# Richard Melvin Schoen

### Quick Info

Celina, Ohio, USA

**Richard Schoen**is an American mathematician known for his work in differential geometry and geometric analysis.

### Biography

**Richard Schoen**is known to his friends and colleagues as Rick Schoen. His parents were Arnold Peter Schoen and Rosemary Heitkamp. He studied at the University of Dayton, in Dayton, Ohio, the local university for his home town of Celina being less than 100 km to the south east. He graduated from Dayton with a B.S. in 1972, receiving the University of Dayton's Distinguished Alumnus Award and the Special Achievement Award, and then, in the same year, he entered Stanford University to undertake graduate studies. He was awarded a National Science Foundation Graduate Fellowship to fund his studies from 1972 to 1975. He explained how he came to have two thesis advisors in an article he wrote in tribute to Shing-Tung Yau:-

I first met Shing-Tung Yau in 1973 when I was a second year Ph.D. student at Stanford University and he was a newly arrived faculty member. We became mathematically involved through a reading course I was doing with Leon Simon on minimal hypersurfaces. This led to a three-way joint work on properties of stable minimal hypersurfaces. I continued to work with both Yau and Leon while I was a student and was officially their joint Ph.D. student. I spent several hours a day working with (mostly learning from) Yau when I was a student. He was interested in anything geometric, and he had ideas for approaching a vast range of problems. This was an incredible opportunity for me, and it gave me a great start on my research career. We wrote two more joint papers while I was a student.These two joint papers were

*Curvature estimates for minimal hypersurfaces*(1975), and

*Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature*(1976). Schoen considered that he was very lucky to have the opportunity to work with Yau in these early days:-

I have vivid memories of Yau from the early times: his tremendous dedication to his work (he was in his office day and night including weekends), his amazing breadth of knowledge and technique, his openness and generosity with his time.Schoen left Stanford in 1976 to take up the position of Instructor in Mathematics at the University of California, Berkeley. He was awarded his Ph.D. from Stanford University in 1977 for his thesis

*Existence and Regularity Theorems for some Geometric Variational Problems*. Soon Yau also arrived in Berkeley and their collaboration continued:-

I left Stanford in 1976 to take up a two year instructorship in Berkeley. Yau came to Berkeley during my second year, and we continued our collaboration. It was in Berkeley that we began our work on scalar curvature and the positive mass theorem.It was in Berkeley that Schoen met Doris Helga Fischer-Colbrie. She had been born in Vienna, Austria, in January 1949 and had been educated at the University of California, Berkeley, receiving a BA in 1971 and a Master's degree two years later. She had been a Teaching Assistant at Berkeley while working for her doctorate advised by Blaine Lawson but, when Schoen arrived in Berkeley, she was a Research Assistant. She was awarded a Ph.D. in 1978 for her thesis

*Minimal Varieties: Theorems on Global Comportment and Local Existence*. Later, on 29 October 1983, Richard Schoen and Doris Fischer-Colbrie were married; they had two children Alan (born 29 February 1984) and Lucy (born 25 April 1988). Before they married, they wrote a joint paper

*The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature*which appeared in print in 1980. In it they studied minimal surfaces in three-dimensional manifolds which, on each compact set, minimize area up to second order. In the same year of 1980, Doris published the single-authored paper

*Some rigidity theorems for minimal submanifolds of the sphere*.

Schoen spent two years as an Instructor at Berkeley, then in 1978 he took up an appointment as an Assistant professor at New York University. He spent two years in this post and during these years he made major breakthroughs with the work that he and Shing-Tung Yau had begun at Berkeley:-

We expanded this work substantially over the next few years, and I remember wonderful times working together at Stanford during the summers of 1978 and 1979. During the 1979-80 academic year Yau organized a special year at the Institute for Advanced Study. This was another formative period in my career since there was so much going on in a wide variety of directions. I learned a lot and did some work that I am still proud of.As indicated by this quote, Schoen was a Visiting Member at the Institute for Advanced Study, Princeton, during 1979-80, a visit that was funded by a Sloan Postdoctoral Fellowship. Justin Corvino and Daniel Pollack describe the advances made by Schoen at this time [1]:-

[A] watershed for the mathematical development of relativity is the celebrated work of Rick Schoen and S-T Yau from the late seventies on the 'Positive Mass Theorem'. Not only did their work employ serious tools of geometric analysis, including partial differential equations and geometric measure theory, to resolve a question motivated by gravitational physics, but they also established a link between the positivity of the mass of an isolated gravitational system and the relationship between positive scalar curvature and topology, a topic of interest to a broad range of mathematicians. In the early eighties, Schoen brought the Positive Mass Theorem to bear on the resolution of the famous Yamabe problem, providing more evidence to support the development of the mathematical theory of the constraint equations, and inspiring many others to do so.In 1980 Schoen returned to the University of California, Berkeley, when he was appointed as Professor. He spent eight years at the University of California, the first four being at Berkeley and the remaining three, 1984-87, being at San Diego. During these seven years, Schoen was invited to address the International Congress of Mathematicians twice. He give a 45 minute invited lecture at the Congress held in Warsaw in 1983, then gave one of the plenary lectures at the Congress held in Berkeley in August 1986. At the Berkeley Congress he gave the lecture

*New Developments in the Theory of Geometric Partial Differential Equations*. Dennis DeTurck describes the content of Schoen's lecture:-

