Curvature Estimates for Graphs Over Riemannian Domains
| Autor: | Fabiani Aguiar Coswosck, Francisco Fontenele |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Mean curvature
Second fundamental form 010102 general mathematics Riemannian manifold Curvature 01 natural sciences Combinatorics Differential geometry 0103 physical sciences Mathematics::Differential Geometry 010307 mathematical physics Geometry and Topology Sectional curvature 0101 mathematics Ricci curvature Mathematics Scalar curvature |
| Zdroj: | The Journal of Geometric Analysis. 31:5687-5720 |
| ISSN: | 1559-002X 1050-6926 |
| DOI: | 10.1007/s12220-020-00497-w |
| Popis: | Let $$M^n$$ be a complete n-dimensional Riemannian manifold and $$\Gamma _f$$ 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 $${\mathbb {R}}^3$$ which is a graph over an open disk in the plane, we obtain in this work upper estimates for $$\inf |R|$$ , $$\inf |A|$$ and $$\inf |H_k|$$ , where R, |A| and $$H_k$$ are, respectively, the scalar curvature, the norm of the second fundamental form and the k-th mean curvature of $$\Gamma _f$$ . From our estimates we obtain several results for graphs over complete manifolds. For example, we prove that if $$M^n,\;n\ge 3,$$ is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constant c, and $$\Gamma _f$$ is a graph over M with Ricci curvature less than c, then $$\inf |A|\le 3(n-2)\sqrt{-c}$$ . This result generalizes and improves a theorem of Chern for entire graphs in $$\mathbb R^{n+1}$$ . |
| Databáze: | OpenAIRE |
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