Stationary harmonic maps into complete Riemannian manifold

The stationary for harmonic maps is considered from a Riemannian manifoldM into a complete Riemannian manifoldN without boundary, and it is proved that its singular set is contained inQ ¹ 2MQ ³ Project supported partially by the Development Foundation Science of Shanghai, China.     

Stationary biharmonic maps from Rm into a Riemannian manifold

For m ≥ 5, we prove that a stationary extrinsic (or intrinsic, respectively) biharmonic map u ∈ W^2,2(Ω, N) from Ω ? R^m into a Riemnanian manifold N is smooth away from a closed set of (m ? 4)‐dimensional Hausdorff measure zero. © 2003 Wiley Periodicals, Inc.     

A Trudinger type inequality for maps into a Riemannian manifold

The tension field of a map into a Riemannian manifold is the equivalent to the Laplacian of a function. However in contrast to the latter, the tension field is given by a nonlinear differential operator. Nevertheless, it permits an extension of a well-known Trudinger inequality that involves an Orlicz space for a function with exponential growth.     

Harmonic maps from a Riemannian manifold with a pole into an Hadamard manifold with negative sectional curvatures

In this paper we show a nonexistence result for harmonic maps with a rotational nondegeneracy condition from a Riemannian manifoldM with polep ₀ to a negatively curved Hadamard manifold under the condition that the metric tensor ofM is bounded and that the sectional curvature ofM at a pointp is bounded from below by −c dist(p ₀,p)⁻² (c: a positive constant) as dist(p ₀,p)→∞. Partly supported by Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan     

Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds

We investigate p-harmonic maps, p ≥ 2, from a complete non-compact manifold into a non-positively curved target. First, we establish a uniqueness result for the p-harmonic representative in the homotopy class of a constant map. Next, we derive a Caccioppoli inequality for the energy density of a p-harmonic map and we prove a companion Liouville type theorem, provided the domain manifold supports a Sobolev–Poincaré inequality. Finally, we obtain energy estimates for a p-harmonic map converging, with a certain speed, to a given point.      

Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

In this article, we show that, for a biharmonic hypersurface (M, g) of a Riemannian manifold (N, h) of non-positive Ricci curvature, if ò_M|H|² v_g < ￥{\int_M\vert H\vert^2 v_g<\infty}, where H is the mean curvature of (M, g) in (N, h), then (M, g) is minimal in (N, h). Thus, for a counter example (M, g) in the case of hypersurfaces to the generalized Chen’s conjecture (cf. Sect. 1), it holds that ò_M|H|² v_g=￥{\int_M\vert H\vert^2 v_g=\infty} .     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

Partial regularity of energy minimizing harmonic maps into a complete manifold

In this paper, we study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at mostn–2, wheren is the dimension of the domain. We will also give an example of an energy minimizing map from surface to surface that has a singular point. Thus then–2 dimension estimate is optimal, in contrast to then–3 dimension estimate of Schoen-Uhlenbeck [SU] for compact targets.     

On the homotopy class of maps with finite p-energy into non-positively curved manifolds

We prove that a map f : M → N with finite p-energy, p > 2, from a complete manifold (M, á , ñ ){\left(M,\left\langle ,\right\rangle \right)} into a non-positively curved, compact manifold N is homotopic to a constant, provided the negative part of the Ricci curvature of the domain manifold is small in a suitable spectral sense. The result relies on a Liouville-type theorem for finite q-energy, p-harmonic maps under spectral assumptions.     

Biharmonic Riemannian submersions from 3-manifolds

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa (Kyushu J Math 52(1):167?C185, 1998) and Jiang (Chin Ann Math Ser. 8A:376?C383, 1987) states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper, Caddeo et?al. (Isr J Math 130:109?C123, 2002) showed that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form into a surface is biharmonic if and only if it is harmonic.     

On the existence of Hermitian-harmonic maps from complete Hermitian to complete Riemannian manifolds

On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is not in divergence form.The case of noncompact complete preimage and target manifolds is considered. We give conditions for existence and uniqueness of Hermitian-harmonic maps and solutions of the corresponding parabolic system, which observe the non-divergence form of the underlying equations. Numerous examples illustrate the theoretical results and the fundamental difference to harmonic maps.Support from the research focus Globale Methoden in der komplexen Geometrie under the auspices of Deutsche Forschungsgemeinschaft is gratefully acknowledged.Acknowledgement We are grateful to Wolf von Wahl (University of Bayreuth) for his suggestion to investigate Hermitian-harmonic maps on noncompact manifolds.Dedicated to Prof. E. Heinz on the occasion of his 80^th birthday     

Harmonic and conformally Killing forms on complete Riemannian manifold

We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms.     

Rotationally symmetric harmonic maps from a ball into a warped product manifold

This paper deals with the existence problem for rotationally symmetric harmonic maps from an Euclidean unit ball B ⁿ or ⁿ into a warped product manifold N_f=[0, r₀)x_fS^n–1.