Hajime Urakawa

Harmonic maps into Lie groups and homogeneous spaces

Abstract The tension field of a smooth map of an arbitrary Riemannian manifold into any homogeneous space (G/K, h) with an invariant Riemannian metric h is calculated. As its applications, a characterization of harmonic maps into any homogeneous space is given and all harmonic maps of the standard Euclidean space (Rm, g0) into a Lie group (G, h) with left invariant Riemannian metric h, of the form ƒ (x1, …, xm) = exp(x1X1) … exp(xmXm) are determined.

BIHARMONIC LAGRANGIAN SUBMANIFOLDS IN K Ä HLER MANIFOLDS

We give necessary and sufficient conditions for a Lagrangian submanifold of a Kahler manifold to be biharmonic. Furthermore, we classify biharmonic PNMC Lagrangian submanifolds in the complex space forms.

Classification of harmonic mappings of constant energy density into spheres

We classify all eigenmaps and isometric minimal immersions of a flat torus into the unit sphere using the parametrization theorem (cf. [2], [14]) for range-equivalence classes of all eigenmaps of an arbitrary compact homogeneous Riemannian manifold into the unit sphere.

On exponential Yang-Mills connections

In this paper, we study a new functional, i.e., the exponential Yang-Mills functional Y Me on the space of all smooth connections ▽ of a vector bundle E over a compact Riemannian manifold (M, g) which is defined by Y Me (▽) = ∫M exp(12 || R▽||2) vg, where || R▽ || is the curvature tensor of a connection ▽. A critical point of Y Me is called an exponential Yang-Mills connection. If || R▽ || is constant, a smooth connection ▽ is an exponential Yang-Mills connection if it is a Yang-Mills one. We show for any vectori bundle E, that the functional Y Me admits a minimising connection ▽ which is Cα-Holder continuous for all 0 < α <1. We show the existence theorem of a smooth exponential Yang-Mills connection and study its properties and the second variation formula.

Biharmonic Homogeneous Submanifolds in Compact Symmetric Spaces

This paper is a survey of our recent works on biharmonic homogeneous submanifolds in compact symmetric spaces (Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups (in preparation), Biharmonic homogeneous hypersurfaces in compact symmetric spaces. Differ Geom Appl 43, 155–179 (2015)) [12, 13]. We give a necessary and sufficient condition for an isometric immersion whose tension field is parallel to be biharmonic. By this criterion, we study biharmonic orbits of commutative Hermann actions in compact symmetric spaces, and give some classifications.