Spectral and convex geometry of α-harmonic maps and sharp Morse index rigidity
Abstract
We develop a weighted scalar spectral geometric description of Sacks-Uhlenbeck α-harmonic maps into round spheres and identify its convex geometry realization for α-harmonic diffeomorphisms onto the round $2$-sphere. More precisely, we establish a one-to-one correspondence, modulo orientation preserving isometries of 𝕊2, between orientation preserving α-harmonic diffeomorphisms onto the round $2$-sphere and positive Codazzi tensors satisfying a determinant bound and a compatibility condition with the prescribed metric on 𝕊2. These tensors are the radii tensors of auxiliary strictly convex surfaces, and the weighted scalar forms in spectral geometry equal −α times the second variations of the enclosed volumes with respect to the support functions. Using these spectral geometry and convex geometry results, we prove a sharp Morse index characterization of isometric equatorial parametrizations among α-harmonic $2$-spheres in the round n-sphere. We also prove that every α-harmonic diffeomorphism between unit round $2$-spheres is orthogonal for all α>1. Equivalently, the only admissible positive Codazzi tensor whose reconstructed metric on 𝕊2 has Gaussian curvature $1$ is (κ+2)α−1g̅, where g̅ is the unit round metric.
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BibTeX
@misc{gao2026spectral,
title = {{Spectral and convex geometry of $α$-harmonic maps and sharp Morse index rigidity}},
author = {Rui Gao and Miaomiao Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00642},
eprint = {2611.00642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00642},
}
RIS
TY - UNPB
TI - Spectral and convex geometry of α-harmonic maps and sharp Morse index rigidity
AU - Rui Gao
AU - Miaomiao Zhu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00642
UR - https://arxiv.org/abs/2611.00642
AB - We develop a weighted scalar spectral geometric description of Sacks-Uhlenbeck α-harmonic maps into round spheres and identify its convex geometry realization for α-harmonic diffeomorphisms onto the round $2$-sphere. More precisely, we establish a one-to-one correspondence, modulo orientation preserving isometries of 𝕊^2, between orientation preserving α-harmonic diffeomorphisms onto the round $2$-sphere and positive Codazzi tensors satisfying a determinant bound and a compatibility condition with the prescribed metric on 𝕊^2. These tensors are the radii tensors of auxiliary strictly convex surfaces, and the weighted scalar forms in spectral geometry equal −α times the second variations of the enclosed volumes with respect to the support functions. Using these spectral geometry and convex geometry results, we prove a sharp Morse index characterization of isometric equatorial parametrizations among α-harmonic $2$-spheres in the round n-sphere. We also prove that every α-harmonic diffeomorphism between unit round $2$-spheres is orthogonal for all α>1. Equivalently, the only admissible positive Codazzi tensor whose reconstructed metric on 𝕊^2 has Gaussian curvature $1$ is (κ+2)^α−1g̅, where g̅ is the unit round metric.
ER -