Toponogov comparison and the collar theorem for complete surfaces with an appendix on the level sets of distance functions

TL;DR

This paper extends the collar theorem to complete, non-compact surfaces by proving new triangle comparison theorems and analyzes the structure of level sets of distance functions, with implications for geometric analysis.

Contribution

It removes the compactness requirement in the collar theorem and introduces new Toponogov-type comparison theorems and results on level set rectifiability.

Findings

Elimination of the compactness hypothesis in the collar theorem.

Development of new Toponogov-type triangle comparison theorems.

Proof that level sets of distance functions are simple closed Lipschitz curves on certain surfaces.

Abstract

In the 1970s, the collar theorem was proven, establishing the existence of uniform tubular neighborhoods of simple closed geodesics on compact surfaces, whose widths depend only on the lengths of the geodesics and the lower bound of the curvature, but not on the surface. In this paper, we improve this result by eliminating the compactness hypothesis. To achieve this result, we needed to prove new Toponogov-type triangle comparison theorems. We also add a new theorem to the literature on the rectifiabilty of the level sets of the distance function, with the corollary that on thin infinite cylinders with geodesic boundary all sets of constant distance to the boundary are simple closed Lipschitz curves.