Toshiki Mabuchi

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). "${\displaystyle K}$-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

• Mabuchi, Toshiki; Mukai, Shigeru, eds. (1993). Einstein Metrics and Yang-Mills Connections. Lecture Notes in Pure and Applied Mathematics. Vol. 145. CRC Press. ISBN 978-0-8247-9069-1.
• ——; Noguchi, Junjiro; Ochiai, Takushiro, eds. (1994). Geometry and Analysis on Complex Manifolds: Festschrift for Professor S. Kobayashi's 60th Birthday. World Scientific. ISBN 978-981-02-2067-9.