Above all, he is famous for discovering[2] that every conformal class on a smooth compact manifold is represented by a Riemannian metric of constant scalar curvature.
After coming back to Japan, Etsuko Yamabe and her daughters lived with the benefits of Hidehiko's social security and of funds raised privately by her and her husband's friends in the United States of America.
This assertion, which naturally generalizes the uniformization of Riemann surfaces to arbitrary dimensions, is completely correct, as is the broad outline of Yamabe's proof.
However, Yamabe's argument contains a subtle analytic mistake arising form the failure of certain natural inclusions of Sobolev spaces to be compact.
This mistake was only corrected in stages, on a case-by-case basis, first by Trudinger ("Remarks Concerning the Conformal Deformation of Metrics to Constant Scalar Curvature", Ann.
9: 55 (1976) 269–296), and finally, in full generality, by Schoen ("Conformal Deformation of a Riemannian Metric to Constant Scalar Curvature," Journal of Differential Geometry 20 (1984) 478-495).
In this sense, the influence of Yamabe's 1960 paper in the Osaka Journal has become such a universal fixture of current mathematical thought that it is often implicitly referred to without an explicit citation.