Frank Morgan

Barbosa and do Carmo [BdC] proved that a compact, stable, oriented, immersed constant-mean-curvature surface S in R³ is umbilic and hence a round sphere. The proof works for hypersurfaces in Rⁿ as well. The proof was simplified by Wente [W], generalized to cones by Morgan and Ritoré [MR], incorrectly generalized to warped products by Montiel [M], and generalized to smooth elliptic integrands by Palmer [P]. Tashiro [T] generalized the fact that umbilic hypersurfaces are round. Locally constant normal variations show that stable implies connected.

Here we give a streamlined version of the proof without passing through the Minkowski formulae. Continue reading ‘Stable Immersions Round’ »