# Gradient Flow Line Near Birth-Death Critical Points

## TL;DR

This paper proves that near a birth-death critical point in gradient flows, two Morse critical points are uniquely connected by a gradient trajectory, using Whitney normal form, Conley index, and adiabatic limit analysis.

## Contribution

It provides a self-contained proof of the folklore theorem relating critical points and gradient trajectories near birth-death points.

## Key findings

- Two Morse critical points are connected by a unique gradient trajectory.
- The proof employs Whitney normal form, Conley index, and adiabatic limit analysis.
- The result clarifies the structure of gradient flows near birth-death critical points.

## Abstract

Near a birth-death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time-shift. The proof is based on the Whitney normal form, a Conley index construction, and an adiabatic limit analysis for an associated fast-slow differential equation.