A Riemannian four vertex theorem for surfaces with boundary

Ghomi, Mohammad

doi:10.1090/S0002-9939-2010-10507-5

A Riemannian four vertex theorem for surfaces with boundary
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Proc. Amer. Math. Soc. 139 (2011), 293-303 Request permission

Abstract:

We prove that every metric of constant curvature on a compact surface $M$ with boundary $\partial M$ induces at least four vertices, i.e., local extrema of geodesic curvature on a connected component of $\partial M$ if, and only if, $M$ is simply connected. Indeed, when $M$ is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on $M$ with only two critical points of geodesic curvature on each component of $\partial M$. With few exceptions, these metrics are obtained by removing the singularities and a perturbation of flat structures on closed surfaces.

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