PreprintCC BY 4.0arXiv
The second variation of 2-spheres in the 2n-sphere
Abstract
We prove sharp upper and lower bounds for the Morse index and nullity of linearly full branched minimal 2-spheres in the round 2n-sphere. In particular, we show that the Morse index of a linearly full minimal 2-sphere in the round 2n-sphere is at least n(n-1)(2n+1), with equality if and only if the 2-sphere is twistor-equivalent to the Boruvka sphere. Our techniques also apply more generally to give bounds for the Morse index and nullity for totally isotropic surfaces in the 2n-sphere.
Related
- The Jacobi Spectrum and Global Second Variation of Pansu Spheres
- Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres
- An improved second pinching theorem for minimal hypersurfaces with constant scalar curvature in spheres
- Second Jacobi eigenvalues and spectral rigidity for surfaces in Berger spheres
- The Second Gap for Minimal Hypersurfaces in the Unit Sphere with Constant Cubic Trace
BibTeX
@misc{ball2026second,
title = {{The second variation of 2-spheres in the 2n-sphere}},
author = {Gavin Ball and Jesse Madnick},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17887},
eprint = {2609.17887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17887},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The second variation of 2-spheres in the 2n-sphere
AU - Gavin Ball
AU - Jesse Madnick
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17887
UR - https://arxiv.org/abs/2609.17887
AB - We prove sharp upper and lower bounds for the Morse index and nullity of linearly full branched minimal 2-spheres in the round 2n-sphere. In particular, we show that the Morse index of a linearly full minimal 2-sphere in the round 2n-sphere is at least n(n-1)(2n+1), with equality if and only if the 2-sphere is twistor-equivalent to the Boruvka sphere. Our techniques also apply more generally to give bounds for the Morse index and nullity for totally isotropic surfaces in the 2n-sphere.
ER -