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

Abstract

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