Simons inequality

2020 Mathematics Subject Classification: Primary: 53A10 [MSN][ZBL]

An inequality proved by Simons in his fundamental work [Si] on minimal varieties, which played a pivotal role in the solution of the Bernstein problem. The inequality bounds from below the Laplacian of the square norm of the second fundamental form of a minimal hypersurface $\Sigma$ in a general Riemannian manifold $N$ of dimension $n+1$. More precisely, if $A$ denotes the second fundamental form of $\Sigma$ and $|A|$ its Hilbert-Schmidt norm, the inequality states that, at every point $p\in \Sigma$, \[ \Delta_\Sigma |A|^2 (p) \geq - C (1 + |A|^2 (p))^2 \] where $\Delta_\Sigma$ is the Laplace operator on $\Sigma$ and the constant $C$ depends upon $n$ and the Riemannian curvature of the ambient manifold $N$ at the point $p$. When $N$ is the Euclidean space, a more precise form of the inequality is \[ \Delta_\Sigma |A|^2 \geq - 2 |A|^4 + 2 \left(1+\frac{2}{n}\right) |\nabla_\Sigma |A||^2 \] (see Lemma 2.1 of [CM] for a proof and [SSY] for the case of general ambient manifolds). Moreover, the inequality is an identity in the special case of $2$-dimensional minimal surfaces of $\mathbb R^3$ (cf. [CM]).

The inequality was used by Simon in [Si] to show, among other things, that stable minimal hypercones of $\mathbb R^{n+1}$ must be planar for $n\leq 6$ and it was subsequently used to infer curvature estimates for stable minimal hypersurfaces, generalizing the classical work of Heinz [He], cf. [SSY], [CS] and [SS]. Simons also pointed out that there is a nonplanar stable minimal hypercone in $\mathbb R^8$, cf. Simons cone.

References

[CS]	H. I. Choi, R. Schoen, "The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature", Invent. Math., 81, (1985) pp. 387-394.
[CM]	T. H. Colding, W. P. Minicozzi III, "A course in minimal surfaces", Graduate Studies in Mathematics, AMS, (2011).
[He]	E. Heinz, "Ueber die Loesungen der Minimalflaechengleichung" Nachr. Akad. Wiss. Goettingen Math. Phys. K1 ii, (1952) pp. 51-56
[SS]	R. Schoen, L. Simon, "Regularity of stable minimal hypersurfaces" Comm. Pure App. Math., 34, (1981), pp. 741-797.
[SSY]	R. Schoen, L. Simon, S. T. Yau, "Curvature estimates for minimal hypersurfaces" Acta Math., 132 (1975) pp. 275-288
[Si]	J. Simons, "Minimal varieties in riemannian manifolds" Ann. of Math., 88 (1968) pp. 62-105 MR233295 Zbl 0181.49702