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Abstract:
An area minimizing double bubble in $\mathbb R^n$ is given by two (not necessarily connected) regions which have two prescribed $n$-dimensional volumes whose combined boundary has least $(n\!-\!1)$-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard double bubble, composed of three spherical caps. This has now been proven for $n=2,3,4$, but is, for general volumes, unknown for $ n\ge 5$. Here, for arbitrary $n$, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This provides an alternative line of reasoning for part of the proof of the double bubble theorem in $\mathbb R^3$, as well as some new component bounds in $\mathbb R^n$.
