Regularity, Quantitative Geometry and Curvature Bounds

Abstract

In this thesis, we study the regularity theory and quantitative geometry in

Riemannian geometry and conformal geometry.

The thesis consists of two parts.

The first part

concentrates on my work on the regularity theorems for collapsed spaces with Ricci curvature bounds.

Specifically, we establish a quantitative nilpotent structure theorem and a new $\epsilon$-regularity theorem for collapsed manifolds with Ricci curvature bounds. As applications, we also study the structure of the collapsed Gromov-Hausdorff limits with bounded Ricci curvature.

The focus of the second part is

the connection between quantitative geometry of Kleinian groups and positivity of non-local curvature in conformal geometry. Precisely, let $(M^n,g)$ be a closed locally conformally flat with positive scalar curvature. We prove that, if $Q_{2\gamma}$-curvature is positive for $1<\gamma<2$, the limit set of the corresponding Kleinian group has Hausdorff dimension less than $\frac{n-2\gamma}{2}$, which is sharp. In dimension $3$, the positivity of $Q_3$ implies a conformal sphere theorem. In dimension $4$ and $5$, we obtain a topological classification theorem for the conformally flat manifolds with positive $Q_{2\gamma}$-curvature.