Connected sums of closed Riemannian manifolds and fourth order conformal invariants

TL;DR

This paper explores a fourth order analogue of the Yamabe problem in conformal geometry, focusing on how Paneitz constants and invariants behave under connected sum operations to understand manifold topology.

Contribution

It introduces the behavior of Paneitz constants and invariants under connected sums, providing initial insights into a higher-order conformal geometry problem.

Findings

Paneitz invariants change predictably under connected sums

Provides formulas for invariants' variation during connected sum operations

Enhances understanding of topology via fourth order conformal invariants

Abstract

In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities change, analogous to how the Yamabe constants and the Yamabe invariants do, under the connected sum operations.