Toshiki Mabuchi (kanji: 満渕俊樹, hiragana: マブチ トシキ, Mabuchi Toshiki, born in 1950) is a Japanese mathematician, specializing in complex differential geometry and algebraic geometry.[1] In 2006 in Madrid he was an invited speaker at the International Congress of Mathematicians.[2] Mabuchi is known for introducing the Mabuchi functional.

Education and career

In 1972 Mabuchi graduated from the University of Tokyo Faculty of Science[1] and became a graduate student in mathematics at the University of California, Berkeley.[3] There he graduated with a Ph.D. in 1977 with thesis C3-Actions and Algebraic Threefolds with Ample Tangent Bundle and advisor Shoshichi Kobayashi[4] As a postdoc Mabuchi was from 1977 to 1978 a guest researcher at the University of Bonn. Since 1978 he is a faculty member of the Department of Mathematics of Osaka University. His research deals with complex differential geometry, extremal Kähler metrics, stability of algebraic varieties, and the Hitchin–Kobayashi correspondence.[1]

In 2006 Toshiki Mabuchi and Takashi Shioya received the Geometry Prize of the Mathematical Society of Japan.

Research contributions

Mabuchi is well-known for his introduction, in 1986, of the Mabuchi energy, which gives a variational interpretation to the problem of Kähler metrics of constant scalar curvature. In particular, the Mabuchi energy is a real-valued function on a Kähler class whose Euler-Lagrange equation is the constant scalar curvature equation. In the case that the Kähler class represents the first Chern class of the complex manifold, one has a relation to the Kähler-Einstein problem, due to the fact that constant scalar curvature metrics in such a Kähler class must be Kähler-Einstein.

Owing to the second variation formulas for the Mabuchi energy, every critical point is stable. Furthermore, if one integrates a holomorphic vector field and pulls back a given Kähler metric by the corresponding one-parameter family of diffeomorphisms, then the corresponding restriction of the Mabuchi energy is a linear function of one real variable; its derivative is the Futaki invariant discovered a few years earlier by Akito Futaki.[5] The Futaki invariant and Mabuchi energy are fundamental in understanding obstructions to the existence of Kähler metrics which are Einstein or which have constant scalar curvature.

A year later, by use of the ∂∂-lemma, Mabuchi considered a natural Riemannian metric on a Kähler class, which allowed him to define length, geodesics, and curvature; the sectional curvature of Mabuchi's metric is nonpositive. Along geodesics in the Kähler class, the Mabuchi energy is convex. So the Mabuchi energy has strong variational properties.

Selected publications

Articles

  • Mabuchi, Toshiki (1986). "-energy maps integrating Futaki invariants". Tohoku Mathematical Journal. 38 (4): 575–593. doi:10.2748/tmj/1178228410. ISSN 0040-8735.
  • Bando, Shigetoshi; Mabuchi, Toshiki (1987). "Uniqueness of Einstein Kähler Metrics Modulo Connected Group Actions". Algebraic Geometry, Sendai, 1985. pp. 11–40. doi:10.2969/aspm/01010011. ISBN 978-4-86497-068-6. ISSN 0920-1971. {{cite book}}: |journal= ignored (help)
  • Mabuchi, Toshiki (1987). "Some symplectic geometry on compact Kähler manifolds. I". Osaka Journal of Mathematics. 24 (2): 227–252.

Books

References

  1. 1 2 3 "Mabuchi Toshiki". J-Global – Japan Science and Technology Agency.
  2. ↑ Mabuchi, Toshiki (2006). "Extremal metrics and stabilities on polarized manifolds". arXiv:math/0603493. (published in vol. 2 of the Proceedings of the ICM, Madrid 2006, pages 813–826)
  3. ↑ Mabuchi, Toshiki (July 25, 2013). "Remembering Professor Shoshichi Kobayashi". (translated form the original Japanese by Hisashi Kobayashi)
  4. ↑ Toshiki Mabuchi at the Mathematics Genealogy Project
  5. ↑ A. Futaki. An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73 (1983), no. 3, 437–443.
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