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\nIn differential geometry, Yamabe–Witten equations are a family of PDEs. They arise in physics as equations of motion of the dilaton and graviton in string theory. They describe a conformal degree of freedom of a conformally invariant field theory. The equations are named after Masao Yamabe and David Witten.
\nDegenerate Yamabe–Witten equations\nYamabe–Witten equations are defined for the following case:
\nLet and be an -dimensional manifold and a differentiable compactly supported function on. is called the Witten–Yamabe conformal factor, where is the scalar curvature.
\nThe Witten–Yamabe conformal transformation of the metric is given by
\nwhere is the volume element. The conformal factor (also called the conformal factor of the conformally invariant field theory) is now defined as:
\nThe following are the Yamabe–Witten equations:
\nYamabe classification\nYamabe classification states that there are only three possibilities for a given manifold:\n The first is the Yamabe I case. There exists a conformal factor so that the scalar curvature and the covariant Laplacian. The Yamabe–Witten equations are solvable.\n The second is the Yamabe II case. There is no conformal factor so that the scalar curvature and the covariant Laplacian. The Yamabe–Witten equations are not solvable.\n The last is the zero case. The Witten–Yamabe conformal factor vanishes. The zero case is not physically significant since there is no conformal invariant field theory with degrees of freedom that is describable by a massless graviton and a scalar field.
\nReferences
\nCategory:Differential geometry\n Category:Partial differential equations\n Category:Mathematical physicsQ:
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