Variational theory for the resonant T-curvature equation

TL;DR

This paper develops a variational framework to analyze the resonant T-curvature equation on 4D manifolds with boundary, establishing existence results via Morse theory and topological methods.

Contribution

It introduces a new Morse theoretical approach at infinity for the T-curvature problem, combining energy estimates and topological techniques to prove existence results.

Findings

Derived sharp energy and gradient estimates.

Established a Morse lemma at infinity.

Proved existence results using topological hypotheses.

Abstract

We study the resonant prescribed T-curvature problem on a compact 4-dimensional Riemannian manifold with boundary. We derive sharp energy and gradient estimates of the associated Euler-Lagrange functional to characterize the critical points at infinity of the associated variational problem under a non-degeneracy on a naturally associated Hamiltonian function. Using this, we derive a Morse type lemma around the critical points at infinity. Using the Morse lemma at infinity, we prove new existence results of Morse theoretical type. Combining the Morse lemma at infinity and the Liouville version of the Barycenter technique of Bahri-Coron[13] developed in [46], we prove new existence results under a topological hypothesis on the boundary of the underlying manifold, and the entry and exit sets at infinity.