The Yamabe problem for Gauss-Bonnet curvatures: a local result around space forms

TL;DR

This paper investigates the Yamabe problem for Gauss-Bonnet curvatures, proving local existence of conformal metrics with constant 2k-Gauss-Bonnet curvature near certain space forms using perturbative methods.

Contribution

It establishes a local solution to the Yamabe problem for Gauss-Bonnet curvatures around non-flat space forms, extending previous variational results.

Findings

Existence of conformal metrics with constant 2k-Gauss-Bonnet curvature near certain space forms.

Affirmative answer to the Yamabe problem for small perturbations of these space forms.

Identification of a neighborhood in the metric space where the problem has solutions.

Abstract

It is shown in the paper "Variational Properties of the Gauss-Bonnet Curvatures" of M.L. Labbi, that metrics with constant 2k-Gauss-Bonnet curvature on a closed n-dimensional manifold, 1<2k<n, are critical points for a certain Hilbert type functional with respect to volume preserving conformal variations. This motivates the corresponding Yamabe problem: is it true that any metric on a closed manifold is conformal to a metric with constant 2k-Gauss-Bonnet curvature? Using perturbative methods we affirmatively answer this question for small perturbations of certain space forms. More precisely, if (X,g) is a non-flat closed space form not isometric to a round sphere, we show the existence of a neighborhood U, of g, in the space of metrics such that any g' in U is conformal to a metric whose 2k-Gauss-Bonnet curvature is constant.