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The geometry of isoperimetric domains of large volume in simply connected homogeneous 3-manifolds

University of Massachusetts, Amherst

Consider a simply connected non-compact homogeneous 3-manifold $X$. For each $V\gt 0$, there exists a smooth compact domain $D(V)$ with volume $V$ whose boundary surface area is minimizing and we call such a boundary area minimizing domain an isoperimetric domain of $X$. One would like to classify all such isoperimetric domains in $X$ for a fixed $V$ up to ambient isometry and to understand the properties of such domains, such as the slope of the "isoperimitric profile of $X$" as volume $V$ tends to infinity. In a recent paper with Mira, Pérez and Ros, the speaker has obtained an number of such geometric results on these questions. We have been able to show that the Cheeger constant $\mathrm{Ch}(X)$ of $X$ is equal to twice the critical mean curvature $H(X)$ of the $X$. As a consequence of our proof of this relation, for any sequence of isopermetric domains in $X$ with volumes tending to infinity, the constant mean curvatures of their boundary surfaces tend to $H(X)$ and their radii tend to infinity. The difficult case in this study is when $X$ is isometric to universal cover of the Lie group $\mathrm{Sl}(2,\mathbb{R})$ equipped with a left invariant metric with a 3-dimensional isometry group. These results together with our recent classification of constant mean curvature spheres in $X$ leads to an isoperimetric inequality for smooth compact immersed surfaces in $X$ with 1 boundary component and with absolute mean curvature function less than or equal to $\mathrm{Ch}(X)/2$.

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