Perturbed cone theorems for proper harmonic maps

We extend the well-known Denjoy-Ahlfors theorem on the number of different asymptotic tracts of holomorphic functions to subharmonic functions on arbitrary Riemannian manifolds. We obtain some new versions of the Liouville theorem for $\p$-harmonic functions without requiring the geodesic completeness requirement of a manifold. Moreover, an upper estimate of the topological index of the height function on a minimal surface in $\R{n}$ has been established and, as a consequence, a new proof of Bernstein's theorem on entire solutions has been derived. Other applications to minimal surfaces are also discussed.

A harmonic morphism between arbitrary Riemannian manifolds is a type of harmonic map. This chapter is devoted to the description of those properties of harmonic maps, which are essential to the development. Harmonic maps are extremals of a natural energy integral; they can be characterized as maps whose tension field vanishes, where the tension field is a natural generalization of the Laplacian. The first three sections in this chapter give the necessary formalism, the basic definitions, examples, and properties of harmonic maps. In Section 3.4, a conservation law involving the stress-energy is given. Harmonic maps from surfaces have special properties and include (branched) minimal immersions, which are discussed in Section 3.5. The chapter ends with a treatment of the second variation.

In this paper we generalize and extend to any Riemannian manifold maximum principles for Euclidean hypersurfaces with vanishing curvature functions obtained by Hounie-Leite.

Recently, the author defined and classified rectifying submanifolds in Euclidean spaces in [12]; extending his earlier work on rectifying curves in Euclidean 3-space done in [6]. In this article, first the author introduces the notion of rectifying submanifolds in an arbitrary Riemannian manifold. Then he defines torqued vector fields on Riemannian manifolds and classifies Riemannian manifolds which admit a torqued vector field. Finally, he characterizes and studies rectifying submanifolds in a Riemannian manifold equipped with a torqued vector field. Some related results and applications are also presented.

We show that the difference between the Morse index of a closed minimal surface as a critical point of the area functional and its Morse index as a critical point of the energy is at most the real dimension of Teichmüller space. This enables us to bound the index of a closed minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold. Our method also yields surprisingly good upper bounds on the index of a minimal surface of finite total curvature in Euclidean space of any dimension.

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ξ, when ∇ξ is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ξ is geodesic, ξ is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures σ are harmonic sections, in the sense of Vergara-Diaz and Wood [25], and in some cases they are also harmonic maps.

In this article, we attempt to describe some of the most important results concerning proper holomorphic mappings between complex spaces. The first two sections deal respectively with the Remmert Proper Mapping Theorem and the Grauert Direct Image Theorem. These are certainly the most important results of what may be called the general theory. Sect. 3 deals with embeddings in CN. Finally, Sect. 4 treats proper maps between bounded domains, which is, perhaps, the domain of greatest recent and current research. We have not dealt with this in as great detail as we might have because of the availability of the excellent survey articles of Bedford [Bed] and Forstneric [FF1].

Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note, we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean background space is replaced by an arbitrary Riemannian manifold. Introducing the notion of ε-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that ε-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth condition at infinity are minimal.

Functional inequalities on arbitrary Riemannian manifolds

The paper is concerned with problems at the intersection of the theory of spatial quasi-conformal mappings and the theory of Riemann surfaces. Theorems on the local behaviour of one class of open discrete mappings with unbounded coefficient of quasi-conformality, which map between arbitrary Riemannian manifolds, are obtained. Such mappings are also shown to extend to isolated points of the boundary of the domain. Some results on the local behaviour of Sobolev and Orlicz-Sobolev classes are obtained as an application. Bibliography: 52 titles.

We suggest a new approach to the statement of boundary value problems for elliptic partial differential equations on arbitrary Riemannian manifolds which is based on the consideration of equivalence classes of functions on a manifold. Using this approach, we establish some interrelation between the solvability of boundary value problems and solvability of exterior boundary problems for the stationary Schrodinger equation. Also we prove the comparison and uniqueness theorems for solutions to boundary value problems in this statement and obtain sufficient conditions for solvability of boundary value problems when the coefficient in the Schrodinger equation is changed.

We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for modelling stationary hypersurfaces under the influence of a non-homogeneous gravitational field defined over an arbitrary Riemannian manifold.

Newtonian dynamical systems that admit normal shift on an arbitrary Riemannian manifold are considered. The determining equations for these systems, which constitute the condition of weak normality, are derived. The extension of the algebra of tensor fields to manifolds is considered.

For any (not necessarily complete) Riemannian manifold, we construct a larger Riemannian metric which is complete and with bounded sectional curvatures. As an application, log-Sobolev inequalities are established on arbitrary Riemannian manifolds with reference measures having smooth and strictly positive densities. In particular, a conjecture of the second-named author (see [J. Math. Anal. Appl. 300 (2004) 426–435]) is solved.

We consider upper estimates for the Green function of the heat equation on an arbitrary smooth connected Riemannian manifold M. We define the Green function G(t, x; y) as the limit of the Green functions G~ for the precompact domains ~CM as ~ ~ M. If the manifold M has boundary, then we will always assume the Neumann homogeneous condition to be fulfilled on the boundary and also consider only those ~ for which a~ is transversed to 3M. Let us denote the geodesic distance between two points x, y~M by Ix yJ and the geodesic ball of radius r with center at x by Br x. If N is a submanifold, then we will denote its volume, corresponding to the dimension, by JN I . THEOREM I. Let the following isoperimetric inequality be fulfilled in a precompact geodesic ball Bpx: for each domain Q ~ Box that has smooth boundary 3Q, transversal to aM,