Morse index characterization of the Veronese projective plane

Following Simons' seminal work in 1968, every closed, connected, embedded minimal surface in the four-sphere that is not totally geodesic has Morse index at least five. In this paper, we prove that equality holds if and only if the surface is congruent to the Veronese projective plane. Relying on recent advances by Bernstein and Ketover, this characterization settles Kusner's 1996 conjecture on the minimizers of the Willmore functional in dimension four for surfaces whose normal bundle has nonzero Euler number. This is always the case, in particular, for embeddings of nonorientable surfaces whose orientation double cover has even genus.