An existence theorem on the isoperimetric ratio over scalar-flat conformal classes

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Chen, Xuezhang^1,2  Jin, Tianling³  Ruan, Yuping^1,2,4 

[1] Nanjing Univ, Dept Math, Nanjing 210093, Peoples R China

[2] Nanjing Univ, IMS, Nanjing 210093, Peoples R China

[3] Hong Kong Univ Sci & Technol, Dept Math, Kowloon, Clear Water Bay, Hong Kong, Peoples R China

[4] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

关键词: Conformal geometry;    Isoperimetric inequality;   

DOI  :  10.1016/j.jde.2020.03.025

来源: Elsevier

【 摘 要 】

Let (M, g) be a smooth compact Riemannian manifold of dimension n with smooth boundary partial derivative M, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean ball, and consequently is achieved, if either (i) 9 <= n <= 11 and partial derivative M has a nonumbilic point; or (ii) 7 <= n <= 9, partial derivative M is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work [12] by the second named author and Xiong. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free