Opera singer Christine triumphs at the gala on the night of the old managers' retirement. Her old childhood friend, Raoul, hears her sing and recalls his love for Christine. At this time, there are rumors of a phantom living at the Opera and he makes himself known to the managers through letters and malevolent acts. Some time after the gala, the Paris Opera performs Faust, with the prima donna Carlotta playing the lead, against the Phantom's wishes. During the performance, Carlotta loses her voice and the grand chandelier plummets into the audience.
Christine is kidnapped by the phantom and is taken to his home in the cellars of the Opera where he identifies himself as Erik. He plans to keep her there for a few days, hoping she will come to love him. But she causes Erik to change his plans when she unmasks him and, to the horror of both, beholds his noseless, lipless, sunken-eyed face which resembles a skull dried up by the centuries, covered in yellowed dead flesh. Fearing that she will leave him, he decides to keep her with him forever, but when Christine requests release after two weeks, he agrees on condition that she wear his ring and be faithful to him.
On the roof of the opera house, Christine tells Raoul that Erik abducted her. Raoul promises to take Christine away to a place where Erik can never find her. Raoul tells Christine he shall act on his promise the next day, to which Christine agrees. She, however, has pity for Erik and will not go until she has sung a song for him one last time. Neither is aware that Erik has been listening to their conversation and that he has become extremely jealous.
The following night, Erik kidnaps Christine during a production of Faust and tries to force Christine to marry him. He states that if she refuses, he will use explosives (which he has planted in the cellars) to destroy the entire opera house. Christine refuses, until she realizes that Erik learned of Raoul's attempt to rescue her and has trapped Raoul in a hot torture chamber (along with the Persian, an old acquaintance of Erik who was going to help Raoul). To save them and the people above in the Opera, Christine agrees to marry Erik. Erik initially tries to drown Raoul and the Persian, using the water which would have been used to douse the explosives. But Christine begs and offers to be his "living bride", promising him not to kill herself after becoming his bride, as she had both contemplated and attempted earlier in the novel. Erik eventually rescues Raoul and the Persian from his torture chamber. When Erik is alone with Christine, he lifts his mask to kiss her on her forehead, and is given a kiss back. Erik reveals that he has never received a kiss (not even from his own mother) nor has been allowed to give one and is overcome with emotion. He and Christine then cry together and their tears "mingle". Erik later expresses that he has never felt so close to another human being.
Erik allows the Persian and Raoul to escape, though not before making Christine promise that she will visit him on his death day, and return the gold ring he gave her. He also makes the Persian promise that afterwards he will go to the newspaper and report his death, as he will die soon and will die "of love". Indeed, some time later Christine returns to Erik's lair, buries him somewhere he'll never be found (by Erik's request) and returns the gold ring. Afterwards, a local newspaper runs the simple note: "Erik is dead".

Question: How does Erik convince Christine to marry him?
Answer: He threatens to set off explosives in the opera house and has trapped Raoul in a torture chamber. 
Alternative answer: He traps Raoul and the Persian