论文标题
特征数字的卷比较定理
A volume comparison theorem for characteristic numbers
论文作者
论文摘要
我们表明，假设RICCI曲率上的下限和Indexitivity Radius riemannian歧管的某些特征数的绝对值，包括所有pontryagin和Chern数字，与体积成比例地界定。证明依赖于Chern-Weil理论适用于根据图表上的欧几里得连接构建的连接，在该图表中，公制张量是谐波并具有限制HölderNorm的连接。 我们将此定理概括为Gromov-Hausdorff封闭的封闭类的粗糙riemannian歧管类，这些歧管是根据Hölder的规律性定义的。假设额外的上部RICCI曲率结合，我们表明欧拉的特征也与体积成比例地界定。此外，我们对贝蒂数量的歧管数字进行了评估，并在截面曲率上具有额外的上限。这是鲍恩结果的结果。
We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern-Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded Hölder norm. We generalize this theorem to a Gromov-Hausdorff closed class of rough Riemannian manifolds defined in terms of Hölder regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.
