How can a mother get legal and sole custody of a child, if the father is not involved?

He has no rights to the child unless the two of you sign an affidavit of paternity (not to be confused with a birth certificate) or has a paternity Order indicating he is the biological father.

File the paperwork with the court and tell them how it is.

First, you state that the child has not yet been born. At least in Washington, no court is going to do anything regarding the child until the child is born. Second, you don't say whether you and the allege father were or are married to each other. If you are not married to each other, paternity is going to have to be established in some fashion after the child is born before the court will do anything. Even once the child is born, the alleged father is not going to have much of any rights to the child until paternity is established. That has to be established by either the father filing a declaration of paternity or through court action. Once paternity is established, then, you can ask the court to address the issue of a parenting plan. In this situation, the mother is likely to be the primary residential parent unless she has something major wrong with her. My major, I am thinking of things like alcoholism, drug addiction, history of domestic violence, etc.

Each parent is presumed to have equal custody of the child, unless there is a court order stating otherwise. Either party can seek a court order awarding joint, primary, or sole custody, or even file a petition to terminate parental rights.

If the parties aren't married, the mother has custody by law in Ohio. Therefore, to obtain any rights, the father will have to file in court. If the parties are married, they have equal custody rights, and the mother will have to file for divorce to obtain superior rights to the father.

When the parents are not married, the "father" has no legal rights whatsoever until his paternity is established by law and specific parental rights are allocated pursuant to a court ordered parenting plan. Nothing can be done to establish those legal rules until the child is born and until mother agrees to have the "father" listed on the birth certificate the only thing he can do is file a paternity case to ask the court to rule that he is the father and give him some specific rights. His threats are legally meaningless and if he does actually attempt to take the child without a court order he will be guilty of kidnapping. The mother should consult an attorney now to prepare for beginning the legal process as soon as the child is born. That will need to be done sooner or later, but it isn't absolutely critical to do anything until the father actually goes to court to obtain some rights.