Continuity mistake: Kate's makeup has her looking well healed while she is still sleeping in the cave, only to return to a swollen eye as she is rescued from the frozen lake. (00:45:05 - 00:56:45)
Other mistake: Despite the up-to-date equipment used in the movie, Kate carries an old style film camera rather than a dslr. It was obviously so that she could develop the film later to realise her 'love'. At any rate, when going through her bag, she finds a roll of 35mm film to develop, but when the film is laid out on the light table for her to examine with a loupe, the film shown there is not 35mm. I think it was 126mm.