An index characterization of the catenoid and index bounds for minimal surfaces in R⁴

Shiu-Yuen Cheng and Johan Tysk

Vol. 134 (1988), No. 2, 251–260

DOI: 10.2140/pjm.1988.134.251

Abstract

The index of a minimal surface is defined to be the number of negative eigenvalues of the operator corresponding to second variation of area. In the present paper, we characterize the catenoid as the only complete oriented minimal surface in R³ of index one with embedded ends. We also obtain upper bounds for the index of minimal surfaces in R⁴, in terms of the total curvature of the surface.

Mathematical Subject Classification 2000

Primary: 53A10

Secondary: 49F10

Milestones

Received: 2 June 1987

Published: 1 October 1988