Some New Results in Geometric Analysis

TL;DR

This thesis advances geometric analysis by studying curve flows on complex curves, curvature-preserving flows on space curves, and geometric structures on Lie groups, providing new insights into their behavior and properties.

Contribution

It introduces new results on curve-shortening flow on figure eights, curvature-preserving flows on space curves, and geometric structures on Lie groups interpolating between Sol and hyperbolic space.

Findings

Analysis of curve-shortening flow on figure eight curves

Examination of point-wise curvature preserving flow on space curves

Description of geometric structures on Lie groups interpolating Sol and hyperbolic space

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.