Joseph Gatti’s vision for his new troupe, United Ballet Theatre (which debuts this weekend), was inspired by a fracture in his foot three years ago: “The injury really opened my eyes to the organization of a company and the care and treatment that needs to be addressed constantly. I wanted to create a company that’s treating the dancers not just as artists but as athletes.” His plan, the Gatti Method, is “to build strength training, conditioning, recovery time and physical therapy into UBT’s daily schedule.”