Summary: In this paper, we study the noncommutative balls $$ \cC_\rho:={(X_1,ldots, X_n)\in B(\cH)^n: \ \omega_\rho(X_1,ldots, X_n)leq 1},\qquad \rho\in (0,\infty], $$ where $\omega_\rho$ is the joint operator radius for $n$-tuples of bounded linear operators on a Hilbert space. In particular, $\omega_1$ is the operator norm, $\omega_2$ is the joint numerical radius, and $\omega_\infty$ is the joint spectral radius. We introduce a Harnack type equivalence relation on $\cC_\rho, \rho>0$, and use it to define a hyperbolic distance $\delta_\rho$ on the Harnack parts (equivalence classes) of $\cC_\rho$. We prove that the open ball $$ [\cC_\rho]_{1}:={(X_1,ldots, X_n)\in B(\cH)^n: \ \omega_\rho(X_1,ldots, X_n)1},\qquad \rho>0, $$ is the Harnack part containing 0 and obtain a concrete formula for the hyperbolic distance, in terms of the reconstruction operator associated with the right creation operators on the full Fock space with $n$ generators. Moreover, we show that the $\delta_\rho$-topology and the usual operator norm topology coincide on $[\cC_\rho]_{1}$. While the open ball $[\cC_\rho]_{1}$ is not a complete metric space with respect to the operator norm topology, we prove that it is a complete metric space with respect to the hyperbolic metric $\delta_\rho$. In the particular case when $\rho=1$ and $\cH=\CC$, the hyperbolic metric $\delta_\rho$ coincides with the Poincar´ e-Bergman distance on the open unit ball of $\CC^n$. We introduce a Carath´ eodory type metric on $[\cC_\infty]_{1} $, the set of all $n$-tuples of operators with joint spectral radius strictly less then 1, by setting $$ d_K(A,B)=\sup_p \|\Re p(A)-\Re p(B)\|,\qquad A,B\in [\cC_\infty]_{1}, $$ where the supremum is taken over all noncommutative polynomials with matrix-valued coefficients $p\in \CC[X_1,ldots, X_n]\otimes M_{m}, m\in \NN$, with $\Re p(0)=I$ and $\Re p(X)\geq 0$ for all $X\in \cC_1$. We obtain a concrete formula for $d_K$ in terms of the free pluriharmonic kernel on the noncommutative ball $[\cC_\infty]_{1}$. We also prove that the metric $d_K$ is complete on $[\cC_\infty]_{1}$ and its topology coincides with the operator norm topology. We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric $\delta_\rho$, the Carath´ eodory metric $d_K$, and the joint operator radius $\omega_\rho, \rho\in (0,\infty]$.
