Research discussion summary (by Ronald)

David, Alvin and I have had several fruitful research discussions through msn last few days. Here is a brief summary.

We're looking for good practical applications for the tools we've developed (Most reviewers commented that our works should be supported by real applications). We've proposed a possible practical application of our "Beltrami Holomorphic flow" method and "Teichmuller extremal map" method.

Basically, the problems we want to solve are:
1. Given two simply-connected open surface S1 and S2. Suppose we know the boundary correspondence between the two surfaces (but we know nothing about the interior: i.e. no registration is done in the interior). How can we define a measure to describe their shape difference?

2. In real situation, we might not know the correspondence for every points on the boundaries. Suppose we only know the correspondence for 2n points on each boundaries of S1 and S2. Again, we do not know the registration in the interior. How can we define a measure to describe their shape difference?

3. Now, we consider a more general situation. Suppose we have two genus zero closed surfaces M1 and M2. Given no information about the correspondence between M1 and M2. How can we define the best measure to describe their shape difference?

We've proposed the following methods to solve each problem:

For 1:
We map S1 and S2 conformally to unit disks D, using Yamabe flow. The boundary correspondence between S1 and S2 gives us a map from circle to circle. Denote it by f: S^1 -> S^1. Extend f to a harmonic map F: D -> D. F can be regarded as the best optimized conformal map that agree with boundary correspondence. F is conformal if S1 and S2 are conformally equivalent. We define the shape index as:
Shape(S1, S2) = \int_D |\mu_F| + a\int_D |H1 - H2(F)| + b\int_D |K1 - K2(F)|
where: \mu_F is the Beltrami coefficient of F; Hi = mean curvature; Ki = Gaussian curvature; a and b are the rescaling parameters.
We can prove that (S1,S2) has the same geometry (same shape up to rigid motion) if and only if Shape(S1,S2) = 0.

For 2:
Again, we map S1 and S2 conformally to unit disks D, using Yamabe flow. The 2n points correspondence between S1 and S2 give us the 2n points correspondence between the circle. We can apply our Teichmuller extremal map method to find an extremal map F:D -> D that matches the 2n points. Define:
Shape(S1, S2) = k + a\int_D |H1 - H2(F)| + b\int_D |K1 - K2(F)|
where: k is the Beltrami coefficient of F; Hi = mean curvature; Ki = Gaussian curvature; a and b are the rescaling parameters.
We can prove that (S1,S2) has the same geometry (same shape up to rigid motion) if and only if Shape(S1,S2) = 0.

For 3:
We map M1 and M2 conformally to unit sphere. We consider the following minimization problem:
E(\mu, T) = \int_S^2 |\mu| + a\int_S^2 |H1(T) - H2(F_mu)| + b\int_S^2 |K1(T) - K2(F_mu)|
where F_mu = quasiconformal map associated with mu; T is a Mobius transformation
Define the shape index as: Shape(M1, M2) = Min E(\mu, T).
We will need to prove that:
(S1,S2) has the same geometry (same shape up to rigid motion) if and only if Shape(S1,S2) = 0.