if you sponsor and immigrant how long are you held liable to keep them at or above minimum wage lifestyle?

The US government does not specifically require you to "keep them at or above minimum wage lifestyle." The government requires you to file an Affidavit of Support with the family-based petition. This Affidavit is a legally binding contract between you and the government, and you agree to repay the government for any means-tested public benefits that the sponsored immigrant takes. When you sign the affidavit of support, you accept legal responsibility for financially supporting the sponsored immigrant(s) until they become U.S. citizens or can be credited with 40 quarters of work. If the immigrant receives any "means-tested public benefits," you are responsible for repaying the cost of those benefits to the agency that provided them. If you do not repay the debt, the agency can sue you in court to get the money owed. When in doubt, ask the benefit provider whether the benefit is a "means-tested public benefit."