Riemann流形上的一类标准共形不变度量  

Canonical conformally invariant metrics on Riemann manifolds

作　　者：施云 SHI Yun(School of Science,Zhejiang University of Science and Technology,Hangzhou 310023,China)

机构地区：[1]浙江科技学院理学院,浙江杭州310023

出　　处：《高校应用数学学报（A辑）》2021年第3期342-348,共7页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(11801508;11971425)。

摘　　要：任意紧Riemann面上都存在一个仅依赖于共形类且拥有常曲率的度量.Harbermann和Jost用Yamabe算子对应的Green函数在数量曲率为正的局部共形平坦流形上构造了一个标准共形不变度量.在此之后,这类标准共形不变度量被推广到了数量曲率为正的球型CR流形上.进一步的,应用相应的Yamabe算子对应的Green函数可以构造数量曲率为正的球型四元切触流形和数量曲率为正的八元切触流形上类似的标准共形不变张量.在四元切触正质量猜测和八元切触正质量猜测成立的前提下,上述共形不变张量是共形不变度量.文中利用Paneitz算子对应的Green函数在局部共形平坦流形上构造了一类上述标准共形不变张量,并且在一定条件(详见定理3.1)下,该标准共形不变张量进一步为标准共形不变度量.Each compact Riemann surface has a metric with constant curvature, which only depends on the conformal class of Riemann surfaces. Harbermann and Jost use the Green function of the Yamabe operator to construct a canonical metric on each scalar positive locally conformally flat manifolds, which only depends on the conformal class of such manifolds. This construction was generalized to the scalar positive spherical CR manifolds. Recently such conformally invariant tensor was also generalized to scalar positive spherical quaternionic contact manifolds and scalar positive octonionic contact manifolds. They become spherical quaternionic contact metric and octionionic contact metric if the quaternionic contact positive mass conjecture and the octionionic contact positive mass conjecture are true, respectively. In this paper, the Green function of the Paneitz operator is used to construct a canonical conformally invariant tensor on locally conformally flat manifolds. Moreover,it becomes a canonical metric under some conditions(cf. Theorem 3.1).

关 键 词：共形不变量 Paneitz算子 GREEN函数 Q曲率 

分 类 号：O184[理学—数学]