1. Richard Schoen is best known for the resolution of the Yamabe problem in 1984 and his works on harmonic maps.
16 Facts About Richard Schoen
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Richard Schoen then received his PhD in 1977 from Stanford University with Leon Simon and Shing-Tung Yau as advisors.
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Richard Schoen is currently Distinguished Professor and Excellence in Teaching Chair at the University of California, Irvine.
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Richard Schoen received an NSF Graduate Research Fellowship in 1972 and a Sloan Research Fellowship in 1979.
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Richard Schoen has been invited to speak at the International Congress of Mathematicians three times, including twice as a Plenary Speaker.
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Richard Schoen was awarded an Honorary Doctor of Science from the University of Warwick in 2015.
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Richard Schoen received the Wolf Prize in Mathematics for 2017, shared with Charles Fefferman.
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Richard Schoen has had over 44 doctoral students, including Hubert Bray, Jose F Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and Andre Neves.
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Richard Schoen has investigated the use of analytic techniques in global differential geometry, with a number of fundamental contributions to the regularity theory of minimal surfaces and harmonic maps.
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In 1983, Richard Schoen obtained similar estimates in the special case of two-dimensional surfaces, making use of the existence of isothermal coordinates.
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Richard Schoen made partial progress towards proving that critical points must exist, which was taken further by Neil Trudinger and Thierry Aubin.
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In 1984, Richard Schoen settled the cases left open by Aubin's work, the decisive point of which rescaled the metric by the Green's function of the Laplace-Beltrami operator.
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The works of Yamabe, Trudinger, Aubin, and Richard Schoen together comprise the solution of the Yamabe problem, which asserts that there is a metric of constant scalar curvature in every conformal class.
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In 1989, Richard Schoen was able to adapt Karen Uhlenbeck's bubbling analysis, developed for other geometric-analytic problems, to the setting of constant scalar curvature.
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Simon Brendle and Richard Schoen were able to prove that the positivity of Mario Micallef and John Moore's "isotropic curvature" is preserved by the Ricci flow in any dimension, a fact independently proven by Huy Nguyen.
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Brendle and Richard Schoen were further able to relate their positivity condition to the positivity of sectional curvature and of curvature operator, which allowed them to exploit then-recent algebraic ideas of Christoph Bohm and Burkhard Wilking, thereby obtaining a new convergence theorem for Ricci flow.