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Cohn-Vossen's inequality

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Relates the integral of Gaussian curvature of surfaces to the Euler characteristic

In differential geometry, Cohn-Vossen's inequality, named after Stefan Cohn-Vossen, relates the integral of Gaussian curvature of a non-compact surface to the Euler characteristic. It is akin to the Gauss–Bonnet theorem for a compact surface.

A divergent path within a Riemannian manifold is a smooth curve in the manifold that is not contained within any compact subset of the manifold. A complete manifold is one in which every divergent path has infinite length with respect to the Riemannian metric on the manifold. Cohn-Vossen's inequality states that in every complete Riemannian 2-manifold S with finite total curvature and finite Euler characteristic, we have

∬ S K d A ≤ 2 π χ ( S ) , {\displaystyle \iint _{S}K\,dA\leq 2\pi \chi (S),}

where K is the Gaussian curvature, dA is the element of area, and χ is the Euler characteristic.

Examples

  • If S is a compact surface (without boundary), then the inequality is an equality by the usual Gauss–Bonnet theorem for compact manifolds.
  • If S has a boundary, then the Gauss–Bonnet theorem gives
∬ S K d A = 2 π χ ( S ) − ∫ ∂ S k g d s {\displaystyle \iint _{S}K\,dA=2\pi \chi (S)-\int _{\partial S}k_{g}\,ds}
where k g {\displaystyle k_{g}} is the geodesic curvature of the boundary, and its integral the total curvature which is necessarily positive for a boundary curve, and the inequality is strict. (A similar result holds when the boundary of S is piecewise smooth.)
  • If S is the plane R, then the curvature of S is zero, and χ(S) = 1, so the inequality is strict: 0 < 2π.

Notes and references

  1. Robert Osserman, A Survey of Minimal Surfaces, Courier Dover Publications, 2002, page 86.

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