Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps

Abstract

We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let $k:[0,\pi]\to\mathbb R$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, \[ (T_kF)(\xi)=\int_{\mathbb S^{n-1}} k\!\left(\arccos\langle\xi,\eta\rangle\right) F(\eta)\,d\sigma(\eta), \] where $\sigma$ is normalized surface measure. For every measurable $F:\mathbb S^{n-1}\to\overline{\mathbb B^n}$, we prove \[ w\!\left(\operatorname{co}T_kF(\mathbb S^{n-1})\right) \le2\lambda_1(k), \] where $w$ denotes mean width, normalized by $w(\overline{\mathbb B^n})=2$, and $\lambda_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(\eta)=Q\eta$ almost everywhere, with $Q\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $|f(r\mathbb B^n)|\le\omega_n r^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.