If your authenticator object is instantiated in your main script then there will be no need to re-instantiate it in other pages. The way I usually develop multi-page apps in Streamlit is by creating a main script/page through which other sub-pages are invoked. This main script is also where the authenticator object is created, which means that all sub-pages will have access to it as well.
If you are using a different multi-page architecture, then I guess you can commit your authenticator object to session state and read it from other pages.