Optimal quantitative stability for the anisotropic Alexandrov soap bubble theorem and Heintze-Karcher inequality

The present paper provides linear quantitative stability estimates for the anisotropic Alexandrov soap bubble theorem. The estimates are sharp and, to the authors' knowledge, provide the first sharp quantitative version of Alexandrov's theorem in the anisotropic setting. Our estimates establish quantitative closeness to the Wulff shape, also providing sharp control of the closeness of the Gauss maps. As shown by the authors for the classical case, such a control can play a crucial role in achieving optimal profiles for arbitrary deviations and stronger measures of closeness. We also obtain sharp control of the so-called (anisotropic) excess. The paper also provides the first sharp quantitative version of the anisotropic Heintze-Karcher inequality. For the Heintze-Karcher inequality, the stability estimates providing closeness of the Gauss maps and control of the excess are new even in the classical (isotropic) setting.