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

On the index of minimal surfaces with free boundary in a half-space