[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.
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.
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.