Multiplicity of solutions to the Yamabe equation on warped products

JOURNAL OF GEOMETRY AND PHYSICS

卷:138

Article

Miguel Ruiz, Juan¹ 

[1] ENES UNAM, Leon 37684, Gto, Mexico

关键词: Yamabe problem;    Bifurcation theory;    Constant scalar curvature;    Nonuniqueness of solutions;   

DOI  :  10.1016/j.geomphys.2018.12.013

来源: Elsevier

【 摘 要 】

Let (M-m, g(M)) be a closed (compact, without boundary) connected manifold with positive scalar curvature and (F-k, h) a closed connected manifold with constant scalar curvature (m >= 3 and k > 3). By a Theorem of Dobarro and Lami Dozo (1987), there are weights f : M -> R+ such that the warped product (M-m x F-k, g(M)+ f(2)h) has constant scalar curvature. We construct paths of warped product metrics (M-m x F-k, g(M) + f(epsilon)(2)h ), epsilon is an element of (0, epsilon(0)), epsilon(0) small, with constant scalar curvature, that exhibit multiplicity of solutions to the Yamabe equation. Moreover, in the case that (F-k, h) has a flat metric we add the constraints of unit volume and fixed constant scalar curvature to the construction of paths of warped metrics (M-m x F-k, g(M) +f(2)h(epsilon)), epsilon is an element of (0, epsilon(0)), that exhibit multiplicity. We use techniques from bifurcation theory along with spectral theory for warped products. (C) 2018 Elsevier B.V. All rights.

【 授权许可】

Free