We develop a new algorithm to automatically register hippocampal (HP) surfaces with complete geometric matching, avoiding the need to manually label landmark features. A good registration depends on a reasonable choice of shape energy that measures the dissimilarity between surfaces. In our work, we first propose a complete shape index using the Beltrami coefficient and curvatures, which measures subtle local differences. The proposed shape energy is zero if and only if two shapes are identical up to a rigid motion. We then seek the best surface registration by minimizing the shape energy. We propose a simple representation of surface diffeomorphisms using Beltrami coefficients, which simplifies the optimization process. We then iteratively minimize the shape energy using the proposed Beltrami Holomorphic flow (BHF) method. Experimental results on 212 HP of normal and diseased (Alzheimer's disease) subjects show our proposed algorithm is effective in registering HP surfaces with complete geometric matching. The proposed shape energy can also capture local shape differences between HP for disease analysis.

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This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowski's theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions.

Let g(t) be a smooth complete solution to the Ricci flow on a noncompact manifold such that g(0) is Kahler. We prove that if |Rm(g(t))| is bounded by a/t for some a > 0, then g(t) is Kahler for t > 0. We prove that there is a constant a(n) > 0 depending only on n such that the following is true: Suppose g(t) is a smooth complete solution to the Kahler-Ricci flow on a non-compact n-dimensional complex manifold such that g(0) has nonnegative holomorphic bisectional curvature and |Rm(g(t))| ≤ a(n)/t, then g(t) has nonnegative holomorphic bisectional curvature for t > 0. These generalize the results by Wan-Xiong Shi. As applications, we prove that (i) any complete noncompact Kahler manifold with nonnegative complex sectional curvature and maximum volume growth is biholomorphic to C^n; and (ii) there is ε(n) > 0 depending only on n such that if (M^n,g_0) is a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature and maximum volume growth and if (1+ε(n))^{−1}h ≤ g_0 ≤ (1+ε(n))h for some Riemannian metric h with bounded curvature, then M is biholomorphic to C^n.

Let X be a compact toric Kaehler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on X. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of X as virtual counts of holomorphic discs with Lagrangian boundary condition L. We prove a formula which equates such open GW invariants with closed GW invariants of certain X-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of X, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono.