(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

Soit M n compacte, orientable, et soit x:M n →R n+1 une immersion a courbure moyenne constante non nulle. Alors x est stable si et seulement si x(M)⊂R n+1 est une sphere S n ⊂R n+1

Stability of hypersurfaces with constant mean curvature

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Almost contact structures and curvature tensors

Let \(x:{\text M}^{n}\rightarrow\,{\text R}^{n+1}\) be an immersion of an orientable, n-dimensional manifold \({\text M}^{n}\) into the euclidean space \({\text R}^{n+1}\). The condition that x has nonzero constant mean curvature H = H 0 is known to be equivalent to the fact that xis a critical point of a variational problem.

Stability of Hypersurfaces with Constant Mean Curvature

Harmonic maps from surfaces to complex projective spaces

Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain

/pdf/a-class-of-almost-contact-riemannian-manifolds-1yxyggdui1.pdf

A class of almost contact riemannian manifolds

Weierstrass formula for surfaces of prescribed mean curvature

In this paper we study a surface of revolution in the Euclidean three space Λ. The generating curve of the surface satisfies a nonlinear differential equation which describes the mean curvature. The purpose of this note is to solve the differential equation by an elementary method. Solutions are represented explicitly by generalized FresnePs integrals which involve the mean curvature. Therefore, for a given continuous function H(s), we can construct a 3-parameter family of surfaces of revolution admitting H(s) as the mean curvature. We shall remark that these computations are different from the one in Delaunay [1]. About 140 years ago, he solved the differential equation under the constancy of the mean curvature and gave a method of geometric constructions for such surfaces. For the proof, he first obtains a solution of an evolute of the generating curve. By making use of this solution, he found a representation formula of the generating curve. Therefore these solutions hold only on some intervals on which the evolute can be defined. It seems not to be simple to obtain a global solution from his method. Our calculations will easily find global solutions and the corollary of the main theorem of this note describes all complete surfaces of revolution with constant mean curvature. During this research I stayed in Kδln and received many nice advices from Professors Peter Dombrowski and Helmut Reckziegel. By their suggestions, my original computations could be simplified and generalized. And also I shall mention that Reckziegel got interested in drawing graphs of these generating curves by making use of the computer and obtained many beautiful pictures. He permitted me with favor to include some of the graphs by his programming in this paper. The author wishes to express his deep gratitude to both of them for helpful

Katsuei Kenmotsu

Papers

A class of almost contact riemannian manifolds

Weierstrass formula for surfaces of prescribed mean curvature

Surfaces of revolution with prescribed mean curvature

Surfaces With Constant Mean Curvature

On minimal immersions of R2 into Pn(C)