## Stable Immersions Round

Barbosa and do Carmo [BdC] proved that *a compact, stable, oriented, immersed constant-mean-curvature surface S in R*

^{3}

*is umbilic and hence a round sphere.*The proof works for hypersurfaces in

**R**

^{n}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} = t^{2}A, V_{t} = t^{3}V; 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)/V^{2} = –(2/9)A/V^{2}.

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

A″|normal = –A^{–2}∫|σ|^{2} ≤ –2A^{–2}H^{2}A = –(2/9)A/V^{2},

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

As scaling derivative of normal derivative,

A″|mixed = –2H/3V = –(2/9)A/V^{2} .

Alternatively, as normal derivative of scaling derivative,

A″|mixed = (2/3)((2/3)(A/V)V–A)/V^{2} = –(2/9)A/V^{2}.

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^{–2}H^{2}A –A^{–1}Ric(n,n) = –(2/9)A/V^{2 }–A^{–1}Ric(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.