A C² Riemannian surface with curvature bounded pointwise by κ admits, on a disc of radius δ, an isothermal coordinate chart whose bi-Lipschitz constant is at most exp(4 δ² κ), for δ²κ bounded below π²/8.
V ., Conformal Invariants: Topics in geometric function theory
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The bi-Lipschitz constant of an isothermal coordinate chart
A C² Riemannian surface with curvature bounded pointwise by κ admits, on a disc of radius δ, an isothermal coordinate chart whose bi-Lipschitz constant is at most exp(4 δ² κ), for δ²κ bounded below π²/8.