On the local Nirenberg problem for σ_k-type curvatures

Philippe Delanoë

Vol. 234 (2008), No. 2, 289–294

DOI: 10.2140/pjm.2008.234.289

Abstract

This note is about the Nirenberg problem for a class of second-order fully nonlinear scalar curvature operators, namely those that are nondegenerate symmetric functions of the eigenvalues of the Schouten tensor. Near the standard metric in its conformal class, we prove the nonconstrainability of their local image, local existence à la Fredholm and local solvability under a symmetry assumption à la Moser. We include a remark on the Kazdan–Warner identities for the σ_k-curvatures.

Keywords

Nirenberg problem, Schouten tensor, fully nonlinear scalar curvature, Kazdan–Warner identities, local image, nonconstrainability, local existence, symmetry

Mathematical Subject Classification 2000

Primary: 35J60, 53C21

Secondary: 58J60

Milestones

Received: 27 February 2007

Revised: 2 October 2007

Accepted: 29 October 2007

Published: 1 February 2008