$M$ and $S$ are two simply-connected surfaces with boundaries $\partial M =\partial S \neq \emptyset$, respectively. Both $M$ and $S$ are disk-like and smooth enough. Here, we assume $M$ is planar (flat surface). We know from the Riemann mapping theorem that conformal mappings from a disk-like surface to a fixed planar simply-connected region not only exist but are also almost unique. Are there any related PDEs whose solution is the conformal map between the two surfaces?
I find an elegant method in the doctoral thesis of Dr. Kennan Crane from Caltech. He have adopted quaternions $\mathbb{H}$ and the so-called spin transformations. For conveniece, I've attached the essential algorithm below (detailed information can be found in Ref. 1 and Ref. 2).
Consider two immersions $f$ , $\tilde{f}$ : $M\to \text{Im}\mathbb{H}$ of a given surface $M$. These surfaces are said to be spin equivalent if their differentials are related by a similarity transformation, i.e., if there exists some function $\lambda$: $M\to \mathbb{H}$ such that
$$\text{d}\tilde{f}(X)=\bar{\lambda}\text{d}f(X)\lambda \tag{1} $$
for all tangent vectors $X$. In this case the induced Riemannian metrics $|\text{d}f|^2$ and $|\text{d}\tilde{f}|^2 = |\lambda|^4 |\text{d}f|^2$ are related by a positive scaling $|\lambda|^4$, hence the two immersions are also conformally equivalent. In particular, we define the quaternionic Dirac operator
$$\text{D}\lambda=-\frac{\text{d}f \wedge \text{d}\lambda}{|\text{d}f|^2} \tag{2}$$
D is a generalization of the gradient operator which expresses first-order derivatives of both scalar- and vector-valued functions on M, and and in local coordinates is equivalent to the standard Dirac operator from physics. Using D, one can easily show that $\text{d}\tilde{f}$ is integrable if and only if $\lambda$ satisfies the integrability condition ($\lambda$ is in the kernel of $\text{D}-\rho$)
$$(\text{D}-\rho)\lambda=0 \tag{3}$$ where $\rho$: $M\to\mathbb{R}$ is a real-valued function, which is related to the curvature deformation.
We can solve the eigenvalue problem $$(\text{D}-\rho)\lambda=\gamma\lambda \tag{4}$$ for the pair ($\gamma$, $\lambda$) with the smallest eigenvalue $\gamma\in\mathbb{R}$, then it follows that $\text{D} \lambda = (\rho+\gamma)\lambda$, i.e., we get an exact solution for Eq. (3) by simply adding a constant shift $\gamma$ to $\rho$.
Overall, the procedure for conformally deforming a surface is:
- pick a scalar function $\rho$ on M;
- solve an eigenvalue problem (Eq. (4)) for the similarity transformation $\lambda$;
- solve a linear system (Eq. (1)) for the new surface $\tilde{f}$.
Unlike the above case, our aim is to find the conformal transformation $\lambda$ for given two surfaces $M$ and $S$, whose boundaries are the same, i.e., $\partial M=\partial S \neq \emptyset$. In my case, $M$ is a flat surface; and $S$ is a curved surface formed by $M$ with a norm displacement (like a bump). The boundaries are overlapped. Any hint on finding the conformal map? Thanks.