Yamabe problems for formally self-adjoint, conformally covariant, polydifferential operators

TL;DR

This paper discusses recent advances in Yamabe problems involving conformally covariant, polydifferential operators, highlighting uniqueness and nonuniqueness results, and outlining open questions for future research.

Contribution

It provides new insights into Yamabe problems for polydifferential operators, including specific results on the sphere and general cases, and suggests directions for solving these problems.

Findings

Uniqueness results on the sphere

Nonuniqueness in general cases

Open questions for future research

Abstract

Formally self-adjoint, conformally covariant, polydifferential operators provide a general framework for studying variational problems, such as prescribing the scalar, $Q$-, or $\sigma_2$-curvatures, within a conformal class. We describe recent progress on Yamabe problems for such operators, including uniqueness results on the sphere and nonuniqueness results in general. We also highlight a number of open questions related to these operators, some of which constitute a possible blueprint for the general solution of the Yamabe problem for polydifferential operators.