A birth certificate, by itself, does not establish legal paternity.
When the parents sign the Recognition of Parentage (ROP) in the hospital, the father’s name will automatically be on the birth certificate. If you sign the ROP after leaving the hospital, the father’s name can be added to the birth certificate.
You can add the father’s name through the MN Dept. of Health, Office of the State Registrar and there is no fee to update the child’s record.