arXiv subjects
Publications and source records attributed to J. -H. Eschenburg.
In this article we investigate some properties of equivariant embeddings of a symmetric Kählerian manifold. Motivated by a theorem of Cartan and Wallach on equivariant embeddings of symmetric spaces we characterize these embeddings in the special case of $\mathbb{CP}^n$. Further, we verify that if a equivariant embedding has parallel plurimean curvature then it is the extrinsically symmetric one.
Real flag manifolds are the isotropy orbits of noncompact symmetric spaces $G/K$. Any such manifold $M$ enjoys two very peculiar geometric properties: It carries a transitive action of the (noncompact) Lie group $G$, and it is embedded in euclidean space as a taut submanifold. The aim of the paper is to link these two properties by showing that the gradient flow of any height function is a one-parameter subgroup of $G$, where the gradient is defined with respect to a suitable homogeneous metric $s$ on $M$; this generalizes the Kaehler metric on adjoint orbits (the so-called complex flag manifolds).
We investigate the local geometry of a class of Kähler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $α$, which we call the pluri-mean curvature. We show that these Kähler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic.