arXiv subjects
Publications and source records attributed to Katsuei Kenmotsu.
We provide an elementary proof of a lemma that plays an important role in the classification of parallel mean curvature surfaces in two-dimensional complex space forms.
Two necessary conditions for the induced metrics of parallel mean curvature surfaces in a complex space form of complex two-dimension are observed. One is similar to the Ricci condition of the classical surface theory in Euclidean three-space and the other is related to the Liouville metric. Conversely, we prove that a special type of the Liouville metric on a domain in the Euclidean two-plane whose Gaussian curvature satisfies the differential equation similar to the Ricci condition is explicitly determined by an elliptic function. We have isometric immersions from a simply connected two-dimensional Riemannian manifold with the special type of the Liouville metric satisfying the Ricci condition to the complex hyperbolic plane with parallel mean curvature vector.
The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: When the curvature of the ambient space is positive, there exists a unique such surface up to rigid motions of the target space. On the other hand, when the curvature of the ambient space is negative, there are `non-trivial' complete parallel mean curvature surfaces generated by Jacobi elliptic functions and they exhaust such surfaces.
We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane
We prove that any piece of a rotational hypersurface with prescribed mean curvature function in a Euclidean space can be uniquely extended infinitely, which generalizes the results by Euler and Delaunay for surfaces of revolution with constant mean curvautre. Next, we prove the same kind of theorem for generalized rotational hypersurfaces of O(l+1) x O(m+1)-type. The key lemmas in this paper show the existence of solutions for singular initial value problems which arise from the analysis of ordinary differential equations of generating curves of those hypersurfaces.