Some New Results in Geometric Analysis

TL;DR

The thesis investigates how geometric objects evolve under nonlinear flows. It analyzes curve-shortening flow on figure-eight curves, establishes a curvature-preserving flow on space curves, and develops an interpolation framework among Sol and hyperbolic geometries via the one-parameter family $G_\alpha$ of Lie groups, including detailed numerical studies. A key result is that constant-curvature torsion flows are integrable (equivalent to the $m^2$KDV equation) with $L^2$-stable helices, complemented by numerical experiments. In the interpolation chapter, strong control is obtained for the $G_{1/2}$ case, proving a Geodesic-Minimizing criterion (small or perfect) and providing a Bounding Box and Monotonicity framework for the cut locus; the work connects curve evolution, integrable PDEs, and Thurston geometries with explicit period-function analyses.

Abstract

This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preserving flow on space curves. Lastly, we present an abridgment of our work on a family of three-dimensional Lie groups, which, when equipped with canonical left-invariant metrics, interpolate between Sol and hyperbolic space.

Figures (21)

• Figure 1: The local geometry of the $x$-coordinate preserving modification.
• Figure 2: Some typical examples for each class of figure-eight curve
• Figure 3: The function $\theta(x,t)$ gives us an evolution of closed curves, evolving according to the equation in Lemma 1.
• Figure 4: We are comparing the flow of a portion of the $\theta$ curve with the red step-function in the proof of Theorem 1.
• Figure 5: This is the graph of the torsion $\tau_1$ over two periods.

Theorems & Definitions (125)

• Theorem
• Conjecture
• Theorem
• Lemma 1: G
• proof
• Lemma 2: Also found in G
• proof
• Lemma 3: See ANG or GH
• Definition 1
• Definition 2