On the topology and index of minimal/Bryant framed surfaces

TL;DR

This paper investigates the topology and Morse index of framed minimal and Bryant surfaces, establishing bounds and exploring their correspondence, thereby advancing understanding of their geometric and topological properties.

Contribution

It provides a lower bound on the Morse index of framed surfaces in terms of genus, ends, and branch points, and describes their correspondence between Euclidean and hyperbolic geometries.

Findings

Lower bound on Morse index as a linear function of topological features.

Description of the Lawson correspondence between Euclidean minimal and Bryant surfaces.

Generalization of previous results by Chodosh and the first author.

Abstract

We study framed surfaces, which are a class of Euclidean minimal and hyperbolic CMC-1 surfaces that generalize immersed minimal surfaces in $\mathbb{R}^3$ and Bryant surfaces. For this class we prove a lower bound on the (unrestricted) Morse index by a linear function of the genus, number of ends and number of branch points (counting multiplicity), generalizing a result by Chodosh and the first author. We include as well a description of the 1-to-1 correspondence between Euclidean minimal and Bryant surfaces, known in the literature as Lawson's correspondence.