Professor Gage has worked on a range of problems in differential geometry, including isoperimetric inequality problems such as a proof of Gehring's conjecture on linked spheres and eigenvalue estimates on Riemannian manifolds. Beginning in 1983 he was one of the first to study the "curve shortening" or "flow by mean curvature" problem, which lead eventually, through the work of many researchers, to the theorem that all simple closed curves which are deformed at a rate proportional to their curvature will shrink smoothly to points while becoming asymptotically circular. Since then he and his students have studied versions of the problem in Minkowski geometries, where the unit ball is an arbitrary convex set, and where the theory has applications to crystal growth and material science. Similar dynamical flows are being studied for their relevance to image enhancement.
Professor Gage's current interests are focused on understanding the properties of the curve shortening flow in general Finsler metrics of which Minkowski geometries are a special case.