The Aviator

In Houston, 1913, nine-year-old Howard Hughes is warned by his mother of the diseases to which she is afraid he will succumb. Fourteen years later, he begins to direct the movie Hell's Angels. However, after the release of The Jazz Singer, the first partially talking film, Hughes becomes obsessed with shooting his film realistically, and decides to convert the movie to a sound film. Despite the film being a hit, Hughes remains unsatisfied with the end result and orders the film to be recut after its Hollywood premiere. He becomes romantically involved with actress Katharine Hepburn, who helps to ease the symptoms of his worsening obsessive-compulsive disorder (OCD).
In 1935, Hughes test flies the H-1 Racer, pushing it to a new speed record. Three years later, he breaks the world record by flying around the world in four days. He subsequently purchases majority interest in Transcontinental &amp; Western Air, aggravating Juan Trippe, company rival and chairman of Pan American World Airways (Pan Am). Trippe gets his friend, Senator Owen Brewster, to introduce the Community Airline Bill, giving Pan Am exclusivity on international air travel. As Hughes' fame grows, he is linked to various starlets, causing a jealous Hepburn to break up with him following her announcement that she has fallen in love with fellow actor Spencer Tracy. Hughes quickly finds a new love interest with 15-year-old Faith Domergue, and later actress Ava Gardner.
Hughes secures a contract with the Army Air Forces for two projects: a spy aircraft and a troop transport unit. In 1946, with the "Spruce Goose" flying boat still in construction, Hughes finishes the XF-11 reconnaissance aircraft and takes it for a test flight. However, one of the engines fails midflight, and the aircraft crashes in Beverly Hills, with Hughes getting severely injured. With the end of WWII, the army cancels its order for the H-4 Hercules, although Hughes still continues the development with his own money. When he is discharged, he is informed that he must choose between funding the airlines or his 'flying boat'. Hughes orders Noah Dietrich to mortgage the TWA assets so he can continue the development.
Hughes becomes increasingly paranoid, planting microphones and tapping Gardner's phone lines to keep track of her. His home is searched by the FBI for incriminating evidence of war profiteering, provoking a powerful psychological trauma on Hughes, with the men searching his possessions and tracking dirt through his house. Brewster privately offers to drop the charges if Hughes sells the TWA to Trippe, but is rejected. With Hughes in a deep depression, Trippe has Brewster summon him for a Senate investigation, certain that Hughes will not show up. With Hughes shut away for nearly three months, Gardner visits him and personally grooms and dresses him in preparation for the hearing.
Hughes defends himself against Brewster's charges and accuses Trippe of bribing the senator. Hughes concludes by announcing that he has committed to completing the H-4 aircraft, and that he will leave the country if he could not get it to fly. After successfully flying the aircraft, Hughes speaks with Dietrich and his engineer, Glenn Odekirk, about a new jetliner for TWA. However, the sight of men in germ-resistant suits causes Hughes to have a mental breakdown. As Odekirk hides him in a restroom while Dietrich fetches a doctor, Hughes begins to have flashbacks of his childhood, his obsession for aviation, and his ambition for success, repeating the phrase, "the way of the future".

Question: Which actress does Howard become romantically involved with?
Answer: Katharine Hepburn