What are district courts in u. s. territories called?

They are called "territorial courts". District courts only exist in US federal judicial courts, that are located in 50 US states, District of Columbia and Puerto Rico. These territorial courts are operating in District Court of Virgin Islands, District Court of Guam and District Court of Northern Mariana Islands. In the territory of American Samoa there is no federal courts and federal law matters are adjudicated to District Court of Hawaii or to District Court for District of Columbia. They are established under Article Four of the United States Constitution, the Territorial Clause, by US Congress.