- Yamabe problem
The Yamabe problem in
differential geometry takes its name from the mathematicianHidehiko Yamabe . Although Yamabe claimed to have a solution in 1960, a critical errorin his proof was discovered in 1968. The combined work ofNeil Trudinger ,Thierry Aubin , andRichard Schoen provided a complete solution to the problem as of 1984.The Yamabe problem is the following: given a smooth,
compact manifold "M" of dimension n geq 3 with aRiemannian metric g, does there exist a metric g' conformal to g for which thescalar curvature of g' is constant? In other words, does there exist a smooth function "f" on "M" for whichthe metric g' = e^{2f}g has constant scalar curvature? The answer is now known to be yes, and was proved using techniques fromdifferential geometry ,functional analysis andpartial 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 M,g)which is not compact, does there exist a conformal metric of constant scalar curvature that is also complete? The answer is well-known to be no, due to counterexamplesgiven by
Jin Zhiren .See also
*
Yamabe flow
*Yamabe invariant References
* J. Lee and T. Parker, "The Yamabe problem", Bull. Amer. Math. Soc. 17, 37-81 (1987).
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