The author surveys recent work on nonlinear elliptic partial differential equations which arise from geometric sources, concentrating especially on the Yamabe problem and the theory of harmonic mappings. For the former, an outline is given of the recent solution of Yamabe's conjecture (that every metric on a compact manifold is pointwise conformally equivalent to one with constant scalar curvature), including the use of the positive mass theorem and a discussion of regularity of weak solutions of Yamabe's equation.The solution of the Yamabe problem on compact manifolds, which Schoen discussed in this lecture, is one of his greatest achievements. He solved this problem in 1984. He had been awarded a MacArthur Fellowship in August 1983 (two awards went to mathematicians with Karen Uhlenbeck receiving one at the same time) and he held this Fellowship until 1988. He had returned to Stanford University in 1987 and continues to work at Stanford as the Anne T and Robert M Bass Professor of Humanities and Sciences. Honours came rapidly: he was elected to the American Academy of Arts and Sciences in 1988; and he was awarded the Bôcher Memorial Prize by the American Mathematical Society in 1989:-

... for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a Riemannian metric to constant scalar curvature".Further honours acknowledged his many achievements: he was elected to the National Academy of Sciences in 1991, he became a fellow of American Association for Advancement of Science in 1995, and in the following year he was awarded a Guggenheim Fellowship. The Stanford University News described his work in the following terms when they announced his election to the National Academy of Sciences on 30 April 1991 [2]:-

Schoen, 40, continues his research in differential geometry, nonlinear partial differential equations and the calculus of variations. He constructs and analyzes geometric objects that optimize certain physical or geometric energies. For example, he has developed new ways to understand surfaces of least area spanning a curve in three-dimensional space - the mathematical model for soap films. Schoen's ideas have been applied to a wide range of mathematical problems, from general relativity to questions about rigidity for lattice subgroups of algebraic groups.We must not give the impression that Schoen's major research contributions stopped in the 1990s. Far from it and, just to give one example of recent highly significant work, let us note his achievement in 2007 when, in collaboration with Simon Brendle, he proved the differentiable sphere theorem. This is a fundamental result in the theory of manifolds with positive sectional curvature.

In addition to the visiting positions which we mentioned above, Schoen was a Visiting Member of the Institute for Advanced Study, Princeton, in the Spring of 1984, a Visiting Professor at the Courant Institute, New York University in academic year 1989-90, Distinguished Visiting Professor at the Institute for Advanced Study, Princeton, in academic year 1992-93, and a Visiting Professor at Harvard University in the autumn of 1999.

Schoen has published two important books, both in collaboration with Shing-Tung Yau, which were based on lecture courses. In 1994 they published

*Lectures on differential geometry*. We give the first and last paragraphs from a review by Man Chun Leung:-

As the authors note in their introduction, the book under review was written for the lecture series given at Princeton University in 1983 and at the University of California, San Diego, in 1984 and 1985. The book contains significant results in differential geometry and global analysis; many of them are the works of the authors. The main topics are differential equations on a manifold and the relation between curvature and topology of a Riemannian manifold. There are nine chapters in the book, with the last three chapters more like appendices, which focus on problems concerning different areas of differential geometry.The second book is

...

The book under review is very well written. Readers will find comprehensive and detailed discussions of many significant results in geometric analysis. The book is both useful as a reference book for researchers and as a course book for graduate students. With details of proofs and background materials presented in a concise and delightful way, the book provides access to some of the most exciting areas in differential geometry.

*Lectures on harmonic maps*(1997). We give a short quote from a detailed review by John C Wood:-

This is a very useful contribution to the literature on harmonic maps. It is not an elementary textbook on harmonic maps ... It is rather a collection of some of the most important topics in, and applications of, harmonic maps, skewed towards the authors' interests. ... this is a book that everyone interested in harmonic maps will benefit from reading.Other important contributions to mathematics by Schoen include his editorial work. He serves on the editorial boards of: the

*Journal of Differential Geometry*,

*Communications in Analysis and Geometry*,

*Communications in Partial Differential Equations*,

*Calculus of Variations and Partial Differential Equations*, and

*Communications in Contemporary Mathematics*. He also does important work for the American Mathematical Society serving on various committees. For example he serves on the Committee to Select the Winner of the E H Moore Research Article Prize, the Committee for National Awards and Public Representation, and the Committee to award Steele Prizes.

Few lecturers receive such praise from their undergraduate students as Schoen does. We give just one short quote from many:-

Best Professor I've had so far at Stanford. His lectures are very lucid and compliment the text.We end this biography by quoting from [1] written by two mathematicians who received their doctorates with Schoen as advisor:-

The mathematical influence of Richard M Schoen can be measured in many ways. His research has fundamentally shaped geometric analysis, and his results form many cornerstones within geometry, partial differential equations and general relativity. Evidence of his influence includes the large number of his students who continue to work in these areas. As two of these students, the authors of this contribution are exceedingly grateful for Rick's mathematical insight and generosity.

### References (show)

- J Corvino and D Pollack, Scalar Curvature and the Einstein Constraint Equations. http://fangdian.net/pdf/1102.5050v1
- National Academy of Sciences elects three Stanford scholars,
*Stanford University News Service*(7 May 1991).

### Additional Resources (show)

Other websites about Richard Schoen:

### Honours (show)

Honours awarded to Richard Schoen

Written by J J O'Connor and E F Robertson

Last Update July 2011

Last Update July 2011