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 $\mathbb{CP}^2$ 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 $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $\beta(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying a positivity condition; it follows from \cite{CGY03} that if $0 \leq \beta(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result is a gap result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq \beta(M^4,[g]) < 4(1 + \epsilon)$ for $\epsilon > 0$ small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to pass from the bounds on $\beta$ to pointwise curvature information. We also prove a lower bound for $\beta(M)$ 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 $(\mathbb{CP}^2, g_{FS})$ and $({S}^2\times{S}^2,g_{p})$ as model cases. A Moser-type iteration argument plays an important role in the case of

$(\mathbb{CP}^2, g_{FS})$ and the convergence theory of Bach-flat metrics is of particular importance in the case of $({S}^2\times{S}^2,g_{p})$.