Critical metrics for log-determinant functionals in conformal geometry

Abstract

We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in $\href{https://doi.org/10.1090/S0002-9939-1991-1050018-8}{[10]}$, extending Polyakov’s formula. These correspond to solutions of elliptic equations of Liouville type that are quasilinear, of mixed orders and of critical type. After studying existence, asymptotic behaviour and uniqueness of fundamental solutions, we prove a quantization property under blow-up, and then derive existence results via critical point theory.