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The harmonic Bergman kernel$Q$_{Ω}for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator$F$_{Ω}║$F$_{Ω}║<1 in operator norm. Suppose that Ω is the image of a univalent analytic function ø in the unit disk with ø' ($z$)=1+ψ ($z$) where ψ(0)=0. We show that if the function ψ belongs to a space$D$_{$s$}($D$),$s$>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for$Q$_{Ω}also converges pointwise in $$\bar \Omega \times \bar \Omega \backslash \Delta (\partial \Omega )$$ , and the rate of convergence can be estimated. The proof uses the eigenfunctions of the Friedrichs operator as well as a formula due to Lenard on projections in Hilbert spaces. As an application, we show that for every$s$>0 there exists a constant$C$_{$s$}>0 such that if ║ψ║_{$D$}_{s}($D$)≤$C$_{$s$}, then the biharmonic Green function for Ω=ø ($D$) is positive.

We calculate the local Fourier transforms for connections on the formal punctured disk, reproducing the results of J. Fang and C. Sabbah using a different method. Our method is similar to Fang’s, but more direct.

Shape registration has a wide range of applications in geometric modeling, medical imaging, and computer vision. This paper focuses on the registration of the genus-3 vestibular systems and studies the geometric differences between the normal and Adolescent Idiopathic Scoliosis (AIS) groups. The non-trivial topology of the VS poses great technical challenges to the geometric analysis. To tackle these challenges, we present an effective and practical solution to register the vestibular systems. We first extract six geodesic landmarks for the VS, which are stable, intrinsic, and insensitive to the VS’s resolution and tessellation. Moreover, they are highly consistent regardless of the AIS and normal groups. The detected geodesic landmarks partition the VS into three patches, a topological annulus and two topological disks. For each pair of patches of the AIS subject and the control, we compute a bijective map using the holomorphic 1-form and harmonic map techniques. With a carefully designed boundary condition, the three individual maps can be glued in a seamless manner so that the resulting registration is a homeomorphism with exact landmark matching. Our method is robust, automatic and efficient. It takes only a few seconds on a low-end PC, which significantly outperforms the non-rigid ICP algorithm. We conducted a student’s t-test on the test data. Computational results show that using the mean curvature measure E_H, our method can distinguish the AIS subjects and the normal subjects.

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