# Isoperimetric inequality in CAT(0) surfaces

I am looking for a version of the (2-dimensional) isoperimetric inequality for globally CAT(0) (in particular simply-connected) surfaces. I am particularly interested in characterizing the disk of maximal area as bounded by a curve of constant geodesic curvature, and also being able to prescribe in advance the center of mass of the optimal disk.

1) That the disk of maximal area on a smooth surface (if the maximum is attained...) has constant geodesic curvature, was stated by Steiner and proved by Minding, see a historical account on page 1200 of

Osserman, Robert, The isoperimetric inequality, Bull. Am. Math. Soc. 84, 1182-1238 (1978). ZBL0411.52006.

For some metrics there are no closed curves of constant geodesic curvatures, ibid. So the optimal disk does not always exist. (I don't know if there are examples with curvature everywhere negative.) On the other hand, on a surface $dr^2 + f(r) d\phi^2$ with negative curvature tending to 0 as $r$ tends to infinity there is no disk of maximum area with a given perimeter.

Also, the domain of maximum area with a given perimeter (if exists) might be non-simply connected, but every component of the boundary must have constant geodesic curvature anyway...

2) Without saying anything about the maximizer, there is the isoperimetric inequality for surfaces of negative curvature, proved by Weil:

Weil, A., Sur les surfaces à courbure négative., C. R. 182, 1069-1071 (1926). ZBL52.0712.05. and, independently, Beckenbach and Radó:

Beckenbach, E. F.; Rad\'o, T., On the isoperimetric inequality., Bulletin A. M. S. 39, 208 (1933). ZBL59.0506.07.

It was later generalized by Alexandrov to the $\mathrm{CAT}(0)$-surfaces, see Section 2.2 of "Geometric inequalities" by Burago and Zalgaller.

3) Is there the notion of a geodesic curvature on $\mathrm{CAT}(0)$-spaces?

Curves in $\mathrm{CAT}(0)$-spaces satisfy the Euclidean isoperimetric inequality (they don't bound disks of the area larger than the Euclidean circle with the same perimeter). This is sort of folklore result. The preprint of Lytchak and Wenger "Isoperimetric characterization of upper curvature bounds" arXiv/1611.05261 proves also the converse: if a metric space satisfies the Euclidean isoperimetric inequality, then it is $\mathrm{CAT}(0)$.

• I am familiar with Alexandrov's article. Unfortunately he does not address the issue of the disk of maximal area. – Mikhail Katz May 9 '17 at 11:06
• I don't think you mean "every component of the boundary must be geodesic anyway". Do you mean "of constant mean curvature"? – Mikhail Katz May 9 '17 at 11:59
• Yes, I meant constant geodesic curvature. I edited the answer. – Ivan Izmestiev May 9 '17 at 15:38
• Ivan, thanks, but we are interested in optimal disks with prescribed center of mass in an Alexandrov surface of CAT(0) type, rather than in the Euclidean isoperimetric inequality in more general spaces. – Mikhail Katz May 9 '17 at 17:01
• Well, if you prescribe the center of mass, the boundary of the disk will not have constant geodesic curvature. – Ivan Izmestiev May 9 '17 at 17:19