We establish an interior C 2 estimate for k+ 1 convex solutions to Dirichlet problems of k-Hessian equations. We also use such estimate to obtain a rigidity theorem for k+ 1 convex entire solutions of k-Hessian equations in Euclidean space.

We give a proof of the Hard Lefschetz Theorem for orbifolds that does not involve intersection homology. We use a foliated version of the Hard Lefschetz Theorem due to El Kacimi.

In this paper we prove the existence and uniqueness of the form-type equation on Khler manifolds of nonnegative orthogonal bisectional curvature.

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the _k functions on the space of its hermitian metrics.

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

Suppose M is a m-dimensional submanifold without umbilic points in the (m+ p)-dimensional unit sphere S m+ p. Four basic invariants of M m under the Mbius transformation group of S m+ p are a symmetric positive definite 2-form g called the Mbius metric, a section B of the normal bundle called the Mbius second fundamental form, a 1-form called the Mbius form, and a symmetric (0, 2) tensor A called the Blaschke tensor. In the Mbius geometry of submanifolds, the most important examples of Mbius minimal submanifolds (also called Willmore submanifolds) are Willmore tori and Veronese submanifolds. In this paper, several fundamental inequalities of the Mbius geometry of submanifolds are established and the Mbius characterizations of Willmore tori and Veronese submanifolds are presented by using Mbius invariants.

We study the rigidity of polyhedral surfaces using variational principles. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.

The LaplaceBeltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the LaplaceBeltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The LaplaceBeltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach.

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P Bowers and K Stephenson in [Mem. Amer. Math. Soc. 170, no. 805, Amer. Math. Soc.(2004)] as a generalization of Andreev and Thurstons circle packing. They conjectured that inversive distance circle packings are rigid. We prove this conjecture using recent work of R Guo [Trans. Amer. Math. Soc. 363 (2011) 47574776] on the variational principle associated to the inversive distance circle packing. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures that we introduced in [arXiv 0612.5714], verifying a conjecture in [arXiv 0612.5714]. As a consequence, we show that the discrete Laplacian operator determines a spherical polyhedral metric.

We prove theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature. In particular, a conjecture of J. Wolf [<i>J. Differential Geometry</i>, 2, 421-446 (1968)] is proved.

The chapter discusses the group actions on R³. The Smith conjecture has many equivalent forms, and each of these forms has various consequences and generalizations. The Smith conjecture is a structure theorem about symmetries of the product of a compact surface with an interval. Here the interval may be closed or open. The usual Smith conjecture is equivalent to proving the smooth <i>Z_n</i> actions on <i>S</i>² [0, 1] are conjugate to actions that preserve the product structure. This generalized Smith conjecture represents the belief that all the symmetries of the product of a compact surface with an interval actually arise from the symmetries of the surface extended trivially to the product structure. The chapter presents how to apply minimal surfaces to study the generalized Smith conjecture on <i>S</i>² <i>I</i>, where <i>I</i> is an open or closed interval. A geodesic version of the loop theorem and a new equivariant

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far.

In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime (Wang and Yau in Phys Rev Lett 102 (2): 021101, 2009; Commun Math Phys 288 (3): 919942, 2009), the anti-de Sitter spacetime (Chen et al. in Commun Anal Geom, 2016. arXiv: 1603.02975), or the Schwarzschild spacetime (Chen et al. in Adv Theor Math Phys 22 (1): 123, 2018). In each case, the reference spacetime admits a conformal KillingYano 2-form which facilitates the application of the Minkowski formula in Wang et al.(J Differ Geom 105 (2): 249290, 2017) to estimate the quasi-local energy. As a consequence of the positive mass theorems in Liu and Yau (J Am Math Soc 19 (1): 181204, 2006) and Shi and Tam (Class Quantum Gravity 24 (9): 23572366, 2007) and the above estimate, we obtain rigidity theorems which characterize the Minkowski spacetime and the hyperbolic space.

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Unit tangent bundle of a surface carries various information of tangent vector fields on that surface. For 2-spheres (ie genus-zero closed surfaces), the unit tangent bundle is a closed 3-manifold that has non-trivial topology and cannot be embedded in R3. Therefore it cannot be constructed by existing mesh generation algorithms directly. This work aims at the first discrete construction of unit tangent bundles over 2-spheres using tetrahedral meshes. We propose a two-stage algorithm for the construction, which starts from constructing two local bundles and then combines them into a global bundle.

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure--a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.

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In this follow-up of our earlier two works D (11.1)(arXiv: 1406.0929 [math. DG]) and D (11.2)(arXiv: 1412.0771 [hep-th]) in the D-project, we study further the notion of adifferentiable map from an Azumaya/matrix manifold to a real manifold'. A conjecture is made that the notion of differentiable maps from Azumaya/matrix manifolds as defined in D (11.1) is equivalent to one defined through the contravariant ring-homomorphisms alone. A proof of this conjecture for the smooth (ie C^{\infty} ) case is given in this note. Thus, at least in the smooth case, our setting for D-branes in the realm of differential geometry is completely parallel to that in the realm of algebraic geometry, cf.\arXiv: 0709.1515 [math. AG] and arXiv: 0809.2121 [math. AG]. A related conjecture on such maps to C^{\infty} , as a C^{\infty} -manifold, and its proof in the C^{\infty} case is also given. As a by-product, a conjecture on a division lemma in the finitely differentiable case that generalizes the division lemma in the smooth case from Malgrange is given in the end, as well as other comments on the conjectures in the general C^{\infty} case. We remark that there are similar conjectures in general and theorems in the smooth case for the fermionic/super generalization of the notion.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature dimension inequality C D E(n, K), which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are the first kind of results in this direction for unbounded Laplacian on graphs.

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In 1976, the author (see the open problem section in [3]) asked the following question: if p is a function defined on i? 3, what is the condition on p so that there is a sphere in R3 whose mean curvature is given by p. There were important works by Treibergs-Wei [2] and others on this question. Recently in a lecture that I gave in the Chinese University of Hong Kong, I realized that my work with R. Schoen on the existence of black hole can be used to settle the prescribed mean curvature question for a large class of p.

We discuss a method for deriving differential equations for the prepotential and the mirror map arising from a pair of Calabi-Yau manifold. Examples with one Kahler modulus are given. Here we nd that the differential equations we derive almost entirely characterize the prepotential and the mirror map in question. Some of the results here have been announced previously in

Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on discrete exterior calculus methods using hybrid domains. We then analyze the geometric structure of curvature operators in a piecewise-flat lattice.

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