Spectral and geometric bounds on 2-orbifold diffeomorphism type

On the differential form spectrum of geometrically finite orbifolds

We derive lower bounds for the essential spectrum of the Hodge-Laplacian on geometrically finite orbifolds and their suborbifolds.

Orbifold finiteness under geometric and spectral constraints

The class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.

Orbifold Biquotients of SU(3)

One of the main methods of constructing new spaces with positive or almost positive curvature is the study of biquotients first studied in detail by Eschenburg. We classify orbifold biquotients of the Lie Group $SU(3)$, and construct a new example of a 5-dimensional orbifold with almost positive curvature. Furthermore, we extend the work of Florit and Ziller on the geometric properties of the orbifolds $SU(3)//T^2$.

On the automorphism group of foliations with geometric transverse structure

Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphism group of a transversely holomorphic foliation or a riemannian foliation is a strong ILH Lie goup in the sense of Omori.

A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows

It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, and solutions to the Ricci flow admit temporal analyticity.

$C^0$-微分同胚极限下可整流流的推前的平坦收敛

Flat Convergence of Pushforwards of Rectifiable Currents Under $C^0-$Diffeomorphism Limits

本文研究紧黎曼流形上微分同胚的 $C^0$-极限对微分形式与电流的作用，证明拉回的一致收敛、推前的弱-* 收敛以及可整流流推前的平坦范数收敛，并应用于辛同胚群等几何结构在 $C^0$ 扰动下的稳定性分析。

Geometric convergence of the Kähler-Ricci flow on complex surfaces of general type

We show that on smooth minimal surfaces of general type, the Kähler-Ricci flow starting at any initial Kähler metric converges in the Gromov-Hausdorff sense to a Kähler-Einstein orbifold surface. In particular, the diameter of the evolving metrics is uniformly bounded for all time and the Kähler-Ricci flow contracts all the holomorphic spheres with $(-2)$ self-intersection number to isolated orbifold points. Our estimates do not require a priori the existence of an orbifold Kähler-Einstein metric on the canonical model.

关于正曲率Eschenburg轨道形的几何与拓扑

On the Geometry and Topology of Positively Curved Eschenburg Orbifolds

本文研究正截面曲率对Eschenburg 6-轨道形奇异集和上同调环的限制，证实了Florit-Ziller猜想，并计算了所有Eschenburg轨道形的上同调环，揭示了正曲率情形下的独特上同调行为。

Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness

Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m<7 which admit a Riemannian metric with sectional curvature bounded in absolute value by C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. Thus in these dimensions the lower positive bound on volume in Cheeger's Finiteness Theorem can be replaced by a purely topological condition, simply-connectedness. In dimension 4 instead of simply-connectedness here only non-vanishing of the Euler characteristic has to be required. A…

幂零分级 Lie 群上半经典微积分的几何不变性

Geometric invariance of the semi-classical calculus on nilpotent graded Lie groups

本文研究了幂零分级 Lie 群上半经典伪微分算子在保持过滤的 Pansu 可微微分同胚拉回下的几何不变性，并在过滤流形设定下解释了其几何意义。

微分同胚群的信息几何

Information geometry of diffeomorphism groups

本文综述微分同胚群上与信息几何相关的各类度量，引入对应无穷维流形的拓扑、光滑结构及动力学，展示信息几何成为微分同胚群几何分析应用新领域的价值。

Spectra of orbifolds with cyclic fundamental groups

We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamental group. The differential operators considered are Laplace-Beltrami operators twisted by characters of the corresponding fundamental group. To prove the characterization, we first give an explicit description of their spectra by using generating functions. We also include many isospectral examples.

Morita Equivalence and Spectral Triples on Noncommutative Orbifolds

Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical orbifold groupoids, a Morita equivalence for the crossed product spectral triples is developed. Noncommutative orbifolds are Morita equivalence classes of the crossed product spectral triples. As a special case of this Morita theory one can study freeness of the $G$-action on the …

概率分布的几何高斯逼近

Geometric Gaussian Approximations of Probability Distributions

本文研究通过微分同胚或黎曼指数映射构造的几何高斯逼近，证明其可逼近任意概率分布，并探讨对给定分布族寻找共同微分同胚以获得一致高质量逼近的可能性。