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In this paper, we propose a novel low dimensional manifold model (LDMM) and apply it to some image processing problems. LDMM is based on the fact that the patch manifolds of many natural images have low dimensional structure. Based on this fact, the dimension of the patch manifold is used as a regularization to recover the image. The key step in LDMM is to solve a Laplace-Beltrami equation over a point cloud which is solved by the point integral method. The point integral method enforces the sample point constraints correctly and gives better results than the standard graph Laplacian. Numerical simulations in image denoising, inpainting and super-resolution problems show that LDMM is a powerful method in image processing.

The present article surmmarizes our recent results about Calabi–Yau threefolds with infinite fundamental group. This class of Calabi–Yau manifolds is relatively simple yet rich enough to display the essential complexities of Calabi–Yau geometries, and thus it provides good testing-grounds for general theories and conjectures.

Let $M^n$ be a complete noncompact K\"ahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $\mathbb{C}^n$. This confirms the uniformization conjecture of Yau under the assumption $M$ has maximal volume growth.