Bounds on the topology and index of minimal surfaces. (English) Zbl 1428.53018

Meeks, William H.; Pérez, Joaquín; Ros, Antonio

Acta Math. 223, No. 1, 113-149 (2019).

Summary: We prove that for every non-negative integer \(g\), there exists a bound on the number of ends of a complete, embedded minimal surface \(M\) in \(\mathbb{R}^3\) of genus \(g\) and finite topology. This bound on the finite number of ends when \(M\) has at least two ends implies that \(M\) has finite stability index which is bounded by a constant that only depends on its genus.

MSC:

53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05	Minimal surfaces and optimization
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

minimal surface; index of stability; curvature estimates; finite total curvature; minimal lamination; removable singularity