Flows in Flatland: A Romance of Few Dimensions

TL;DR

This paper explores flows on 2D manifolds with boundary, emphasizing Morse theory and boundary effects to understand traversing flows in low-dimensional spaces.

Contribution

It introduces a novel approach to Morse theory on manifolds with boundary, focusing on boundary effects in the context of flows on surfaces.

Findings

General results on traversing flows on manifolds with boundary

Application of Morse theory to 2D surfaces with boundary

Clarification of boundary effects in flow analysis

Abstract

In this paper, we present our general results about traversing flows on manifolds with boundary in the context of the flows on surfaces with boundary. We take advantage of the relative simplicity of $2D$-worlds to explain and popularize our approach to the Morse theory on smooth manifolds with boundary, in which the boundary effects take the central stage.