Maybe you are thinking of the kind of work that people such as Jamie Sethian do. They use level set methods to study the evolution of surfaces under various PDE in which the topology of the surface can change, usually because one passes through a critical point of the function whose level sets give the evolving surface. Try looking at his book Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science.
Also, while I'm not aware of a book, per se, geometric measure theorists do tend to use level set methods in the study of mean curvature flow and inverse mean curvature flow. You might take a look a the papers of Brian White and Tom Ilmanen in particular.