Richard Streit Hamilton (January 10, 1943 – September 29, 2024) was an American mathematician who served as the Davies Professor of Mathematics at Columbia University.
There, he interacted with James Eells, who with Joseph Sampson had recently published a paper introducing harmonic map heat flow.
After publishing his first paper on the topic, Hamilton moved to University of California, San Diego in the mid-1980s, joining Richard Schoen and Shing-Tung Yau in the group working on geometric analysis.
[7] In July 2010, Perelman turned down the award and prize money, saying that he believed his contribution in proving the Poincaré conjecture was no greater than that of Hamilton.
"[10] In 2003 he received the Clay Research Award for "his introduction of the Ricci flow equation and his development of it into one of the most powerful tools in geometry and topology".
[H82b][15] In 2011, the million-dollar Shaw Prize was split equally between Hamilton and Demetrios Christodoulou "for their highly innovative works on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.
In 1986, Peter Li and Shing-Tung Yau discovered a new method for applying the maximum principle to control the solutions of the heat equation.
[H93a] His result required the stronger assumption that the underlying closed Riemannian manifold has nonnegative sectional curvature and parallel Ricci tensor (such as the flat torus or the Fubini–Study metric on complex projective space).
[22] Hamilton also found that Li and Yau's calculations were directly transferable to derive Harnack inequalities for the scalar curvature along a positively-curved Ricci flow on a two-dimensional closed manifold.
[29] With the use of such maps, they were able to derive a number of purely geometric corollaries, such as restrictions on the topology of precompact open subsets with simply-connected boundary inside complete Riemannian manifolds of nonnegative Ricci curvature.
[30] In 1987, Matthew Grayson proved a complementary result, showing that for any smoothly embedded circle in the plane, the corresponding mean curvature flow eventually becomes convex.
[31] In combination with Gage and Hamilton's result, one has essentially a complete description of the asymptotic behavior of the mean curvature flow of embedded circles in the plane.
[H82b] He also put this into the general setting of a parameter-dependent section of a vector bundle over a closed manifold which satisfies a heat equation, giving both strong and weak formulations.
Such results are notable in highly restricting the topology of such manifolds; the space forms of positive curvature are largely understood.
There are other corollaries, such as the fact that the topological space of Riemannian metrics of positive Ricci curvature on a closed smooth 3-manifold is path-connected.
An important consequence is that, in three dimensions, a limiting Ricci flow as produced by the compactness theory automatically has nonnegative curvature.
These methods were extended by Grigori Perelman, who due to his noncollapsing theorem was able to verify the preconditions of Hamilton's compactness theory in a number of new contexts.
[23][22] In 1997, Hamilton was able to combine his developed methods to define Ricci flow with surgery for four-dimensional Riemannian manifolds of positive isotropic curvature.
As a consequence, he obtained a result which classifies the smooth four-dimensional manifolds which support Riemannian metrics of positive isotropic curvature.