Ricci流及其几何应用

This project studies some problems inspired by the geometry of Ricci flow in two directions. The first direction concentrates on the construction and classification of ancient solutions of Ricci flow. We want to extend Fateev's general explicit construction on 3-sphere to high dimensional spheres, as well as an explicit noncompact ancient solution. Then try to classify those which admit curvature conditions or symmetry by group actions, such as noncompact ancient solutions of type II with positive curvature operater or homogeneous ancient solutions on spheres or complex projective spaces. The second direction is to study Bohm-Wilking's invariant cone under Ricci flow. Suppose the Riemannian curvature tensor of an initial metric lies on the boundary of Bohm-Wilking invariant cone, consider if the Ricci flow converges to a symmetric space. Extend Brendle-Schoen's strong maximum principles to cover the case of Bohm-Wilking and investigate the connection with Riemannian holonomy group. Mean while, applying this extension to verify Bohm-Wilking's conjecture on the regidity of Einstein manifolds whose Riemannian curvature tensor lies on the boundary of invariant cone of Ricci flow. Also derive an explicit expression of Harnack inequalities for general invariant cone, especially for Riemannian 3-manifolds with non-negative Ricci curvature. Find high dimensional generalization of Hamilton's pinching estimates for four-manifolds with positive isotropic curvature.

本项目主要研究Ricci流中两方面的几何问题. 第一, 研究Ricci流的远古解的具体构造及分类. 首先将Fateev三维球上的一般远古解推广至高维球面. 尝试构造非紧的远古解. 对于具有曲率限制或拥有对称的远古解进行分类, 如具有正曲率算子的解或球面和复射影空间的齐性解. 第二, 研究Bohm-Wilking不变锥。若初始度量在锥的边界，考察Ricci流是否收敛于对称空间。 推广Brendle-Schoen的强极值原理至Bohm-Wilking不变锥的情形并考察与和乐群之间的联系。同时尝试证明Bohm-Wilking关于Einstein流形刚性的一个猜测。 在曲率满足Bohm-Wilking锥的条件下， 推广Hamilton的关于Ricci流的Harnack不等式， 特别是具有非负Ricci 曲率三维流形。推广Hamilton的关于四维流形具正迷向曲率的挤压估计至高维。