A modified mean curvature flow in Euclidean space and soap bubbles in symmetric spaces (Q1663763)

A soap bubble in an oriented $(n+1)$-dimensional space $\tilde{M}$ is the image by an immersion of an $n$-dimensional compact oriented submanifold. The author recalls results previously obtained for soap bubbles in products of hyperbolic spaces possibly with a Euclidean factor. The aim of the author in this paper is to study the behaviour of flows starting from small spheres. The results become more precise when the system is associated to an irreducible symmetric space of rank greater than one. In that case, the author shows how the spherical soap bubbles are constructed namely from the limit hypersurfaces of the modified mean curvature flows starting from small spheres. The author then investigates the shape and the mean curvature of the spherical soap bubbles. These results are formally stated as two theorems (A and B) followed by two corollaries (C and D). Corollary D allows for a description of soap bubbles in the rank-two case (specifically in the cases of $\mathfrak{a}_2$, $\mathfrak{b}_2$ and $\mathfrak{g}_2$). An important notion introduced to investigate these questions is that of a weighted root system (it is incompletely defined in the paper). The author associates the modified mean curvature flow in a Euclidean space associated to that system. The author then proceeds to recall the necessary definitions and results on the volume-preserving curvature flow and the mean curvatures of hypersurfaces invariant under the isotropy action. The rest of the paper is devoted to proving Theorems A and B followed by the two corollaries.