New applications of the Ahlfors Laplacian

TL;DR

This paper explores new applications of the Ahlfors Laplacian in differential geometry, specifically in the study of Ricci tensors on Riemannian manifolds and spacelike hypersurfaces in Lorentzian manifolds, leading to insights in geometric analysis and general relativity.

Contribution

It introduces novel uses of the Ahlfors Laplacian for orthogonal decompositions in geometric contexts involving Ricci tensors and hypersurfaces, advancing methods in geometric analysis and Einstein constraint equations.

Findings

Orthogonal decomposition of Ricci tensor traceless part

Orthogonal expansion of second fundamental form in Lorentzian manifolds

Applications to constructing solutions of Einstein constraint equations

Abstract

In this article, we consider an orthogonal decomposition of the traceless part of the Ricci tensor of a compact Riemannian manifold and study its application to the geometry of compact almost Ricci solitons. In addition, we consider an orthogonal expansion of the traceless part of the second fundamental form of a compact spacelike hypersurface in a Lorentzian manifold and study its application to the problem of constructing solutions of general relativistic constraint equations in vacuum. In these two cases, we use the well-known Ahlfors Laplacian.