| Abstract: |
| Ciraolo, Maggi (2015) proved that if the scalar mean curvature of a surface is sufficiently uniformly close to the mean curvature of a sphere, then the surface must be quantitatively close to a union of spheres. Their argument quantified a compactness theorem for a uniform mean curvature deficit. Delgadino, Maggi, Mihaila, Neumayer (2017) proved an analogous compactness theorem for the anistropic mean curvature with an $L^2$ deficit from the corresponding Wulff shape, but did not provide quantitative estimates. We will discuss these two arguments, and we will give ideas of the difficulties and new constructions needed for quantitative estimates in the anisotropic regime. |
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