Volume 22, issue 6 (2018)

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A formal Riemannian structure on conformal classes and uniqueness for the $\sigma_2$–Yamabe problem

Matthew Gursky and Jeffrey Streets

Geometry & Topology 22 (2018) 3501–3573
Abstract

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the ${\sigma }_{2}$–Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.

Keywords
fully nonlinear Yamabe problem, uniqueness
Mathematical Subject Classification 2010
Primary: 58J05
Secondary: 53C44, 58B20