Your own domain
Use an address you already own.
After you keep an app you can give it a name of your choosing (family-books.uarpie.app instead of quiet-harbor-4471.uarpie.app) and, on Personal and above, attach a domain you own.
Adding a domain
From the app's Address tab, or ask your AI ("use books.example.com for the family app"). You get exactly one DNS record to add at your domain provider, a CNAME. Once it resolves, a certificate is issued automatically and renewed forever.
While it is pending
Status shows Waiting for DNS. It usually takes minutes, sometimes an hour. The uarpie.app address keeps working throughout.
Screening
Because free hosting with real certificates attracts abuse, some domains are reviewed before they are attached. If yours is, you are emailed when it clears.