Weitzenböck formulas for Riemannian foliations

In this paper we study the interplay between adiabatic limits of a Riemannian foliation and the classical Weitzenböck formula. For the leafwise part, our study leads to a vanishing result for the first order term of differential spectral sequence associated with the foliation. For the transversal part we obtain a Weitzenböck type formula which is an extension of the previous formula for basic forms due to Ph. Tondeur, M. Min-Oo, and E. Ruh, and is also more general than a Weitzenböck formula for transverse fiber bundle due to Y. Kordyukov.     

Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds

We prove the Bochner–Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li–Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry–Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.     

Gap Phenomena for p-Harmonic Maps

In this paper, we generalize to p-harmonic mapssome gap results known for harmonic maps. In particular, we prove that,under a certain level of energy depending on the curvature of the domainand target manifolds, the only p-harmonic maps are theconstant ones. The main tools are Bochner–Weitzenböck andReilly-type formulas involving the p-Laplace operator.     

Scalar curvature rigidity of hyperbolic product manifolds

This paper presents a scalar curvature rigidity result of real hyperbolic product manifolds in analogy to M. Min–Oos result in [14]. In order to prove this, we consider Dirac bundles obtained from the spinor bundle, and we derive Killing equations trivializing these Dirac bundles. Moreover, an integrated Bochner–Weitzenböck formula is shown which allows the usage of the non–compact Bochner technique.Mathematics Subject Classification (2000): Primary 53C24, Sec. 53C21in final form: 11 August 2003     

Eigenvalue estimates of the Dirac operator depending on the Ricci tensor

We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenb?ck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds. Received: 20 April 2001 / Published online: 5 September 2002     

Sectional Curvature Rigidity of Asymptotically Locally Hyperbolic Manifolds

This paper presents a rigidity result of real hyperbolic quotients inanalogy to Min-Oo's result in Math. Ann. 285(4) (1989),527–539, but without the spin condition. In order to prove this,we use special Killing forms on the exterior form bundle. Moreover, wemake an assumption on the sectional curvature to obtain the necessaryeigenvalue estimates of the curvature endomorphism in theBochner–Weitzenböck formula of ^k M.     

RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.     

Compactification of moduli space of harmonic mappings

We introduce the notion of harmonic nodal maps from the stratified Riemann surfaces into any compact Riemannian manifolds and prove that the space of the energy minimizing nodal maps is sequentially compact. We also give an existence result for the energy minimizing nodal maps. As an application, we obtain a general existence theorem for minimal surfaces with arbitrary genus in any compact Riemannian manifolds. Received: 1 April 1997; revised: 15 April 1998.     

Connexion markovienne,courbure et formule de Weitzenböck sur l'espace des chemins riemanniens

We consider the Markovian connection on the Riemannian path spaces. The curvature is computed explicitly and a Weitzenböck formula is established.     

Geometrically Formal Homogeneous Metrics of Positive Curvature

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.     

Harmonic map heat flow with free boundary

In this paper, we study the harmonic map heat flow with free boundary from a Riemannian surface with smooth boundary into a compact Riemannian manifold. As a consequence, we get at least one disk-type minimal surface in a compact Riemannian manifold without minimal 2-sphere.     

Differential calculus on path and loop spaces II. Irreducibility of Dirichlet forms on loop spaces

We prove the irreducibility of a Dirichlet form on the based loop space on a compact Riemannian manifold. The Dirichlet form is defined by the gradient operator due to Driver and Léandre. We also prove the uniqueness of the ground states of the Schrödinger operator for which the Dirichlet form satisfies the logarithmic Sobolev inequality. This is an extension of the corresponding results of Gross ([28], [29]) to the case of general compact Riemannian manifolds.     

Locally minimizing harmonic maps from noncompact manifolds

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.     

The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

This is a continuation of the Cambridge Tract ``Harmonic maps between Riemannian polyhedra', by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.

     

The existence of harmonic maps from Finsler manifolds of Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler manifold and study the characterisation of harmonic maps,in the spirit of Ishihara.Using heat quation method we show that any map from a compact Finsler manifold M to a compact Riemannian manifold with non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.     

Hyperbolic Manifolds, Harmonic Forms, and Seiberg–Witten Invariants

New estimates are derived concerning the behavior of self-dual harmonic 2-forms on a compact Riemannian 4-manifold with nontrivial Seiberg–Witten invariants. Applications include a vanishing theorem for certain Seiberg–Witten invariants on compact 4-manifolds of constant negative sectional curvature.     

Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space

We study a semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space. Key results are semiboundedness theorem of the Schrödinger operator, Laplace-type asymptotic formula and IMS localization formula. We also make a remark on the semiclassical problem of a Schrödinger operator on a path space over a Riemannian manifold.     

Law of the iterated logarithm for transitiveC 2 Anosov flows and semiflows over maps of the interval

It is proved that a functional law of the iterated logarithm is valid for transitiveC ² Anosov flows on compact Riemannian manifolds when the observable belongs to a certain class of real-valued Hölder functions. The result is equally valid for semiflows over piecewise expanding interval maps that are similar to the Williams' Lorenz-attractor semiflows. Furthermore the observables need only be real-valued Hölder for these semiflows.     

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.     

Harmonic functions under quasi-isometry

Given a complete Riemannian manifold (M, g) with nonnegative sectional curvature outside a compact subset. Let h be another Riemannian metric which is uniformly equivalent to g. It was shown that the dimension of the space of bounded harmonic functions on (M, h) is finite and is the same as of that under metric g, and the dimension of the space spanned by nonnegative harmonic functions on (M, h) is also finite and is the same as of that under metric g. Moreover, bases were constructed for both spaces on (M, h) and precise estimates were established on the asymptotic behavior at infinity for those basic functions.