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Bi-Lipschitz parameterization of surfaces


We consider surfaces Z homeomorphic to the plane with complete, possibly singular Riemannian metrics. If we have ∫ Z K +<2π−ε for the positive and ∫ Z K <C for the negative part of the integral curvature, then Z is L-bi-Lipschitz equivalent to ℝ2 with L depending only on ε>0 and C>0. This result implies a conjecture by J. Fu.

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Correspondence to Mario Bonk.

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Supported by NSF grant DMS-0200566.

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Bonk, M., Lang, U. Bi-Lipschitz parameterization of surfaces. Math. Ann. 327, 135–169 (2003).

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