Ricci flow has been applied to computer vision (e.g., to 3D shape matching),
and to computer graphics (e.g., surface parametrization):

Wei Zeng, Samaras, D., Gu, D. ,
"Ricci Flow for 3D Shape Analysis,"
*IEEE Transactions on Pattern Analysis and Machine Intelligence*, 32(4) 2010 662-677.

Yang, Guo, Luo, Hu, Gu,
"Generalized Discrete Ricci Flow,"
*Computer Graphics Forum*, Volume 28, Issue 7, pages 2005–2014, October 2009.

If you classify the eikonal equation as a geometric evolution equation, then there
are many applications to the computer vision problem known as "shape from shading": deriving
the 3D shape of an object from shading variation. For example:

Anna R. Bruss,
"The eikonal equation: Some results applicable to computer vision,"
*J. Math. Phys.* 23, 890 (1982).

Ron Kimmel , James A. Sethian,
"Optimal Algorithm for Shape from Shading and Path Planning,"
*Journal of Mathematical Imaging and Vision*,Volume 14 Issue 3, May 2001.

And in fact Sethian's book,

*Level Set Methods:
Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science*,
Cambridge University Press, 1996.

is filled with applications of the eikonal equation (and its variants):
"The text includes applications from physics, chemistry, fluid mechanics, combustion, image processing, material science, seismology, fabrication of microelectronic components, computer vision and control theory."