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 \(\mathbb R^{n+1}\).

假设\（M ^ n \）是一个完整的n维黎曼流形，并且\（\ Gamma _f \）是在（（M ^ n \）度量球上定义的\（C ^ 2 \）函数f的图）。就像海因茨针对\（{\ mathbb {R}} ^ 3 \）中曲面的平均值和高斯曲率所做的估算一样，我们在这项工作中获得\（\ inf | R | \），\（\ inf | A | \）和\（\ inf | H_k | \）的最高估计值，其中R，| A | 和\（H_k \）分别是标量曲率，第二基本形式的范数和\（\ Gamma _f \）的第k个平均曲率。从我们的估计中，我们可以获得完整流形上图形的几个结果。例如，我们证明如果\（M ^ n，\; n \ ge 3，\）是一个完整的非紧黎曼流形，且其截面曲率在下面受常数c限制，并且\（\ Gamma _f \）是一个图M且Ricci曲率小于c，则\（\ inf | A | \ le 3（n-2）\ sqrt {-c} \）。该结果推广了\（\ mathbb R ^ {n + 1} \）中整个图的Chern定理并对其进行了改进。