标题： CMC叶层的平均曲率的界 显示英文标题

标题： Bounds to the mean curvature of leaves of CMC foliations

摘要： 本文的主要目标是将Barbosa、Kenmotsu和Oshikiri（1991）证明的结果及其思想带入到Ricci曲率有下界的视角中。例如，对于紧致（无边界）黎曼流形$M^{n+1}$上的CMC超曲面叶状结构，其Ricci曲率被下界$-nK_0\leq 0$所限制，并且叶状结构的平均曲率$H$满足$|H|\geq \sqrt{K_0}$，我们证明了$|H|\equiv \sqrt{K_0}$，且所有叶状结构都是全脐的。这特别给出了Barbosa、Kenmotsu和Oshikiri（1991）证明结果的一个推广，在该情况下上述结果是在$K_0=0$的情况下证明的。 我们还得到了以下结论的证明：在紧致（无边界）黎曼流形$M$上，由常平均曲率超曲面组成的叶状结构，其叶的平均曲率$H$满足$|H|\leq \sqrt{K_0}$，该流形的里奇曲率从下方有界为$-nK_0\leq 0$。 此外，如果叶状结构包含一个叶 $L$，其绝对平均曲率为 $|H_L|=\sqrt{K_0}$，则要么 $K_0=0$且 $\mathfrak{F}$的所有叶都是全测地的，要么 $K_0>0$并且存在一个全脐叶。 显示英文摘要

摘要： The main goal of this present paper is to bring the results proved by Barbosa, Kenmotsu and Oshikiri (1991) and its ideas to a perspective where the Ricci curvature is bounded from below. For instance, for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold $M^{n+1}$ with Ricci curvature bounded from below by $-nK_0\leq 0$ and such that the mean curvature $H$ of the leaves of the foliation satisfies $|H|\geq \sqrt{K_0}$, we prove that $|H|\equiv \sqrt{K_0}$ and all the leaves are totally umbilical. This gives, in particular, a generalization for the result proved by Barbosa, Kenmotsu and Oshikiri (1991), where the above result was proved in the case $K_0=0$. We also obtain a proof of the following: for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold $M$ with Ricci curvature bounded from below by $-nK_0\leq 0$, the mean curvature $H$ of the leaves of the foliation satisfies $|H|\leq \sqrt{K_0}$. Furthermore, if the foliation contains a leaf $L$ whose absolute mean curvature is $|H_L|=\sqrt{K_0}$, then either $K_0=0$ and all the leaves of $\mathfrak{F}$ are totally geodesic, or $K_0>0$ and there is a totally umbilical leaf.