完备流形上拉普拉斯算子的L2特征形式

本文研究了完备非紧流行上拉普拉斯算子的L²特征形式.利用应力能量张量的方法,得到在此类流形上拉普拉斯算子的L²特征形式的一些不存在性定理。     

Sobolev Inequalities on Forms and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript>-Cohomology on Complete Riemannian Manifolds

We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable geometric conditions. Moreover, we establish some L ^p,q -estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing theorems of the L ^p,q -cohomology and prove the L ^q -solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions.     

Holomorphically planar conformal vector fields on contact metric manifolds

We study holomorphically planar conformal vector fields (HPCV) on contact metric manifolds under some curvature conditions. In particular, we have studied HPCV fields on (i) contact metric manifolds with pointwise constant ξ-sectional curvature (under this condition M is either K-contact or V is homothetic), (ii) Einstein contact metric manifolds (in this case M becomes K contact), (iii) contact metric manifolds with parallel Ricci tensor (under this condition M is either K-contact Einstein or is locally isometric to E ⁿ⁺¹×S ⁿ (4)).     

Elliptic Symbols

If G is the structure group of a manifold M it is shown how a certain ideal in the character ring of G corresponds to the set of geometric elliptic operators on M. This provides a simple method to construct these operators. For classical structure groups like G = O(n) (Riemannian manifolds), G = SO(n) (oriented Riemannian manifolds), G = U(m) (almost complex manifolds), G = Spin(n) (spin manifolds), or G = Spin^c(n) (spin^c manifolds) this yields well known classical operators like the Euler—deRham operator, signature operator, Cauchy—Riemann operator, or the Dirac operator. For some less well studied structure groups like Spin^h(n) or Sp(q)Sp(1) we can determine the corresponding operators. As applications, we obtain integrality results for such manifolds by applying the Atiyah—Singer Index Theorem to these operators. Finally, we explain how immersions yield interesting structure groups to which one can apply this method. This yields lower bounds on the codimension of immersions in terms of topological data of the manifolds involved.     

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C^*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.     

Brinkman-type Operators on Riemannian Manifolds: Transmission Problems in Lipschitz and <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Domains

In this paper we use the method of boundary integral equations to treat some transmission problems for Brinkman-type operators on Lipschitz and C ¹ domains in Riemannian manifolds.     

Hofer'sL ∞-geometry: energy and stability of Hamiltonian flows,part II

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group Ham ^c (M) of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction onM. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic toM×D ² which are symplectically ruled overD ². When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided thatM is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory ofJ-holomorphic curves in arbitraryM.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongstall paths, not only the homotopic ones) under even more restrictive conditions onM, for example whenM is exact and convex or of dimension 2. The new difficulty is caused by the possibility that there are non-trivial and very short loops in Ham ^c (M). When such lengthminimizing paths do exist, we can extend the Bialy-Polterovich calculation of the Hofer norm on a neighbourhood of the identity (C ^l-flatness).Although it applies to a more restricted class of manifolds, the Hofer-Zehnder capacity seems to be better adapted to the problem at hand, giving sharper estimates in many situations. Also the capacity-area inequality for split cylinders extends more easily to quasi-cylinders in this case. As applications, we generalise Hofer's estimate of the time for which an autonomous flow is length-minimizing to some manifolds other thanR ²ⁿ , and derive new results such as the unboundedness of Hofer's metric on some closed manifolds, and a linear rigidity result.Oblatum 13-X-1994 & 8-V-1995Partially supported by NSERC grant OGP 0092913 and FCAR grant ER-1199Partially supported by NSF grant DMS 9103033 and NSF Visiting Professorship for Women GER 9350075     

Center manifolds for nonuniformly partially hyperbolic diffeomorphisms

We establish the existence of smooth invariant center manifolds for the nonuniformly partially hyperbolic trajectories of a diffeomorphism in a Banach space. This means that the differentials of the diffeomorphism along the trajectory admit a nonuniform exponential trichotomy. We also consider the more general case of sequences of diffeomorphisms, which corresponds to a nonautonomous dynamics with discrete time. In addition, we obtain an optimal regularity for the center manifolds: if the diffeomorphisms are of class C^k then the manifolds are also of class C^k. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the center manifolds, but also for their derivatives up to order k.     

An Existence Theorem of Smooth Nonlocal Center Manifolds for Systems Close to a System with a Homoclinic Loop

Summary. In this paper we give a proof of the existence of smooth nonlocal center manifolds for systems close to a system with a homoclinic orbit to a saddle-type equilibrium point. Our proof is based on a consideration of some class of the boundary value problems (see Section 3). We obtain estimates for solutions of the boundary value problems that allow us to prove the theorem on the center manifolds at the C ¹ -assumptions for the smoothness of systems. Received June 4, 1997; final revision received April 24, 1998     

Sur la dynamique unidimensionnelle en régularité intermédiaire

Using probabilistic methods, we prove new rigidity results for groups and pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class (i.e. between C ¹ and C ²). In particular, we show some generalizations of Denjoy theorem and the classical Kopell lemma for abelian groups. These techniques are also applied to the study of codimension-1 foliations. For instance, we obtain several generalized versions of Sacksteder theorem in class C ¹. We conclude with some remarks about the stationary measure.     

Nonnegatively curved fixed point homogeneous manifolds in low dimensions

Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M ^G . We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M ^G . Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.     

On a Class of Fully Nonlinear Elliptic Equations on Closed Hermitian Manifolds II: L∞ Estimate

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of the cone condition, we derive the L^∞ estimate directly. As an application, we solve the complex quotient equations on closed Kähler manifolds. © 2016 Wiley Periodicals, Inc.     

Invariant manifolds for differential equations

In this paper we discuss the concept ‘generalized exponential dichotomy’ and give the existence ofC ^k invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces. Also we give a classification of invariant manifolds and an estimate of the locality radius of invariant manifolds.     

Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

L p -cohomology of negatively curved manifolds

We compute theL ^p -cohomology spaces of some negatively curved manifolds. We deal with two cases: manifolds with finite volume and sufficiently pinched negative curvature, and conformally compact manifolds. This paper has been (partially) supported by the European Commission through the Research Training Network HPRN-CT-1999-00118 “Geometric Analysis”.     

On the geometry of model almost complex manifolds with boundary

We study some special almost complex structures on strictly pseudoconvex domains in ℝ^{2 n} . They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.     

Concavity of minimal L2 integrals related to multiplier ideal sheaves on weakly pseudoconvex Kähler manifolds

In this paper, we present the concavity of the minimal L² integrals related to multiplier ideal sheaves on the weakly pseudoconvex Kähler manifolds which implies the sharp effectiveness results of the strong openness conjecture and a conjecture posed by Demailly and Kollár (2001) on weakly pseudoconvex Kähler manifolds. We obtain the relation between the concavity and the L² extension theorem.

     

Cohomology of bundles on homological Hopf manifolds

We discuss the properties of complex manifolds having rational homology of S ¹ × S ²ⁿ⁻¹ including those constructed by Hopf, Kodaira and Brieskorn-van de Ven. We extend certain previously known properties of cohomology of bundles on such manifolds. As an application we consider degeneration of Hodge-de Rham spectral sequence in this non Kahler setting.     

Optimal transportation on non-compact manifolds

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type d ^r , r > 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.