Curvature Estimates for Graphs Over Riemannian Domains
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Curvature Estimates for Graphs Over Riemannian Domains Fabiani Aguiar Coswosck1 · Francisco Fontenele2 Received: 9 May 2017 © Mathematica Josephina, Inc. 2020
Abstract Let M n be a complete n-dimensional Riemannian manifold and f the graph of a C 2 -function f defined on a metric ball of M n . In the same spirit of the estimates obtained by Heinz for the mean and Gaussian curvatures of a surface in R3 which is a graph over an open disk in the plane, we obtain in this work upper estimates for inf |R|, inf |A| and inf |Hk |, where R, |A| and Hk are, respectively, the scalar curvature, the norm of the second fundamental form and the k-th mean curvature of f . From our estimates we obtain several results for graphs over complete manifolds. For example, we prove that if M n , n ≥ 3, is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constant c,√and f is a graph over M with Ricci curvature less than c, then inf |A| ≤ 3(n − 2) −c. This result generalizes and improves a theorem of Chern for entire graphs in Rn+1 . Keywords Graphs over Riemannian domains · Scalar curvature · Higher order mean curvatures · Norm of the second fundamental form Mathematics Subject Classification Primary 53C42 · Secondary 53A10
1 Introduction In a well known paper, Heinz [19] obtained the following estimates for the mean curvature H and Gaussian curvature K of a surface in R3 which is the graph of a smooth function defined on an open disk of radius r in the plane:
Fabiani Coswosck: Supported by CAPES (Brazil). Francisco Fontenele: Partially supported by CNPq (Brazil).
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Francisco Fontenele [email protected] Fabiani Aguiar Coswosck [email protected]
1
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, RJ, Brazil
2
Departamento de Geometria, Universidade Federal Fluminense, Niterói, RJ, Brazil
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F. Coswosck
inf |H | ≤
1 , r
inf |K | ≤
3e2 . r2
(1.1)
Independently, Chern [5] and Flanders [11] extended the first of the above inequalities to higher dimensions. As an immediate consequence one has that an entire constant mean curvature graph in Rn+1 is minimal (see Corollary 1.2 for a generalization of this result). Finn [10] generalized the first inequality of (1.1) to a broader class of domains of the plane. More precisely, he showed that if f : D ⊂ R2 → R is a smooth function defined on a bounded domain D with smooth boundary ∂ D, then the mean curvature H of the graph of f satisfies inf |H | ≤
l , 2A
where A is the area of D and l the length of ∂ D. Generalizing this estimate, as well as the estimate of Heinz–Chern–Flanders referred to above, Salavessa [23,24] proved that if f : M n → R is a smooth function defined on a Riemannian manifold M n then, for each oriented compact domain D ⊂ M with smooth boundary ∂ D, the mean curvature H of the graph of f satisfies inf |H | ≤ D
1 A(∂ D) , n V (D)
(1.2)
where A(∂ D) and V (D) are, respectively, the area of ∂ D and the volume of D, with respect to the metric of M (see in [1] a version of the first ineq
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