Stable constant mean curvature surfaces minimize area
It has become customary to describe an extremal configuration for a stationary variational problem as ``stable'' if the second variation is nonnegative, and as ``strictly stable'' if the second variation is positive definite. These can be unfortunate misnomers, just as the designation ``extremal'' can be a misnomer, as in general the positivity of second variation for an extremal need not imply that the extremal yield a local minimum, even in a weak sense. For the area functional however, a more favorable situation prevails; in view of the special properties of foliations of a neighborhood of a minimal surface by extremals, positivity of second variation implies significant minimizing properties for the surface. The present paper concerns surfaces of constant mean curvature \(H\) (not necessarily zero), and continues earlier work of \textit{B. White} [J. Reine Angew. Math. 457, 203-218 (1994; Zbl 0808.49037)] in establishing for them properties analogous to those known for minimal surfaces. Such surfaces can be viewed as extremals for the problem (i) of minimizing area \(A\) under variations that preserve a volume \(V\) subtended by the surface, or alternatively as extremals for the problem (ii) of minimizing the expression \(A+HV\) under unrestricted variations. These two problems lead to different second variations. In the present work, the author extends the foliations procedure for minimal surfaces to give a new proof of a result of White [loc. cit.] that a strictly stable extremal yields a local minimum for (ii). This also yields for (i) that a strictly stable extremal surface locally minimizers area among surfaces enclosing the same volume. The existence proof for the required foliation extends the corresponding result of \textit{W. H. Meeks} and \textit{S.-T. Yau} [Math. Z. 179, 151-168 (1982; Zbl 0479.49026)] for minimal surfaces. Additionally, the author provides an elegant proof that for the problem (i) an immersed strictly stable extremal locally minimizes area with respect to \(C^1\) deformations that are graphs over its surface.
scientific article; zbMATH DE number 1026459