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CracksNow – Find latest Software cracking downloads. Read about the latest Software crack and keygens and find best Software cracks and keygens by genre. Find newest cracks and keygens and new Software cracks and keygens that you can not get on the web.Yamabe–Witten equations and the existence problem for the minimal two-sphere
In 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.
Degenerate Yamabe–Witten equations
Yamabe–Witten equations are defined for the following case:
Let and be an -dimensional manifold and a differentiable compactly supported function on. is called the Witten–Yamabe conformal factor, where is the scalar curvature.
The Witten–Yamabe conformal transformation of the metric is given by
where is the volume element. The conformal factor (also called the conformal factor of the conformally invariant field theory) is now defined as:
The following are the Yamabe–Witten equations:
Yamabe classification
Yamabe classification states that there are only three possibilities for a given manifold:
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.
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.
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.
References
Category:Differential geometry
Category:Partial differential equations
Category:Mathematical physicsQ:
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