On Riesz groups

In this paper we investigate projective 4-dimensional manifolds X whose tangent bundles T_X are numerically effective and give an almost complete classification. An important technical tool is the “Mori theory” of projective manifolds X whose canonical bundles K_X are not numerically effective.     

Unstable products of smooth curves

We give examples of smooth manifolds with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kähler classes that do not admit any constant scalar curvature Kähler metrics. These manifolds also have unstable Hilbert and Chow points. We compare this to the work of Song-Weinkove on the J-flow.     

K-Semistability of cscK Manifolds with Transcendental Cohomology Class

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson–Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.     

On the quasidiagonality of Roe algebras

Let X be a noncompact discrete metric space with bounded geometry. Associated with X are two C*-algebras, the so-called uniform Roe algebra B*(X) and coarse Roe algebra C*(X), which arose from the index theory on noncompact complete Riemannian manifolds. In this paper, we describe the quasidiagonality of B*(X) and C*(X) in terms of coarse geometric invariants. Some necessary and suficient conditions are given, which involve the Fredholm index and coarse connectedness of metric spaces.     

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature \(h\). In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds \(X\) with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification \(\overline{X}\).     

Beispiele homogener Mannigfaltigkeiten

Let X be a connected compact complex manifold. We call X a homogeneous manifold if there exists a complex Liegroup G of biholomorphic automorphisms of X such that for any two points p, qX there is a gG with: g(p)=q. We give some non-trivial examples of homogeneous manifolds. All examples are non-kaehlerian manifolds. Such examples have been given first by Iwasawa (as far as we know). The last section of this note contains a proposition, which shows that only very special non-semisimple Liegroups are able to act transitivily in the sense stated above, on a compact complex manifold. This is remarkable also in contrast to the semisimple case (cf. [3]).     

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results

In this paper we prove rigorous results on persistence of invariant tori and their whiskers. The proofs are based on the parameterization method of [X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J. 52 (2) (2003) 283-328; X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2) (2003) 329-360]. The invariant manifolds results proved here include as particular cases of the usual (strong) stable and (strong) unstable manifolds, but also include other non-resonant manifolds. The method lends itself to numerical implementations whose analysis and implementation is studied in [A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, preprint, 2005; A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical implementation and examples, preprint, 2005]. The results are stated as a posteriori results. Namely, that if one has an approximate solution which is not degenerate, then, one has a true solution not too far from the approximate one. This can be used to validate the results of numerical computations.     

On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra , the orbits of the action Φ of on given by are pseudo‐Hessian manifolds, where . We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.     

线性流形上矩阵方程B^TXB=D的反对称解

该文讨论了两类线性流形上矩阵方程B^TXB=D的反对称解和反对称最佳逼近解存在的条件,给出了通解的一般表达式,同时解决了解对给定矩阵的唯一最佳逼近问题.     

线性流形上两类矩阵反问题的最小二乘解

给定广义自反矩阵R,S,即R=R=R-1,S=S=S-1,若复矩阵X满足条件RXS=X（或RXS=X）,则称其为（R,S）-对称矩阵（或（R,S）-斜对称矩阵）.分别讨论了线性流形上（R,S）-对称矩阵和（R,S）-斜对称矩阵约束下矩阵方程MZN=E的最小二乘问题,得到了通解表达式.     

Foliations of the Sasakian Space R2n+1 by Minimal Slant Submanifolds

Examples of slant submanifolds in the Sasakian space R²ⁿ⁺¹ are obtained as the leaves of a harmonic, Riemannian 3-dimensional foliation. With the exception of the anti-invariant ones, these leaves are all locally homogeneous manifolds with negative scalar curvature, whose Ricci tensor satisfies (S)(X, X) = 0 for all tangent vector fields.     

Almost nonnegative curvature operator and cohomology rings

We give a survey of results on the construction of and obstructions to metrics of almost nonnegative curvature operator on closed manifolds and results on the cohomology rings of closed, simply-connected manifolds with a lower curvature and upper diameter bound. The latter is motivated by a question of Grove whether these condition imply finiteness of rational homotopy types. This question has answers by F. Fang–X. Rong, B. Totaro and recently A. Dessai and the present author.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

（0,2）型张量场诱导的线性变换的显表达式提取方法

对于黎曼流形（M,g）上（0,2）型光滑张量场R诱导出的线性变换R＊：X（M）→X（M）,其隐表达式为g（R＊（X）,Y）=S（X,Y）,X,Y∈X（M）,研究了从局部上提取出R＊（X）的显表达式的方法.提出一种所谓g正定提取方法,并将这种方法进一步应用到度量保形变换下的线性变换Φ＊（X）,L＊（X）.     

K(a)hler Manifolds with Almost Non-negative Ricci Curvature

Compact K(a)hler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell's work, if M is a compact K(a)hler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition (M) ≌ X1 × … × Xm, where Xj is a Calabi-Yau manifold, or a hyperK(a)hler manifold, or Xj satisfies Ho(Xj,Ωp) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature K(a)hler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ε ＞ 0, there exists a K(a)hler structure (Je,ge) on M such that the volume Volge(M) ＜ V, the sectional curvature |K(gε)| ＜ Λ2, and the Ricci-tensor Ric(gε)＞ -εgε, where ∨ and Λ are two constants independent of ε. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, (X) ≌ X1 × … × Xs, where Xi is a Calabi-Yau manifold, or a hyperK(a)hler manifold, or Xi satisfies Ho(Xi, Ωp) = {0}, p ＞ 0.     

p-Harmonic obstacle problems

Summary We develop an interior partial regularity theory for vector valued Sobolev functions which locally minimize degenerate variational integrals under the additional side condition that all comparison maps take their values in the closure of a smooth region of the target space. Our results apply to the case of penergy minimizing mappings X Y between Riemannian manifolds including target manifolds Y with nonvoid boundary.     

Variational analysis of a perturbed free boundary problem

We consider maps u:X→Y of Riemannian manifolds which are locally energy minimizing subject to a constraint of the form Im(u) where M denotes a smooth bounded domain contained in a regular ball B of the target manifold. In general local minima are regular up to a set of vanishing Hausdorff measure, here we show for star-shaped obstacles M singular points can be exluded.     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group.E is a holomorphic vectorbundle of rank r with trivial pull-back to W=C~n-{0}.We prove the existence of a non-vanishingsection of L(?)E for some line bundle on X and study the vector bundles filtration structure of E.These generalize the results of D.Mall about structure theorem of such a vector bundle E.     

Minimal total absolute curvature for immersions

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.This research was partially supported by National Science Foundation grant GP-7952X1.