Calabi metric on the space of k-Hessian potentials
Abstract
Let (X,ω) be a compact Kähler manifold of complex dimension n, and let 1≤ k≤ n. We consider the space of smooth k-admissible potentials introduced in the study of the Hessian-cscK equation by Guo--Smith--Tong. The complex k-Hessian equation identifies this space, modulo constants, with the space of smooth positive densities of fixed total mass. We compute the differential of the Hessian density map and use it to pull back the Fisher--Rao metric. We then show that the square-root Hessian density map φlongmapsto 2√((ωφk∧ωn−k)/ωn) is a global Riemannian isometry onto the positive part of the L2(X,ωn)-sphere of radius 2√V, where V=∫Xωn. For k=n, this recovers the spherical description of the classical Calabi metric. We also derive the curvature, geodesic and diameter formulas, and prove a quantitative local comparison between adjacent Hessian--Fisher distances.
Related
- Neumann problem for Hessian equation and geometric inequalities in hyperbolic space
- Overdetermined problems for sum Hessian equations in the space forms
- Metric entropy of Kähler potentials
- The Kuranishi space of joint deformations of Calabi--Yau manifolds and holomorphic vector bundles
- No Shortcuts to Infinity: Weil-Petersson Distance in Calabi-Yau Complex Structure Moduli Space
BibTeX
@misc{zhang2026calabi,
title = {{Calabi metric on the space of k-Hessian potentials}},
author = {Linwei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19653},
eprint = {2610.19653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19653},
}
RIS
TY - UNPB
TI - Calabi metric on the space of k-Hessian potentials
AU - Linwei Zhang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.19653
UR - https://arxiv.org/abs/2610.19653
AB - Let (X,ω) be a compact Kähler manifold of complex dimension n, and let 1≤ k≤ n. We consider the space of smooth k-admissible potentials introduced in the study of the Hessian-cscK equation by Guo--Smith--Tong. The complex k-Hessian equation identifies this space, modulo constants, with the space of smooth positive densities of fixed total mass. We compute the differential of the Hessian density map and use it to pull back the Fisher--Rao metric. We then show that the square-root Hessian density map φlongmapsto 2√((ω_φ^k∧ω^n−k)/ω^n) is a global Riemannian isometry onto the positive part of the L^2(X,ω^n)-sphere of radius 2√V, where V=∫_Xω^n. For k=n, this recovers the spherical description of the classical Calabi metric. We also derive the curvature, geodesic and diameter formulas, and prove a quantitative local comparison between adjacent Hessian--Fisher distances.
ER -