Toshiki Mabuchi (

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: 満渕俊樹,

hiragana is a Japanese syllabary, part of the Japanese writing system, along with ''katakana'' as well as ''kanji''. It is a phonetic lettering system. The word ''hiragana'' literally means "flowing" or "simple" kana ("simple" originally as contrast ...

: マブチトシキ, Mabuchi Toshiki, born in 1950) is a Japanese mathematician, specializing in complex differential geometry and algebraic geometry. In 2006 in Madrid he was an invited speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

. Mabuchi is known for introducing the Mabuchi functional.

Education and career

In 1972 Mabuchi graduated from the University of Tokyo Faculty of Science and became a graduate student in mathematics at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

. There he graduated with a Ph.D. in 1977 with thesis ''C3-Actions and Algebraic Threefolds with Ample Tangent Bundle'' and advisor

Shoshichi Kobayashi was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie alge ...

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 , abbreviated as , is a public research university located in Osaka Prefecture, Japan. It is one of Japan's former Imperial Universities and a Designated National University listed as a "Top Type" university in the Top Global University Project. ...

. His research deals with complex differential geometry, extremal

Kähler metric Kähler may refer to: ;People * Alexander Kähler (born 1960), German television journalist * Birgit Kähler (born 1970), German high jumper *Erich Kähler (1906–2000), German mathematician *Heinz Kähler (1905–1974), German art historian and a ...

s, stability of algebraic varieties, and the Hitchin–Kobayashi correspondence. 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 In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...

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.A. Futaki. ''An obstruction to the existence of Einstein Kähler metrics.'' Invent. Math. 73 (1983), no. 3, 437–443. 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 In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

on a Kähler class, which allowed him to define length,

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

s, and

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...

; the

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

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

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Books

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References

{{DEFAULTSORT:Mabuchi, Toshiki Differential geometers University of Tokyo alumni University of California, Berkeley alumni Academic staff of Osaka University 1950 births Living people 20th-century Japanese mathematicians 21st-century Japanese mathematicians