Cohn-Vossen's inequality explained

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^[1]

\iint _{S} K dA \le 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

\iint _{S} K dA = 2 \pi\chi(S) - \int _{\partial} k _{gds}

where

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 R², then the curvature of S is zero, and χ(S) = 1, so the inequality is strict: 0 < 2.

Notes and references

• 1556908. Cohn-Vossen. Stefan. Kürzeste Wege und Totalkrümmung auf Flächen. Compositio Mathematica. 2. 1935. 69–13. Stefan Cohn-Vossen. 0011.22501. 61.0789.01.
• 0094452. Huber. Alfred. On subharmonic functions and differential geometry in the large. Commentarii Mathematici Helvetici. 32. 1957. 13–72. 1. 10.1007/BF02564570. 0080.15001. 2027/mdp.39015095254580. free.
• Encyclopedia: 1919432. Li. Peter. Curvature and function theory on Riemannian manifolds. Surveys in Differential Geometry. 375–432. 7. International Press. Somerville, MA. 2000. Peter Li (mathematician). 1-57146-069-1. Shing-Tung Yau. Yau. S.-T.. 10.4310/SDG.2002.v7.n1.a13. free. 1066.53084.
• Book: 2028047. Shiohama. Katsuhiro. Shioya. Takashi. Tanaka. Minoru. The geometry of total curvature on complete open surfaces. Cambridge Tracts in Mathematics. 159. Cambridge University Press. Cambridge. 2003. 0-521-45054-3. 10.1017/CBO9780511543159. 1086.53056.

External links

• Gauss–Bonnet theorem, in the Encyclopedia of Mathematics, including a brief account of Cohn-Vossen's inequality

Notes and References

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