发表机构
Universität Potsdam（波茨坦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文探讨格罗莫夫平均平均曲率猜想中是否需额外假设边界平均曲率下界，证实二维情形无需该假设但三维及以上维数需该假设
AI 中文摘要
格罗莫夫猜想：对于带边界的紧致黎曼流形$X$，其总平均曲率$\n\na\nb\ns\n\nt\n\ni\n\nn\n\n$\n\nt\n\nh\n\ne\n\n\int_{\partial X} H$的上界仅由$\partial X$的内蕴几何及$X$的标量曲率下界决定。此前该猜想的相关结果还额外要求边界的平均曲率下界。本文研究这一额外假设是否必要：在二维情形，证明无需测地曲率下界，通过边界长度及曲面高斯曲率下界估计其总测地曲率，从而在二维无额外假设下证实格罗莫夫猜想；但给出的例子表明，在$n \geq 3$维情形，平均曲率下界是真正必需的。
英文摘要
Gromov conjectured that for a compact Riemannian manifold $X$ with boundary, the total mean curvature $\int_{\partial X} H$ is bounded above by a constant depending only on the intrinsic geometry of $\partial X$ and a lower bound on the scalar curvature of $X$. Previous results towards this conjecture require, in addition, a lower bound on the mean curvature of the boundary. In the present paper, we investigate whether this extra assumption is necessary. In dimension $2$, we show that no lower bound on the geodesic curvature is needed. We estimate the total geodesic curvature of the boundary in terms of its length and a lower bound for the Gauss curvature of the surface. This confirms Gromov's conjecture in $2$~dimensions without any extra assumptions. In contrast, we give examples showing that a lower bound on the mean curvature is genuinely needed in dimensions $n \ge 3$.
Comments11 pages, 3 figures
