Boundary operators associated to the $\sigma_k$-curvature

TL;DR

This paper investigates conformal deformation problems involving the $\sigma_k$-curvature on manifolds with boundary, establishing a Dirichlet principle and an Obata-type theorem, and introduces new conformally covariant multilinear operators.

Contribution

It provides new theoretical results for boundary value problems related to $\sigma_k$-curvature and introduces novel conformally covariant multilinear operators as technical tools.

Findings

Proved a Dirichlet principle for prescribed $\sigma_k$-curvature with fixed boundary metric.

Established an Obata-type theorem on the upper hemisphere.

Developed conformally covariant multilinear operators for analysis.

Abstract

We study conformal deformation problems on manifolds with boundary which include prescribing $\sigma_k\equiv0$ in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear operators as a key technical tool.