View source for Simons inequality

{{MSC|53A10}}
[[Category:Differential geometry]]
[[Category:Partial differential equations]]
{{TEX|done}}

An inequality proved by Simons in his fundamental work {{Cite|Si}} on minimal varities, which played a pivotal role in the solution of the [[Bernstein problem in differential geometry|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 {{Cite|CM}} for a proof and {{Cite|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. {{Cite|CM}}).

The inequality was used by Simon in {{Cite|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 {{Cite|He}}, cf. {{Cite|SSY}}, {{Cite|CS}} and {{Cite|SS}}. Simons also pointed out that there is a nonplanar stable minimal hypercone in $\mathbb R^8$, cf. [[Simons cone]].

====References====
{|
|-
|valign="top"|{{Ref|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.
|-
|valign="top"|{{Ref|CM}}|| T. H. Colding, W. P. Minicozzi III, "A course in minimal surfaces", Graduate Studies in Mathematics, AMS, (2011).
|-
|valign="top"|{{Ref|He}}|| E. Heinz, "Ueber die  Loesungen   der Minimalflaechengleichung" ''Nachr. Akad. Wiss. Goettingen Math.  Phys.'' K1 ii, (1952) pp. 51-56
|-
|valign="top"|{{Ref|SS}}|| R. Schoen, L. Simon, "Regularity of stable minimal hypersurfaces" ''Comm. Pure App. Math.'', '''34''', (1981), pp. 741-797.
|-
|valign="top"|{{Ref|SSY}}|| R. Schoen, L. Simon, S. T. Yau, "Curvature estimates for minimal hypersurfaces" ''Acta Math.'', '''132''' (1975) pp. 275-288
|-
|valign="top"|{{Ref|Si}}||  J. Simons, "Minimal varieties in riemannian manifolds" ''Ann. of  Math.'', '''88''' (1968) pp. 62-105 {{MR|233295}} {{ZBL|0181.49702}}
|-
|}