Richard Schoen

Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis.

Born in Celina, Ohio, and a 1968 graduate of Fort Recovery High School, he received his B.S. from the University of Dayton in mathematics.

Schoen received an NSF Graduate Research Fellowship in 1972 and a Sloan Research Fellowship in 1979.

In 1976, Schoen and Shing-Tung Yau used Yau's earlier Liouville theorems to extend the rigidity phenomena found earlier by James Eells and Joseph Sampson to noncompact settings.

By identifying a certain interplay of the Bochner identity for harmonic maps together with the second variation of area formula for minimal hypersurfaces, they also identified some novel conditions on the domain leading to the same conclusion.

These rigidity theorems are complemented by their existence theorem for harmonic maps on noncompact domains, as a simple corollary of Richard Hamilton's resolution of the Dirichlet boundary-value problem.

As a consequence they found some striking geometric results, such as that certain noncompact manifolds do not admit any complete metrics of nonnegative Ricci curvature.

He then received his PhD in 1977 from Stanford University.

In 1979, Schoen and his former doctoral supervisor, Shing-Tung Yau, made a number of highly influential contributions to the study of positive scalar curvature.

By an elementary but novel combination of the Gauss equation, the formula for second variation of area, and the Gauss-Bonnet theorem, Schoen and Yau were able to rule out the existence of several types of stable minimal surfaces in three-dimensional manifolds of positive scalar curvature.

By contrasting this result with an analytically deep theorem of theirs establishing the existence of such surfaces, they were able to achieve constraints on which manifolds can admit a metric of positive scalar curvature.

Schoen and Doris Fischer-Colbrie later undertook a broader study of stable minimal surfaces in 3-dimensional manifolds, using instead an analysis of the stability operator and its spectral properties.

An inductive argument based upon the existence of stable minimal hypersurfaces allowed them to extend their results to higher dimensions.

Further analytic techniques facilitated the application of topological surgeries on manifolds which admit metrics of positive scalar curvature, showing that the class of such manifolds is topologically rich.

Mikhael Gromov and H. Blaine Lawson obtained similar results by other methods, also undertaking a deeper analysis of topological consequences.

By an extension of their techniques to noncompact manifolds, Schoen and Yau were able to settle the important Riemannian case of the positive mass theorem in general relativity, which can be viewed as a statement about the geometric behavior near infinity of noncompact manifolds with positive scalar curvature.

In two papers from the 1980s, Schoen and Karen Uhlenbeck made a foundational contribution to the regularity theory of energy-minimizing harmonic maps.

The techniques they developed, making extensive use of monotonicity formulas, have been very influential in the field of geometric analysis and have been adapted to a number of other problems.

Fundamental conclusions of theirs include compactness theorems for sets of harmonic maps and control over the size of corresponding singular sets.

Leon Simon applied such results to obtain a clear picture of the small-scale geometry of energy-minimizing harmonic maps.

Later, Mikhael Gromov had the insight that an extension of the theory of harmonic maps, to allow values in metric spaces rather than Riemannian manifolds, would have a number of significant applications, with analogues of the classical Eells−Sampson rigidity theorem giving novel rigidity theorems for lattices.

The intense analytical details of such a theory were worked out by Schoen.

Further foundations of this new context for harmonic maps were laid out by Schoen and Nicholas Korevaar.

Schoen is a 1983 MacArthur Fellow.

He has been invited to speak at the International Congress of Mathematicians (ICM) three times, including twice as a Plenary Speaker.

In 1983 he was an Invited Speaker at the ICM in Warsaw, in 1986 he was a Plenary Speaker at the ICM in Berkeley, and in 2010 he was a Plenary Speaker at the ICM in Hyderabad.

He is best known for the resolution of the Yamabe problem in 1984.

After faculty positions at the Courant Institute, NYU, University of California, Berkeley, and University of California, San Diego, he was Professor at Stanford University from 1987–2014, as Bass Professor of Humanities and Sciences since 1992.

He is currently Distinguished Professor and Excellence in Teaching Chair at the University of California, Irvine.

His surname is pronounced "Shane."

He was elected to the American Academy of Arts and Sciences in 1988 and to the National Academy of Sciences in 1991, became Fellow of the American Association for the Advancement of Science in 1995, and won a Guggenheim Fellowship in 1996.

For his work on the Yamabe problem, Schoen was awarded the Bôcher Memorial Prize in 1989.

In 2012 he became a Fellow of the American Mathematical Society.

He received the 2014–15 Dean’s Award for Lifetime Achievements in Teaching from Stanford University.

In 2015, he was elected Vice President of the American Mathematical Society.

He was awarded an Honorary Doctor of Science from the University of Warwick in 2015.

He received the Wolf Prize in Mathematics for 2017, shared with Charles Fefferman.

In the same year, he was awarded the Heinz Hopf Prize, the Lobachevsky Medal and Prize by Kazan Federal University, and the Rolf Schock Prize.

He has had over 44 doctoral students, including Hubert Bray, José F. Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and André Neves.

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.