Estimates for stable minimal surfaces in three-dimensional manifolds. (English) Zbl 0532.53042

Schoen, Richard

Semin. on minimal submanifolds, Ann. Math. Stud. 103, 111-126 (1983).

[For the entire collection see Zbl 0521.00014.]
Let \(N^ 3\) be a three-dimensional Riemannian manifold and f: \(M^ 2\to N^ 3\) a stable minimal surface in \(M^ 3\). The author derives various clever estimates for geometrical quantities associated to this situation. In particular, if N has nonnegative Ricci curvature, an estimate is obtained that implies immediately that M is totally geodesic, or does not exist in case the Ricci curvature is everywhere positive [D. Fischer-Colbrie and the author, Commun. Pure Appl. Math. 33, 199-211 (1980; Zbl 0439.53060)], and for the case \(N=R^ 3\), cf. the reviewer and C. K. Peng [Bull. Am. Math. Soc., New Ser. 1, 903-906 (1979; Zbl 0442.53013)].

Reviewer: M.P.do Carmo

MSC:

53C42

Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

stable minimal surface; totally geodesic; Ricci curvature

Citations:

Zbl 0521.00014; Zbl 0439.53060; Zbl 0442.53013