On the mean curvature of surface with boundary (Q757829)

The present study is concerned with the mean curvature of surfaces located in the Euclidean space \({\mathbb{R}}^ 3\). A great deal of interests has long been focussed upon the question in the scope of global analysis, under what condition such surfaces can possess an everywhere constant mean curvature at all. The case of compact surface, namely, of the surface without boundary was treated mainly from the standpoint of isoperimetric problem, and the satisfactory results seem to have been achieved about in the last decade. As for the case of surface with boundary, on the other hand, the theory remains still in its infancy and has much room to be investigated. Our contribution in this paper will be, above all, (a) completion of a criterion for constancy of mean curvature with respect to volume and surface-area in the conditioned variation arguments up to its dual; and (b) presentation of a new convexity theorem valid for every surface with boundary that has a constant mean curvature. To be precise, {\S} 1 explains our situation, setting the problem and prepares the tools used. In {\S} 2 we derive the first variations of volume- and area-functionals in our own terms. We state and prove, in {\S} 3, our main theorems in this paper, among which are included a duality theorem for a conditioned critical point problem in calculus of variations, as well as the convexity theorem, which answers a question raised by \textit{M. Koiso} [cf. ``Stability of surfaces with constant mean curvature in \({\mathbb{R}}^ 3\)'', Res. Rep. No.MPI/87-17, 1-29, Max-Planck-Institut f. Math. (Bonn/Ger. 1987)]. Partial reason for our restriction of the ambient space to \({\mathbb{R}}^ 3\), not to \({\mathbb{R}}^ n\) (n\(\geq 3)\) lies in an attempt to adopt a few ideas together with their wordings from classical physics, relevant to the soap bubble experiment by blowing the tube. We expose such point of view in {\S} 4, by virtue of which we have been able to obtain somewhat exacter information for the convexity than in {\S} 3.