In mathematics, in the field of differential geometry, the Yamabe invariant, also referred to as the sigma constant, is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe. Used by Vincent Moncrief and Arthur Fischer to study reduced Hamiltonian for Einstein's equations.
Definition Let M {\displaystyle M} be a compact smooth manifold (without boundary) of dimension n ≥ 2 {\displaystyle n\geq 2} . The normalized Einstein–Hilbert functional E {\displaystyle {\mathcal {E}}} assigns to each Riemannian metric g {\displaystyle g} on M {\displaystyle M} a real number as follows:
E ( g ) = ∫ M R g d V g ( ∫ M d V g ) n − 2 n , {\displaystyle {\mathcal {E}}(g)={\frac {\int _{M}R_{g}\,dV_{g}}{\left(\int _{M}\,dV_{g}\right)^{\frac {n-2}{n}}}},}
where R g {\displaystyle R_{g}} is the scalar curvature of g {\displaystyle g} and d V g {\displaystyle dV_{g}} is the volume density associated to the metric g {\displaystyle g} . The exponent in the denominator is chosen so that the functional is scale-invariant: for every positive real constant c {\displaystyle c} , it satisfies E ( c g ) = E ( g ) {\displaystyle {\mathcal {E}}(cg)={\mathcal {E}}(g)} . We may think of E ( g ) {\displaystyle {\mathcal {E}}(g)} as measuring the average scalar curvature of g {\displaystyle g} over M {\displaystyle M} . It was conjectured by Yamabe that every conformal class of metrics contains a metric of constant scalar curvature (the so-called Yamabe problem); it was proven by Yamabe, Trudinger, Aubin, and Schoen that a minimum value of E ( g ) {\displaystyle {\mathcal {E}}(g)} is attained in each conformal class of metrics, and in particular this minimum is achieved by a metric of constant scalar curvature. We define
Y ( g ) = inf f E ( e 2 f g ) , {\displaystyle Y(g)=\inf _{f}{\mathcal {E}}(e^{2f}g),}
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