The goal of urban building mesh simplification is to generate a compact representation of a building from a given mesh. Local smoothness and sharp contours of urban buildings are important features for converting unstructured data into solid models, which should be preserved during the simplification. In this paper, we present a general method to filter and simplify 3D building mesh models, capable of preserving piecewise planar structures and sharp features. Given a building mesh model, a mesh filtering technique is firstly designed to yield piecewise planar regions and extract crease contours. The planar regions are used to constrain the simplification of the mesh. Mesh decimation is achieved through a series of edge collapse operations, which uses regional structural constraints and local geometric error metrics to handle planar and non-planar areas respectively. The proposed method preserves the mesh structure with meaningful levels of detail while reducing the number of faces. The effectiveness of this method is evaluated on various building models generated from different observation scales, and the performance is validated by extensive comparisons to state-of-the-art techniques.

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We give a detailed proof of Siu’s theorem on extendibility of holomorphic vector bundles of rank larger than one, and prove a corresponding extension theorem for holomorphic sprays. We apply this result to study ellipticity properties of complements of compact subsets in Stein manifolds. In particular we show that the complement of a closed ball in $\mathbb{C}^{n}, n \geq3$ , is not subelliptic.

This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary minimal surfaces. Moreover, we describe estimates for the first Steklov eigenvalue on such free boundary minimal surfaces and various smooth compactness results. Finally, we mention some sharp area bounds for free boundary minimal submanifolds and related questions.

We prove a sharp inequality for hypersurfaces in the ndimensional Anti-deSitter-Schwarzschild manifold for general $n \ge  3$. This inequality generalizes the classical Minkowski inequality [19] for surfaces in the three dimensional Euclidean space. The proof relies on a new monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established in [4].

We construct SYZ mirrors of the local Calabi--Yau manifolds of type $\widetilde{A}$ by developing an equivariant SYZ theory for the toric Calabi--Yau manifolds of infinite-type. The equations for the SYZ mirrors involve the Riemann theta functions and generating functions of the open Gromov--Witten invariants. We obtain explicit formulae for the generating functions which are open analogs of the Yau--Zaslow formula in dimension $2$, and show that they have nice modular properties. We also relate the SYZ mirror pairs with mirror symmetry for the abelian varieties and hypersurfaces therein.

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.

Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a selfsimilar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009) [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.

We study the space of Killing fields on the four dimensional AdS spacetime AdS 3,1 . Two subsets S and O are identified: S (the spinor Killing fields) is constructed from imaginary Killing spinors, and O (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of O is properly contained in that of S . This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. In [10], Chru\'sciel, Maerten and Tod proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the "rest mass" of an asymptotically AdS spacetime is then defined by minimizing the observer energy, and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime.