  We study the Morse index of minimal surfaces with free boundary in a
half-space. We improve previous estimates relating the Neumann index to the
Dirichlet index and use this to answer a question of Ambrozio, Buzano,
Carlotto, and Sharp concerning the non-existence of index two embedded minimal
surfaces with free boundary in a half-space. We also give a simplified proof of
a result of Chodosh and Maximo concerning lower bounds for the index of the
Costa deformation family.
