Evolution Equations in Geometry

Partial differential equations have been used for a long time to model the evolution of physical systems in time, the theory of their solutions has often been developed in close correspondence to a progressive understanding and continuing development of the underlying physical models. Often the partial differential equation links the physical phenomenon to a geometrical model: A soapfilm at rest is modelled by the nonlinear elliptic minimal surface equation, representing a hypersurface of vanishing extrinsic mean curvature in the surrounding space. The vacuum in General Relativity is modelled by a Lorentzian manifold of vanishing intrinsic average curvature as described by the Einstein field equations for the metric. The Einstein equations can be interpreted as a hyperbolic evolution system for the induced metric and extrinsic curvature of a 3-dimesional spacelike hypersurface creating the 4-dimensional spacetime through time.

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