Infinitesimal transformations on noncompact manifolds

Summary The object of study in this paper is the A_x-operator associated to a Killing field (resp. to an infinitesimal analytic transformation) on Riemannian (resp. Kähler) complete but noncompact manifolds. We recall (§ 1) Kostant's and Lichnéroivicz's Theorems for that operator in the compact case and we dedicate, § 2 and § 3 to study all the simply-connected noncompact manifolds in which similar results to that theorems do not hold, we give examples of all these cases. § 4 is dedicated to give sufficient conditions in order to ensure the validity of similar results to that above mentioned, on complete noncompact manifolds. The hypothesis used refer to vector fields whose norm are either pointwise bounded or globally bounded.Partially supported by C.A.I.C.Y.T. 1085-84.     

Gradient estimate for exponentially harmonic functions on complete Riemannian manifolds

The notion of exponentially harmonic maps was introduced by Eells and Lemaire (Proceedings of the Banach Center Semester on PDE, pp. 1990–1991, 1990). In this note, by using the maximum principle we get the gradient estimate of exponentially harmonic functions, and then derive a Liouville type theorem for bounded exponentially harmonic functions on a complete Riemannian manifold with nonnegative Ricci curvature and sectional curvature bounded below.     

Av-operator on complete foliated Riemannian manifolds

We give a generalization of the result obtained by C. Currás-Bosch. We consider the Av-operator associated to a transverse Killing fieldν on a complete foliated Riemannian manifold (M, ℱ, g). Under a certain assumption, we prove that, for eachx ∈M, (Av) _x belongs to the Lie algebra of the linear holonomy group ψv(x). A special case of our result, the version of the foliation by points, implies the results given by B. Kostant (compact case) and C. Currás-Bosch (non-compact case).     

Groups of transformations of Riemannian manifolds

This paper is a survey of the articles reviewed by the journalMatematika from January 1971 through August 1989 on groups of transformations of Riemannian manifolds and their applications.Translated fromItogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 22, 1990, pp. 97–165.     

Quasilinear elliptic inequalities on complete Riemannian manifolds

We prove maximum and comparison principles for weak distributional solutions of quasilinear, possibly singular or degenerate, elliptic differential inequalities in divergence form on complete Riemannian manifolds. A new definition of ellipticity for nonlinear operators on Riemannian manifolds is introduced, covering the standard important examples. As an application, uniqueness results for some related boundary value problems are presented.     

Hardy type inequalities on complete Riemannian manifolds

In this paper, we show that complete Riemannian manifolds with asymptotically non-negative Ricci curvature in which some Hardy type inequalities hold are not far from the Euclidean space.     

Harmonic transforms of complete Riemannian manifolds

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.     

Conditional extremals in complete Riemannian manifolds

Conditional extremal curves in a complete Riemannian manifold M are defined as the critical points of the squared L² distance between the tangent vector field of a curve and a so-called prior vector field. We prove that this L² distance satisfies the Palais-Smale condition on the space of absolutely continuous curves joining two submanifolds of M, and thus establish the existence of critical points. We also prove a Morse index theorem in the case where the two submanifolds are single points, and use the Morse inequalities to place lower bounds on the number of critical points of each index.     

Transversal harmonic transformations for Riemannian foliations

The main purpose of the present paper is to study geometric properties of transversal (infinitesimal) harmonic transformations for Riemannian foliations. For the point foliation these notions are discussed in [14]. Especially we treat transversal infinitesimal harmonic transformations from the standpoint of λ-automorphisms. Our results extend those obtained in [6, 7, 15] for the case of harmonic foliations. Mathematics Subject Classifications (2000): Primary 53C20, Secondary 57R30.     

Heat semi-group and generalized flows on complete Riemannian manifolds

We will use the heat semi-group to regularize functions and vector fields on Riemannian manifolds in order to develop Di Perna–Lions theory in this setting. Malliavin?s point of view of the bundle of orthonormal frames on Brownian motions will play a fundamental role. As a byproduct we will construct diffusion processes associated to an elliptic operator with singular drift.