p-Laplace equations in conformal geometry

TL;DR

This paper introduces p-Laplace equations related to the intermediate Schouten curvature in conformal geometry, offering new tools for understanding manifold topology and geometry, with applications to Betti numbers, singular sets, and homotopy groups.

Contribution

It develops p-Laplace equations for the intermediate Schouten curvature, extending nonlinear potential theory tools in conformal geometry for topological and geometric analysis.

Findings

Positivity of intermediate Schouten curvature implies Betti number vanishing.

Nonnegative intermediate Schouten curvature enables estimates on singular set dimensions.

Results extend Schoen-Yau's work on homotopy groups and potential theory.

Abstract

In this paper we introduce the p-Laplace equations for the intermediate Schouten curvature in conformal geometry. These p-Laplace equations provide more tools for the study of geometry and topology of manifolds. First, the positivity of the intermediate Schouten curvature yields the vanishing of Betti numbers on locally conformally flat manifolds as consequences of the B\"{o}chner formula as in the works of Nayatani and Guan-Lin-Wang. Secondly and more interestingly, when the intermediate Schouten curvature is nonnegative, these p-Laplace equations facilitate the geometric applications of p-superharmonic functions and the nonlinear potential theory. This leads to the estimates on Hausdorff dimension of singular sets and vanishing of homotopy groups that is inspired by and extends the work of Schoen-Yau. In the forthcoming paper we will present our results on the asymptotic behavior of p-superharmonic functions at singularities.