Minimal surfaces and surfaces with constant mean curvature in the Euclidean space \(R^ 3\) are both critical points of the area function for certain variations; in the first case, the authors considers compactly supported variations and in the second case, they consider those variations that, in addition, preserve a certain “volume”. Thus, problems on stability and indices of their Jacobi operators L, acting on spaces determined by the corresponding variations, arise naturally. Here the index of L in a bounded domain is the number (with multiplicities) of negative eigenvalues of L; the definition is extended to a complete surface by taking the supremum of the indices over all bounded subdomains of the surface.
For complete and orientable minimal surfaces in \(R^ 3\) it has been shown that the index is finite iff the total curvature is finite. [See D. Fischer-Colbrie, Invent. Math. 82, 121-132 (1985; Zbl 0573.53038), R. Gulliver and H. Lawson jun., Geometric measure theory and the calculus of variations, Proc. Summer Inst., Arcata/Calif 1984, Proc. Symp. Pure Math. 44, 213-237 (1986; Zbl 0592.53005), and R. Gulliver, ibid., 207-211 (1986; Zbl 0592.53006)]. Furthermore, it is easily checked that the catenoid and the Enneper’s surface have index equal to one. This paper gives the first proof that these are the only complete orientable minimal surfaces with index one. It is also proved in the paper that the only stable, complete non-compact, orientable surface with constant mean curvature in \(R^ 3\) is the plane. This result was obtained independently by A. M. da Silveira [Math. Ann. 227, 629- 638 (1987; Zbl 0627.53045)] and D. Palmer (Ph. D. thesis, Stanford).