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We prove the existence of rational maps having smooth degenerate Hermanrings. This answers a question of Eremenko affirmatively. The proof is based onthe construction of smooth Siegel disks by Avila, Buff and Ch\'{e}ritat as wellas the classical Siegel-to-Herman quasiconformal surgery. A crucial ingredientin the proof is the surgery's continuity, which relies on the control of theloss of the area of quadratic filled-in Julia sets by Buff and Ch\'{e}ritat.  As a by-product, we prove the existence of rational maps having a nowheredense Julia set of positive area for which these maps have no irrationallyindifferent periodic points, no Herman rings, and are not renormalizable.

Rational maps with smooth degenerate Herman rings