三维流形上的一个新共形不变量
微分几何
2011-03-22 v1 偏微分方程分析
摘要
通过改进在\s k \s_k \s k ( M 3 , g ) (M^3, g) ( M 3 , g ) ∫ M ∣ R i c − R ˉ 3 g ∣ 2 d v ( g ) ≤ 9 ∫ M ∣ R i c − R 3 g ∣ 2 d v ( g ) , \int_M |Ric-\frac{\bar R} 3 g|^2 dv (g)\le 9\int_M |Ric-\frac{R} 3 g|^2dv(g), ∫ M ∣ R i c − 3 R ˉ g ∣ 2 d v ( g ) ≤ 9 ∫ M ∣ R i c − 3 R g ∣ 2 d v ( g ) , R ˉ = v o l ( g ) − 1 ∫ M R d v ( g ) \bar R=vol (g)^{-1} \int_M R dv(g) R ˉ = v o l ( g ) − 1 ∫ M R d v ( g ) g g g R R R ( M 3 , g ) (M^3,g) ( M 3 , g ) \ds Y ~ ( [ g 0 ] ) : = sup g ∈ C 1 ( [ g 0 ] ) \ds v o l ( g ) ∫ M \s 2 ( g ) d v ( g ) \ds ( ∫ M \s 1 ( g ) d v ( g ) ) 2 , \ds \widetilde Y([g_0]):=\sup_{g\in {\cal C}_1([g_0])}\frac {\ds vol(g)\int_M \s_2(g) dv(g)} {\ds (\int_M \s_1(g) dv(g))^2}, \ds Y ([ g 0 ]) := g ∈ C 1 ([ g 0 ]) sup \ds ( ∫ M \s 1 ( g ) d v ( g ) ) 2 \ds v o l ( g ) ∫ M \s 2 ( g ) d v ( g ) , C 1 ( [ g 0 ] ) : = { g = e − 2 u g 0 ∣ R > 0 } {\cal C}_1([g_0]):=\{g=e^{-2u}g_0\,|\, R>0\} C 1 ([ g 0 ]) := { g = e − 2 u g 0 ∣ R > 0 } Y ~ ( [ g 0 ] ) ≤ 1 / 3 \widetilde Y([g_0])\le 1/3 Y ([ g 0 ]) ≤ 1/3
引用
@article{arxiv.1103.3838,
title = {A new conformal invariant on 3-dimensional manifolds},
author = {Yuxin Ge and Guofang Wang},
journal= {arXiv preprint arXiv:1103.3838},
year = {2011}
}
备注
23 pages