On the topology and index of minimal and Bryant framed surfaces

Abstract We study framed surfaces, which are a class of Euclidean minimal and hyperbolic CMC-1 surfaces that generalize immersed minimal surfaces in $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> 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 (J Differ Geom 123(3):431–459, 2023). 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 .