The scalar curvature flow on Sn-perturbation theorem revisited

• 成果类型: Article
• 署名作者: Chen, Xuezhang; Xu, Xingwang
• 署名单位: National University of Singapore; Nanjing University; Nanjing University; Peking University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0335-6
• 发表日期: 2012
• 页码: 395-506
• 关键词: yamabe flow; global existence; CONVERGENCE; equation; METRICS

摘要

For the problem of finding a geometry on S-n, for n >= 3, with a prescribed scalar curvature, there is a well- known result which is called the perturbation theorem; it is due to Chang and Yang (Duke Math. J. 64, 27-69, 1991). Their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm. It is important to know how large that difference in sup norm can possibly be. Here we consider prescribing scalar curvature problem using the scalar curvature flow. For simplicity, we assume that the given curvature candidate f is a smooth positive Morse function which is non-degenerate in the sense that vertical bar del f vertical bar S-n(2) + (Delta S(n)f)(2) not equal 0. For delta(n) = 2(2/n) when n = 3,4 and delta(n) = 2(2/(n-2)) for n >= 5, we show that if max(S)n f/min(S)n f < delta(n), then f can be realized as the scalar curvature of some conformal metric provided that the degree counting condition holds for f. This shows that the best possible difference in the sup norm is n(n - 1)(delta(n) - 1)/(delta(n) + 1).