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I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive sectional curvature).

In particular, we know that for a compact Riemannian manifold $M$ with smooth boundary, isoperimetric domains exist. That is, for each $V \in (0,vol(M))$ there exists $E\subset M$ such that $Vol(E) = V$ and $Area(\partial E) = \inf\{area(\partial E) \, | \, E\subset M, vol(E)=V\}$. We also know that isoperimetric domains are $C^{1,1}$ (loosely speaking) and the mean curvature is constant almost everywhere (the mean curvature is defined almost everywhere for $C^{1,1}$ surfaces).

It is an open problem to show that Cartan-Hadmard manifolds satisfy a Euclidean isoperimetric inequality, so I'm wondering if these manifolds similarly have the property that isoperimetric domains are convex. So, if $M$ is a compact, convex Cartan-Hadamard manifold, is it known if isoperimetric domains in $M$ are convex?

A less restrictive question would be: if $M$ is a compact, convex Cartan-Hadamard manifold, do isoperimetric domains $E_0 \subset M$ have the property that $$Area(\partial \,\overline{Conv(E_0)}) \leq Area(\partial E_0)$$ where $\partial \,\overline{Conv(E_0)}$ is the boundary of the closure of the convex hull of $E_0$? Note that in dimensions greater than 2, the surface area of the convex hull of a domain may be greater than the surface area of the domain, even in the Euclidean case.

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  • $\begingroup$ Your first question would follow from the second. $\endgroup$ – Anton Petrunin Sep 18 '15 at 13:55
  • $\begingroup$ Certainly if the first question is true (isoperimetric domains are convex), then the second question is true (perimeter of the convex hull of an isoperimetric domain is smaller than the perimeter of an isoperimetric domain), but how does the first question follow from the second? $\endgroup$ – Alec Payne Sep 18 '15 at 20:27
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Using the definition of a Cartan Hadamard as a complete, simply connected manifold having non-positive curvature, a region in a Cartan Hadamard manifold realizing the optimal isoperimetric ratio need not be convex. In fact it does not even need to be connected, even in dimension two.

An explicit example is constructed in Isoperimetric Regions in Nonpositively Curved Manifolds arXiv:1604.02768

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