Estimates of automorphic forms on $\mathrm{SU}(n,1)$

For $n\geq 2$, let $\Gamma\subset \mathrm{SU}((n,1),\mathcal{O}_{K})$ be a torsion-free, finite-index subgroup, where $\mathcal{O}_K$ denotes the ring of integers of a totally imaginary number field $K$ of degree $2$. Let $\mathbb{B}^n$ denote the $n$-dimensional complex ball endowed with the hyperbolic metric, and let $X_{\Gamma}:=\Gamma\backslash \mathbb{B}^n$ denote the quotient space. Furthermore, let $\mu_{\mathrm{hyp}}^{\mathrm{vol}}$ denote the volume form associated to the hyperbolic metric. Let $\Lambda:=\Omega_{\overline{X}_{\Gamma}}^{n}$ denote the line bundle, where $\overline{X}_{\Gamma}:=X_{\Gamma}\cup\lbrace \infty\rbrace$. For any $k\geq 1$, let $\lambda^{k}:=\Lambda^{\otimes k}\otimes O_{\overline{X}_{\Gamma}}((k-1)\infty)$. For any $k\geq 1$, the hyperbolic metric induces a point-wise metric on $H^{0}(\overline{X}_{\Gamma},\lambda^{k})$. For any $k\geq 1$, let $\mathcal{B}_{X_{\Gamma}}^{\lambda^{k}}$ denote the Bergman kernel associated $H^{0}(\overline{X}_{\Gamma},\lambda^{k})$. Then, for $k\gg1$, the first main result of the article, is the following estimate $$ \sup_{z\in \overline{X}_{\Gamma}}\big|\mathcal{B}_{X_{\Gamma}}^{\lambda^{k}}(z,z)\big|_{\mathrm{hyp}}=O_{X_{\Gamma}}(k^{n+1/2}).$$ For any $k\geq 1$, and $z\in X_{\Gamma}$, let $\mu_{\mathrm{Ber},k}(z)$ denote the Bergman metric associated to the line bundle $\lambda^{ k}$, and let $\mu_{\mathrm{ber},k}^{\mathrm{vol}}$ denote the associated volume form. Then, for $k\gg1$, the second main result of the article is the following estimate $$ \sup_{z\in \overline{X}_{\Gamma}}\bigg|\frac{\mu_{\mathrm{Ber},k}^{\mathrm{vol}}(z)}{\mu_{\mathrm{hyp}}^{\mathrm{vol}}(z)}\bigg|=O_{X_{\Gamma}}\big(k^{2(n-1)(n+1)+n+3} \big).$$ Our estimate for the Bergman metric completes our arguments and corrects our estimate from arXiv:2305.11609, for $n=1$.