Browsing by Author "Zhang, Siyi"

In this thesis, we study some problems in four-dimensional conformal geometry. This thesis consists of two main parts: conformally invariant characterization of $\mathrm{CP}2$ 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 $(\mathrm{CP}2,gFS)$. In particular, we introduce a conformal invariant $\beta (M4,[g])\ge 0$ defined on conformal four-manifolds satisfying a positivity condition; it follows from \cite{CGY03} that if , then $M4$ is diffeomorphic to $S4$. Our main result is a gap result showing that if $b2+(M4)>0$ and for $ϵ>0$ small enough, then $M4$ is diffeomorphic to $\mathrm{CP}2$. 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 $(\mathrm{CP}2,gFS)$ and $(S2\times S2,gp)$ as model cases. A Moser-type iteration argument plays an important role in the case of

$(\mathrm{CP}2,gFS)$ and the convergence theory of Bach-flat metrics is of particular importance in the case of $(S2\times S2,gp)$.