Prof. 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.
Prof. 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.
- 1 365 302Gage, Michael E. Minkowski plane geometry and anisotropic curvature flow of curves. Curvature flows and related topics (Levico, 1994), 83--97, GAKUTO Internat. Ser. Math. Sci. Appl., 5, Gakk=otosho, Tokyo, 1995. (Reviewer: Anders Linnér) 58E10 (35K99 53A04)
- 95g:53004 Gage, Michael E.; Li, Yi Evolving plane curves by curvature in relative geometries. II.Duke Math. J. 75 (1994), no. 1, 79--98. (Reviewer: Anders Linnér) 53A04 (58E10)
- 94j:53001 Gage, Michael E. Evolving plane curves by curvature in relative geometries. Duke Math. J. 72 (1993), no. 2, 441--466. (Reviewer: Anders Linnér) 53A04 (58E10)
- 93c:35066 Gage, Michael E. On the size of the blow-up set for a quasilinear parabolic equation.Geometry and nonlinear partial differential equations (Fayetteville, AR, 1990), 41--58, Contemp. Math., 127, Amer. Math. Soc., Providence, RI, 1992. (Reviewer: Zhen Chao Cao) 35K55 (35B05)
- 92g:58021 Gage, Michael E. Deforming curves on convex surfaces to simple closed geodesics.Indiana Univ. Math. J. 39 (1990), no. 4, 1037--1059. 58E10 (53C45)
- 91c:52011 Gage, Michael E. Positive centers and the Bonnesen inequality. Proc. Amer. Math. Soc. 110 (1990), no. 4, 1041--1048. (Reviewer: G. D. Chakerian) 52A40
- 91a:53072 Gage, Michael E. Curve shortening on surfaces. Ann. Sci. École Norm. Sup. (4) 23(1990), no. 2, 229--256. (Reviewer: Dennis M. DeTurck) 53C22 (35K55 58G11)
- 89f:58128 Epstein, C. L.; Gage, Michael The curve shortening flow. Wave motion: theory, modelling, and computation (Berkeley, Calif., 1986), 15--59, Math. Sci. Res. Inst. Publ., 7, Springer, New York, 1987. (Reviewer: R. Osserman) 58G11 (53A04)
- 87m:53003 Gage, M.; Hamilton, R. S. The heat equation shrinking convex plane curves. J. Differential Geom. 23 (1986), no. 1, 69--96. (Reviewer: R. Osserman) 53A04 (35K05 52A40 58E99 58G11)
- 87g:53003 Gage, Michael On an area-preserving evolution equation for plane curves. Nonlinear problems in geometry (Mobile, Ala., 1985), 51--62, Contemp. Math., 51, Amer. Math. Soc., Providence, RI, 1986. (Reviewer: W. J. Firey) 53A04