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Morse index characterization of the Veronese projective plane
Abstract
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.
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BibTeX
@misc{carlotto2026morse,
title = {{Morse index characterization of the Veronese projective plane}},
author = {Alessandro Carlotto and Yangyang Li and Zhihan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35820},
eprint = {2610.35820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35820},
}
RIS
TY - UNPB
TI - Morse index characterization of the Veronese projective plane
AU - Alessandro Carlotto
AU - Yangyang Li
AU - Zhihan Wang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35820
UR - https://arxiv.org/abs/2610.35820
AB - 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.
ER -