TY - JOUR
T1 - An existence theorem on the isoperimetric ratio over scalar-flat conformal classes
AU - Chen, Xuezhang
AU - Jin, Tianling
AU - Ruan, Yuping
N1 - Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/8/15
Y1 - 2020/8/15
N2 - Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂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 ∂M has a nonumbilic point; or (ii) 7≤n≤9, ∂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.
AB - Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂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 ∂M has a nonumbilic point; or (ii) 7≤n≤9, ∂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.
KW - Conformal geometry
KW - Isoperimetric inequality
UR - https://www.webofscience.com/wos/woscc/full-record/WOS:000534486500004
UR - https://openalex.org/W3013203365
UR - https://www.scopus.com/pages/publications/85082178517
U2 - 10.1016/j.jde.2020.03.025
DO - 10.1016/j.jde.2020.03.025
M3 - Journal Article
SN - 0022-0396
VL - 269
SP - 4116
EP - 4136
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 5
ER -