Masatake Kuranishi

Masatake Kuranishi (倉西 正武 Kuranishi Masatake; July 19, 1924 – June 22, 2021)^[1] was a Japanese mathematician who worked on several complex variables, partial differential equations, and differential geometry.

Education and career

Kuranishi received in 1952 his

Kuranishi and Élie Cartan established the eponymous Cartan–Kuranishi Theorem on the continuation of exterior differential forms.^[6] In 1962, based upon the work of Kunihiko Kodaira and Donald Spencer, Kuranishi constructed locally complete deformations of compact complex manifolds.^[7]

In 1982 he made important progress in the embedding problem for CR manifolds (Cauchy–Riemann structures).

In a series of deep papers published in 1982 [Kur I,^[8] II,^[9] III^[10]], Kuranishi developed the theory of harmonic integrals on strongly pseudoconvex CR structures over small balls along the line developed by D. C. Spencer, C. B. Morrey, J. J. Kohn and Nirenberg. He considered a strongly pseudoconvex CR structure on a manifold of real dimension ${\displaystyle 2n-1}$. In [Kur I], he established the a priori estimate for the Neumann boundary problem on the complex associated with the structure, in the case the structure is induced by an embedding in ${\displaystyle \mathbb {C} ^{n}}$ and restricted to a small ball of special type, provided ${\displaystyle 1\leq q\leq n-3}$, where q is the degree of differential forms. In [Kur II], he developed the regularity theorem of solutions of the Neumann boundary problem based on the a priori estimate of [Kur I]. As a significant application of his deep theory, he proved in [Kur III] that, when ${\displaystyle n\geq 5}$, the structure is realized on a neighborhood of a reference point by an embedding in ${\displaystyle \mathbb {C} ^{n}}$.^[11]

Thus, by Kuranishi's work, in real dimension 9 and higher, local embedding of abstract CR structures is true and is also true in real dimension 7 by the work of Akahori.^[12] A simplified presentation of Kuranishi's proof is due to Sidney Webster.^[13] For ${\displaystyle n=2}$ (i.e., real dimension 3), Nirenberg published a counterexample. The local embedding problem remains open in real dimension 5.

Selected publications

• Heisuke Hironaka (ed.): Masatake Kuranishi - Selected Papers, Springer 2010
• Kuranishi: Deformations of compact complex manifolds, Montreal, Presses de l'Universite de Montreal, 1971, 99 pages.
• Kuranishi: Lectures on involutive systems of partial differential equations, Sociedade de matemática de São Paulo, 1967, 75 pages.
• Kuranishi with notes by M.K. Venkatesha Murthy: Lectures on exterior differential systems, Tata Institute of Fundamental Research, 1962.

See also

• Kuranishi structure

References