A note on the stability theorem of J. L. Barbosa and M. Do Carmo for closed surfaces of constant mean curvature

Henry Wente

Vol. 147 (1991), No. 2, 375–379

DOI: 10.2140/pjm.1991.147.375

Abstract

The theorem of Barbosa and do Carmo asserts that the only stable compact hypersurface of constant mean curvature in Rⁿ⁺¹ is the round n-sphere. We present an elementary proof of this fact by considering the 2-parameter family y = s(x + tξ) where x is the constant mean curvature immersion and ξ is the unit normal vector field.

Mathematical Subject Classification 2000

Primary: 53A10

Secondary: 53C42

Milestones

Received: 15 March 1989

Published: 1 February 1991