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We present a new AFEM for the Laplace-Beltrami operator with arbitrarypolynomial degree on parametric surfaces, which are globally $W^1_\infty$ andpiecewise in a suitable Besov class embedded in $C^{1,\alpha}$ with $\alpha \in(0,1]$. The idea is to have the surface sufficiently well resolved in$W^1_\infty$ relative to the current resolution of the PDE in $H^1$. This givesrise to a conditional contraction property of the PDE module. We present asuitable approximation class and discuss its relation to Besov regularity ofthe surface, solution, and forcing. We prove optimal convergence rates for AFEMwhich are dictated by the worst decay rate of the surface error in $W^1_\infty$and PDE error in $H^1$.

High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates