On the topology and index of minimal/Bryant framed surfaces

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.