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The Yamabe problem in differential geometry concerns the existence of Riemannian metrics with constant scalar curvature, and takes its name from the mathematician Hidehiko Yamabe. claimed to have a solution, but discovered a critical error in his proof. The combined work of Neil Trudinger, Thierry Aubin, and Richard Schoen later provided a complete solution to the problem in 1984.〔http://www.math.mcgill.ca/gantumur/math580f12/Yamabe.pdf 〕 The Yamabe problem is the following: given a smooth, compact manifold of dimension with a Riemannian metric , does there exist a metric conformal to for which the scalar curvature of is constant? In other words, does a smooth function exist on for which the metric has constant scalar curvature? The answer is now known to be yes, and was proved using techniques from differential geometry, functional analysis and partial differential equations. ==The non-compact case== A closely related question is the so-called "non-compact Yamabe problem", which asks: on a smooth, complete Riemannian manifold which is not compact, does there exist a conformal metric of constant scalar curvature that is also complete? The answer is no, due to counterexamples given by . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Yamabe problem」の詳細全文を読む スポンサード リンク
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