Some Problems in Four-dimensional Conformal Geometry

Abstract

In this thesis, we study some problems in four-dimensional conformal geometry. This thesis consists of two main parts: conformally invariant characterization of CP2 and conformally invariant gap theorems for Bach-flat metrics. In the first part, we extend the sphere theorem of \cite{CGY03} to give a conformally invariant characterization of (CP2,gFS). In particular, we introduce a conformal invariant β(M4,[g])≥0 defined on conformal four-manifolds satisfying a positivity condition; it follows from \cite{CGY03} that if 0≤β(M4,[g])<4, then M4 is diffeomorphic to S4. Our main result is a gap result showing that if b2+(M4)>0 and 4≤β(M4,[g])<4(1+ϵ) for ϵ>0 small enough, then M4 is diffeomorphic to CP2. The Ricci flow is used in a crucial way to pass from the bounds on β to pointwise curvature information. We also prove a lower bound for β(M) under some topological conditions. In the second part, we extend a conformal gap theorem for Bach-flat metrics with round sphere as model case established in \cite{CQY} to prove conformally invariant gap theorems for Bach-flat 4-manifolds with (CP2,gFS) and (S2×S2,gp) as model cases. A Moser-type iteration argument plays an important role in the case of (CP2,gFS) and the convergence theory of Bach-flat metrics is of particular importance in the case of (S2×S2,gp).