Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I would like to know what is the most general form of the Gauss-Bonnet theorem in the plane for curves. It is well known for that for any piecewise $C^2$ simply connected curve with corners, one has

$\int_{\partial \Omega} \kappa(y) dS(y) = 2\pi$,

where $\kappa$ denotes the curvature of the boundary $\partial \Omega$. This formula continues to hold for any curve for which $\hat n \cdot (1,0)$ (for instance) defines a $BV$ function.

Question: I would like to know to what extent this generalizes. For example, if $\Omega$ is a simply connected, compact set in $\mathbb{R}^2$ whose boundary has finite length and has generalized curvature $\kappa \in L^{\infty}(\partial \Omega)$, does it hold that $\int_{\partial \Omega} \kappa(y) dS(y) = 2\pi$? If not, are there counter examples in the plane where this fails?

share|improve this question
    
How about the boundary of the von-koch snowflake? –  john mangual May 29 '12 at 10:23
    
I am looking for a class of curves, in paritcular rectifiable verifolds with generalized mean curvature in $L^{\infty}(\partial \Omega)$.. –  Dorian May 29 '12 at 10:42

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.