and $S_2$ as points in a shape space, and I want to consider all diffeomorphisms between them that preserve angles throughout. I am wondering if there is a natural metric on this space that can select out the "optimal/shortest" path(s) between the two shapes.

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Suggestive image:
   
   _{(Image from "Shape Analysis of 3d Objects".)}

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This has surely been studied (in the cited paper above in one form). One can imagine using Ricci flow to map both $S_1$ and $S_2$ to canonical surfaces, and so connect them. But I am rather interested in a "shortest path" between $S_1$ and $S_2$, a sort of optimal Ricci-flow-like transformation of one surface to the other, without connecting via intermediate canonical forms.

(Added.) My apologies for the vagueness of this question. Just one more remark. My understanding is that Ricci flow for a surface in $\mathbb{R}^3$ is conformal, which was my motivation for concentrating on conformal diffeomorphisms. Essentially I am seeking a Ricci-like flow that morphs $S_1$ to $S_2$. But the comments have shown that my question is almost too naive in its current form to admit an answer.

Answer 1

Here is how you can get started: Consider a conformal map $f: S_1\to S_2$ between two Riemannian surfaces of genus $0$. Let $0<\lambda(x), x\in S_1$, be the conformal factor of $f$. If $\lambda(x)=1$ everywhere, you should be happy. Thus, you may want to infimize (over all conformal maps $f: S_1\to S_2$), say, the function $L_f(x)=|\log \lambda (x)|$. You have to decide on which norm of this function to use. If you use the sup-norm, which I will denote $C(f)$, then $C(f)\le L$ will mean that $f$ is $e^L$-Lipschitz and, by Arzela-Ascoli, the distance $$ d_\infty(S_1, S_2)=\inf_{f: S_1\to S_2} C(f) $$ is realized by some conformal map $f_\infty: S_1\to S_2$. (Here and below, I am using only conformal maps $f: S_1\to S_2$.) Alternatively, you can use any $L_p$-norm, which I will denote $$ C_p(f)= \|L_f(x)\|_{L_p} $$ ($1\le p\le \infty$), for functions on $S_1$ (using measure coming from the Riemannian area form on $S_1$). A bit more thought shos that the infimum in the $p$-distance $$ d_p(S_1, S_2)=\inf_{f: S_1\to S_2} C_p(f) $$ is again realized by some conformal map.

Now what? The problem is that, as Alexandre said, the space of conformal maps between the two genus 0 surfaces is $PSL(2, {\mathbb C})$, which is not contractible, so there is no way to invoke some convexity arguments to show that infimum is unique, is stable, etc. For instance, if $S_1, S_2$ are round spheres, then infima (for all $p$) are realized by subsets diffeomorphic to $SO(3)$. I think, however (but I do not have a proof of this), that for generic surfaces $S_i$, the infimum is realized by a unique map and, moreover, (setting $p=2$) the map $$ C_2: PSL(2, {\mathbb C})\to {\mathbb R} $$
is a Morse function (its properness is easy to see as I noted above). I do not even have a guess if, generically, there is only one local minimum. Thus, there might be problems with computing $d_2(S_1,S_2)$ numerically.