The $Ï_k$-Yamabe problem revisited

Yuxin Ge; Guofang Wang; Wei Wei

paper

The $I ¨ _{k}$ -Yamabe problem revisited

arXiv:2605.05414

Abstract

In this paper we revisit the $I ¨ _{k}$ -Yamabe problem on $M^{n}$ , namely, finding a conformal metric with constant $I ¨ _{k}$ -scalar curvature. We prove that on a closed manifold $(M, [g_{0}])$ with positive Yamabe constant $Y_{1} (M, [g_{0}]) > 0$ , the $I ¨ _{2}$ -Yamabe constant $Y_{2} (M, [g_{0}]) := g \in [g_{0}], R_{g} > 0 in f \frac{\int _{M} I ¨ _{2} ( g ) d vol ( g )}{vol ( g ) ^{\frac{n - 4}{n}}}$ is achieved by a conformal metric $g \in [g_{0}]$ , which in particular solves the $I ¨ _{2}$ -Yamabe problem, assuming $Y_{2} (M, [g_{0}]) > 0$ . As a consequence, for any $(M, g_{0})$ with $Y_{1} (M, [g_{0}]) >$ 0 and $Y_{2} (M, [g_{0}]) > 0$ one has $g \in [g_{0}], R_{g} > 0 in f \frac{\int _{M} I ¨ _{2} ( g ) d vol ( g )}{vol ( g ) ^{\frac{n - 4}{n}}} = g \in [g_{0}], R_{g} > 0, I ¨ _{2} (g) > 0 in f \frac{\int _{M} I ¨ _{2} ( g ) d vol ( g )}{vol ( g ) ^{\frac{n - 4}{n}}} .$ We also show that these conclusions can fail if the condition $R_{g} > 0$ is removed.

Minor typos corrected, including the proof of Corollary 4.2

The I¨k​-Yamabe problem revisited

Abstract

The $I ¨ _{k}$ -Yamabe problem revisited