*Accepted Paper*

**Inserted:** 23 aug 2023

**Last Updated:** 16 jul 2024

**Journal:** IMRN

**Year:** 2023

**Doi:** https://doi.org/10.1093/imrn/rnae159

**Abstract:**

Given $ n \geq 2 $ and $ k \in \{2, \ldots , n\} $, we study the asymptotic behaviour of sequences of bounded $C^2$-domains of finite total curvature in $ \mathbb{R}^{n+1} $ converging in volume and perimeter, and with the $ k $-th mean curvature functions converging in $ L^1 $ to a constant. Under natural mean convexity hypothesis, and assuming an $ L^\infty $-control on the mean curvature outside a set of vanishing area, we prove that finite unions of mutually tangent balls are the only possible limits. This is the first result where such a uniqueness is proved without assuming uniform bounds on the exterior or interior touching balls.