Calabi metric on the space of k-Hessian potentials

Abstract

Let $(X,ω)$ be a compact Kähler manifold of complex dimension $n$, and let $1\leq k\leq n$. 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. The resulting Riemannian metric is \[ G^{\HC,k}_φ(u,v) =\int_X \bigl(Δ_φ^{(k)}u\bigr) \bigl(Δ_φ^{(k)}v\bigr) \,ω_φ^{k}\wedgeω^{n-k}. \] We then show that the square-root Hessian density map \[ φ\longmapsto 2\sqrt{\frac{ω_φ^{k}\wedgeω^{n-k}}{ω^n}} \] is a global Riemannian isometry onto the positive part of the $L^2(X,ω^n)$-sphere of radius $2\sqrt{V}$, where $V=\int_Xω^n$. 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.