Can I assign my domain (e.g. www.example.com) to Shifter?

Yes, you can!
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Check our document Assign your domain and follow the process there.

If you want to assign example.com (*) to Shifter, you should use a DNS service which supports ALIAS records. e.g. Route 53 on AWS
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Please note the domain without www. or any subdomain is generally called "naked domain" or "apex domain."
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For more details: Assigning root domain to Shifter.