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.

Since S is compact, we may assume that the (constant) inward mean curvature H is positive.

For unit scaling, A_t = t²A, V_t = t³V; when t=1, dA/dt = 2A, dV/dt = 3V, A′ = dA/dV = (2/3)A/V.

For a constant unit normal variation, A′ = 2H.

Equilibrium says that initially A′|scaling =  A′|normal, so A = 3VH.

The stability hypothesis is nonnegative second variation, which implies that

(*)            A″|scaling + A″|normal –2A″|mixed ≥ 0.

A”|scaling = (2/3)((2/3)(A/V)V–A)/V² = –(2/9)A/V².

By the second variation formula (or the fact that dH/dt = -.5|σ|²),

A″|normal = –A^–2∫|σ|² ≤ –2A^–2H²A = –(2/9)A/V²,

with equality if and only if umbilic (hence round sphere).

As scaling derivative of normal derivative,

A″|mixed = –2H/3V =  –(2/9)A/V² .

Alternatively, as normal derivative of scaling derivative,

A″|mixed = (2/3)((2/3)(A/V)V–A)/V² = –(2/9)A/V².

Hence equality holds in (*) and S is a round sphere. QED

Cones [MR]. In cone C over M, same except dH/dt = -.5|σ|² – Ric,

A″|normal = –A^–2∫|σ|² + Ric(n,n) ≤ –2A^–2H²A –A^–1Ric(n,n) = –(2/9)A/V² –A^–1Ric(n,n)

Hence e.g. if Ric_M > n-1, which means Ric is positive except radially 0, stable hypersurface must be completely tangential. Spheres are of course always marginally stable, indeed invariant, under the normal/scaling variation, but need not always be for all variations.

References

[BdC]   J. Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984) 339-353.

[M]   Sebastián Montiel, Stable constant mean curvature hypersurfaces in some Riemannian manifolds, Comment. Math. Helv. 73 (1998), 584-602. The proof is flawed by assuming without justification that φ = f′ is constant (top of p. 596), which holds just for cones.

[MR]   Frank Morgan and Manuel Ritoré, Isoperimetric regions in cones, Trans. Amer. Math. Soc. 354 (2002), 2327-2339.

[P]   Bennett Palmer, Stability of the Wulff shape, Proc. AMS 126 (1998) 3661-3667. See also his arXiv post (2011) on piecewise smooth surfaces.

[T]  Y. Tashiro, Complete Riemannian manifolds and some vectorfields, Trans. Amer. Math. Soc., 117 (1965), 251-275.

[W]   Henry C. Wente, A note on the stability theorem of J. L. Barbosa and M. do Carmo for closed surfaces of constant mean curvature, Pacific J. Math 147 (1991) 375-379.