Many applications have been found for geometric evolution equations.
Yamabe flow was used in shape analysis which is related to medical research. See: "Detection of shape deformities using Yamabe flow and Beltrami coefficients" by Lok Ming Lui at al.
Inverse mean curvature flow was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality. See: http://en.wikipedia.org/wiki/Riemannian_Penrose_inequality