Tunnels, Wells, and Scalar Curvature

This dissertation explores advancements in Riemannian geometry, specifically focusing on how scalar curvature influences the shape and size of mathematical spaces. It introduces new examples related to stability conjectures concerning the characterization of spheres by scalar curvature. The work addresses the necessity of conditions that prevent 'bubbling' in sequences of manifolds and constructs examples that converge in a specific sense (volume-preserving intrinsic flat sense) but not in another (Gromov–Hausdorff sense). This is achieved by refining a technique called the 'tunnel construction' to allow for the attachment of 'tunnels' or 'wells' to manifolds while minimally affecting scalar curvature. The research also builds upon and extends previous work on constructing manifolds with specific properties, including limits with no geodesics and counterexamples to existing conjectures. The study delves into the theoretical aspects of scalar curvature and its implications for classifying geometric structures. It presents novel constructions that challenge existing conjectures and expand the understanding of manifold properties. The dissertation aims to provide a deeper insight into the relationship between curvature, convergence of manifolds, and their topological characteristics, offering non-perturbative counterexamples with complex topologies.

Curriculum

The dissertation is structured into three main chapters, preceded by an introduction and followed by a bibliography. The content covers theoretical background, construction methods, and the presentation of specific mathematical objects and their properties.

1. Chapter 1: Introduction
   • Spheres with Larger Scalar Curvature
   • Hemispheres with Large Scalar Curvature
   • Background
   • Gromov–Hausdorff Convergence
   • Intrinsic Flat Convergence
   • Volume Above Distance Below Convergence
2. Chapter 2: Quantitative Surgery
   • Wells and Tunnels
   • A Submanifold Defined by a Curve
   • Constructing the Curve
   • Adjusting the Curve to Construct a Well
   • Smoothing the Curve that Defines the Well
   • Attaching the Well
   • Constructing a Tunnel
   • Surgery in Higher Codimensions
3. Chapter 3: Constructing Manifolds
   • Manifolds with Shrinking Tunnels
   • Manifolds with Many Wells
   • Sewing Manifolds
   • Intrinsic Flat Limit with No Geodesics
   • Novel Min-Oo Counterexamples
   • Manifold with Volume Constraint and Scalar Curvature greater than n(n-1)