In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K\"ahler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds $\S^{2n-1}\times \S^1$. We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.

We show that a compact Kaehler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu-Yau.

This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user defined field topologies. The user has full control of the number, positions and indices of the singularities, the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric, such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility, such that, it can design N-RoSy fields with single singularity, and mixed-RoSy fields. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.

Automatically discovering high-level facade structures in unorganized 3D point clouds of urban scenes is crucial for applications like digitalization of real cities. However, this problem is challenging due to poor-quality input data, contaminated with severe missing areas, noise and outliers. This work introduces the concept of adaptive partitioning to automatically derive a flexible and hierarchical representation of 3D urban facades. Our key observation is that urban facades are largely governed by concatenated and/or interlaced grids. Hence, unlike previous automatic facade analysis works which are typically restricted to globally rectilinear grids, we propose to automatically partition the facade in an adaptive manner, in which the splitting direction, the number and location of splitting planes are all adaptively determined. Such an adaptive partition operation is performed recursively to generate a hierarchical representation of the facade. We show that the concept of adaptive partitioning is also applicable to flexible and robust analysis of image facades. We evaluate our method on a dozen of LiDAR scans of various complexity and styles, and the image facades from the eTRIMS database and the Ecole Centrale Paris database. A series of applications that benefit from our approach are also demonstrated.

This work presents a technique that allows quick conversion of acquired low-quality data from consumer-level scanning devices to high-quality 3D models with labeled semantic parts and meanwhile their assembly reasonably close to the underlying geometry. This is achieved by a novel structure recovery approach that is essentially local to global and bottom up, enabling the creation of new structures by assembling existing labeled parts with respect to the acquired data. We demonstrate that using only a small-scale shape repository, our part assembly approach is able to faithfully recover a variety of high-level structures from only a single-view scan of man-made objects acquired by the Kinect system, containing a highly noisy, incomplete 3D point cloud and a corresponding RGB image.