截面曲率有上界的4维收缩的梯度Ricci孤立子

基金项目:

正迷向曲率流形上Ricci流的奇点分析

详细信息

On Shrinking 4-Solitons with Sectional Curvature Bounded Above

摘要: 分析梯度 Ricci soliton的几何性质，是运用Ricci流理论去解决微分几何问题的重要一步。在本文中，作者利用标准的极值原理来探讨 4 维的 shrinking 梯度 Ricci soliton的几何性质，获得了soliton的一个重要的曲率估计。具体地说，在一个紧致的 shrinking 4-soliton 上，如果截面曲率有恰当的上界，那么其 Ricci 曲率一定是非负的。如果 soliton 不是紧致的，但是进一步要求数量曲率有界且有正的下界，那么类似的结论成立。特别的，结论中截面曲率的上界是最优的。
- 极值原理.
Abstract: To understand the geometric properties of shrinking gradient Ricci solitons, is an important step as one using the Ricci flow theory to solve the differential geometric problems. In this paper, the author studies the geometric properties of shrinking 4-solitons by using the standard maximum principle, and obtains an important curvature estimate. More precisely, on a compact shrinking 4-soliton, if the sectional curvature has a suitable upper bound, then the Ricci curvature should be nonnegative. When the soliton is non-compact, if in addition, the scalar curvature is bounded and has a positive lower bound, then a similar result will hold. In particular, the upper bound of the sectional curvature is optimal.
- maximum principle.