August 22, 1923|
|Died||November 20, 1960
|Fields||Differential geometry, Group theory|
|Institutions||Osaka University, Princeton University, University of Minnesota, Northwestern University|
|Alma mater||University of Tokyo|
|Doctoral advisor||Shokichi Iyanaga|
|Known for||Hilbert's fifth problem, Yamabe flow, Yamabe invariant, Yamabe problem|
|Influenced||Differential geometry, Group theory|
Hidehiko Yamabe (山辺 英彦? Yamabe Hidehiko, August 22, 1923 in Ashiya, Hyōgo, Japan – November 20, 1960 in Evanston, Illinois) was a Japanese mathematician. Above all, he is famous for discovering that every conformal class on a smooth compact manifold is represented by a Riemannian metric of constant scalar curvature. Other notable contributions include his definitive solution of Hilbert's fifth problem.
Hidehiko Yamabe was born on August 22, 1923 in the city of Ashiya, belonging to the Hyōgo Prefecture, the sixth son of Takehiko and Rei Yamabe. After completing the Senior High School in September 1944, he joined Tokyo University as a student of the Department of Mathematics and graduated in September 1947: his doctoral advisor was Shokichi Iyanaga. He was then associated with the Department of Mathematics at Osaka University until June 1956, even while employed by the Department of Mathematics at Princeton University in Princeton, New Jersey. Shortly before coming to the United States of America, Yamabe married his wife Etsuko, and by 1956 they had two daughters. Yamabe died suddenly of a stroke in November 1960, just months after accepting a full professorship at Northwestern University.
After graduating from the University of Tokyo in 1947, Yamabe became an assistant at Osaka University. From 1952 until 1954 he was an assistant at Princeton University, receiving his Ph.D. from Osaka University while at Princeton. He left Princeton in 1954 to become assistant professor at the University of Minnesota. Except for one year as a professor at Osaka University, he stayed in Minnesota until 1960. Yamabe died suddenly of a stroke in November 1960, just months after accepting a full professorship at Northwestern University.
The Yamabe Memorial Lecture and the Yamabe Symposium
After coming back to Japan, Etsuko Yamabe and her daughters lived with the benefits of Hidehiko's social security and of funds raised privately by her and her husband friends in the United States of America. When she had achieved some financial stability, it was her wish to return the kindness shown to her in a time of great need by setting up funds for an annual lecture, to be alternatively held at Northwestern and Minnesota University: the Yamabe Memorial Lecture was so established, and was able to attract distinguished lecturers as Eugenio Calabi. Further funding permitted the expansion of the lecture to the present state bi-annual Yamabe Symposium.
Half of Yamabe's papers concern the theory of Lie groups and related topics. However, he is best known today for his remarkable posthumous paper, "On a deformation of Riemannian structures on compact manifolds," Osaka Math. J. 12 (1960) 21–37. This paper claims to prove that any Riemannian metric on any compact manifold without boundary is conformal to another metric for which the scalar curvature is constant. This assertion, which naturally generalizes the uniformization of Riemann surfaces to arbitrary dimensions, is completely correct, as is the broad outline of Yamabe's proof. However, Yamabe's argument contains a subtle analytic mistake arising form the failure of certain natural inclusions of Sobolev spaces to be compact. This mistake was only corrected in stages, on a case-by-case basis, first by Trudinger ("Remarks Concerning the Conformal Deformation of Metrics to Constant Scalar Curvature", Ann. Scuola Norm. Sup. Pisa 22 (1968) 265–274), then by Aubin (Équations Différentielles Non Linéaires et Problème de Yamabe, J. Math. Pures Appl. 9: 55 (1976) 269–296), and finally, in full generality, by Schoen ("Conformal Deformation of a Riemannian Metric to Constant Scalar Curvature," Journal of Differential Geometry 20 (1984) 478-495). Yamabe's visionary paper thereby became a cornerstone of modern Riemannnian geometry, and is thus largely responsible for his posthumous fame. For example, as of January 16, 2015, MathSciNet records 186 citations of Yamabe's 1960 paper in the Osaka Journal, compared with only 148 citations of all of his other publications combined. As of January 16, 2015, MathSciNet also lists 997 reviews containing the word "Yamabe." This, of course, is notably larger than the number of papers that explicitly cite any of Yamabe's articles. However, the vast majority of these reviews contain one of the phrases "scalar curvature" or "Yamabe equation," referring to Yamabe's equation governing the behavior of the scalar curvature under conformal rescaling. In this sense, the influence of Yamabe's 1960 paper in the Osaka Journal has become such a universal fixture of current mathematical thought that it is often implicitly referred to without an explicit citation.
- Boas, R. P., ed. (1967), Collected works of Hidehiko Yamabe, Notes on Mathematics and its Applications, New York–London–Paris: Gordon and Breach Science Publishers, pp. XII+142, MR 0223206, Zbl 0153.30502
- According to the Yamabe Symposium Organizing Committee (2008, p. 6)
- Lee and Parker, The Yamabe Problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91.
- According to Goto (1961, p. i): however, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers. For a review of recent claims (however completely ignoring the contributions of Yamabe) and for a new one, see Rosinger (1998, pp. xiii–xiv and pp. 169–170). For a general review, including an historical sketch dealing with all contributors, see the Hilbert's fifth problem entry.
- The content of this section is largely based on the commemoration by Goto (1961, p. i).
- According to Goto (1961, p. i), who also reports that Yamabe precisely suffered from subarachnoid hemorrhage.
- According to Goto (1961, p. i), who also refers that he precisely suffered from subarachnoid hemorrhage.
- According to the University of Minnesota School of Mathematics Newsletter (2008 p. 6).
- According to the University of Minnesota School of Mathematics Newsletter (2008 p. 7).
- According to the University of Minnesota School of Mathematics Newsletter (2008 p. 7): see also the brief historical sketch "History of the Yamabe Memorial Symposium" at the Symposium web page.
- According to Goto (1961, p. i).
- See (Boas 1967).
- Goto, Morikuni (1961), "Hidehiko Yamabe (1923–1960)", Osaka Mathematical Journal, 13 (1): i–ii, MR 0126362, Zbl 0095.00505. Available from Project Euclid.
- Rosinger, Elemér E. (1998), Parametric Lie Group Actions on Global Generalised Solutions of Nonliear PDE. Including a solution to Hilbert's Fifth Problem., Mathematics and Its Applications, 452, Doerdrecht–Boston–London: Kluwer Academic Publishers, pp. xvii+234, ISBN 0-7923-5232-7, MR 1658516, Zbl 0934.35003.
- University of Minnesota, School of Mathematics (January 24, 2012), History of the Yamabe Memorial Symposium, retrieved March 16, 2012.
- Yamabe Symposium Organizing Committee (2008), "Yamabe Symposium: Early History" (PDF), School of Mathematics Newsletter, University of Minnesota, Volume 14 (Spring): 6–7.