Optimal quantitative stability estimates for Alexandrov's Soap Bubble Theorem via Gagliardo-Nirenberg-type interpolation inequalities

The paper provides optimal quantitative stability estimates for the celebrated Alexandrov's Soap Bubble Theorem within the class of $C^{k,\alpha}$ domains, for any $k \ge 1$ and $0 < \alpha \leq 1$, by leveraging Gagliardo-Nirenberg-type interpolation inequalities. Optimal estimates of uniform closeness to a ball are established for $L^r$ deviations of the mean curvature from being constant, for any $r\geq 2$ (more generally, for any $r>1$ such that $r\geq (2N-2)/(N+1)$). For $r>\frac{N-1}{2}$, the stability profile is linear, thus returning the existing results established in the literature through computations for nearly spherical sets. All the stability estimates for $r\le \frac{N-1}{2}$, for which the profile is not linear, are new; even in the particular case $r=2$ (which has been extensively studied, since it is a case of interest for several critical applications), the sharp stability profile that we obtain is new. Interestingly, we also prove that the (non-linear) profile for $r \leq \frac{N-1}{2}$ improves as $k$ becomes larger to such an extent that it becomes formally linear as $k$ goes to $\infty$. Finally, for any $k \geq 1$ and $0< \alpha \leq 1$, we show that our estimates are optimal within the class of $C^{k,\alpha}$ domains, by providing explicit examples.