Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

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).     

Harmonic maps from closed Riemannian manifolds with positive scalar curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.     

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.     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

On f-bi-harmonic maps and bi-f-harmonic maps between Riemannian manifolds

Both bi-harmonic maps and f-harmonic maps have some nice physical motivation and applications.Motivated largely by f-tension field not involving Riemannian curvature tensor, we attempt to formalize some large objects so as to broaden the notions of f-tension field and bi-tension field. We introduce a very large generalization of harmonic maps called f-bi-harmonic maps as the critical points of f-bi-energy functional, and then derive the Euler-Lagrange equation of f-bi-energy functional given by the vanishing of f-bi-tension field.Subsequently, we study some properties of f-bi-harmonic maps between the same dimensional manifolds and give a non-trivial example. Furthermore, we also study the basic properties of f-bi-harmonic maps on a warped product manifold so that we could find some interesting and complicated examples.     

Harmonic maps on Sasakian manifolds

The object of the present work is to study harmonic maps on Sasakian manifolds.     

Harmonic maps on Trans-Sasaki manifolds

In this paper we prove that a (?,J)-holomorphic mapf:M→N (i.e.f _*o?=Jof _*) from a Trans-Sasaki manifold to a nearly Kähler manifold is a harmonic map. We also study the stability of a such map whenM is a compact Trans-Sasaki manifold andN is a Kähler manifold.     

Semi-invariant Riemannian maps from almost Hermitian manifolds

We construct Gauss–Weingarten-like formulas and define O’Neill’s tensors for Riemannian maps between Riemannian manifolds. By using these new formulas, we obtain necessary and sufficient conditions for Riemannian maps to be totally geodesic. Then we introduce semi-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, give examples and investigate the geometry of leaves of the distributions defined by such maps. We also obtain necessary and sufficient conditions for semi-invariant maps to be totally geodesic and find decomposition theorems for the total manifold. Finally, we give a classification result for semi-invariant Riemannian maps with totally umbilical fibers.     

Harmonic functions on Riemannian manifolds with ends

We study the problem of solvability of some boundary value problems on noncompact Riemannian manifolds with ends. We obtain the conditions for existence and uniqueness of solutions to the problems as well as the conditions for the fulfillment of Liouville-type theorems for harmonic functions on the manifolds.     

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.     

The existence of harmonic maps from Finsler manifolds to Riemannian manifolds

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

Harmonic maps between compact Hermitian manifolds

In this paper, we generalize the Bochner-Kodaira formulas to the case of Hermitian com- plex (possibly non-holomorphic) vector bundles over compact Hermitian (possibly non-K¨ahler) mani- folds. As applications, we get the complex analyticity of harmonic maps between compact Hermitian manifolds.     

Liouville-type theorems for CC-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.     

On weakly harmonic maps from Finsler to Riemannian manifolds

We prove global C0,α$C^{0,\alpha }$-estimates for harmonic maps from Finsler manifolds into regular balls of Riemannian target manifolds generalizing results of Giaquinta, Hildebrandt, and Hildebrandt, Jost and Widman from Riemannian to Finsler domains. As consequences we obtain a Liouville theorem for entire harmonic maps on simple Finsler manifolds, and an existence theorem for harmonic maps from Finsler manifolds into regular balls of a Riemannian target.     

Pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature

In this paper, we use heat flow method to prove the existence of pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature, which is a generalization of Eells–Sampson’s existence theorem. Furthermore, when the target manifold has negative sectional curvature, we analyze horizontal energy of geometric homotopy of two pseudo-harmonic maps and obtain that if the image of a pseudo-harmonic map is neither a point nor a closed geodesic, then it is the unique pseudo-harmonic map in the given homotopic class. This is a generalization of Hartman’s theorem.