*Greetings from Oberwolfach from a great conference on algebraic, geometric, and topological combinatorics. Stay tuned for more pictures and updates from Oberwolfach and CERN, and also in case you did not see it already here is the link to the previous post with the sunflower news. And towards the end of this post a connection to Saint-Exupéry’s “Le Petit Prince”.*

Or Hershkovich reported on FB group “diggings in mathematics” about the solution of the Cartan-Hadamard conjecture by Mohammad Ghomi and Joel Spruck. The conjecture, one of the most famous conjectures in Riemannian geometry, is a far reaching extension of the classic isoperimetric inequality for general Riemannian manifolds.

Update (March 22, 2020): Unfortunately, a gap in the proof has now been reported. *Piet Hein who said: “Problems worthy of attack. Prove their worth by fighting back”, and we can hope that the gap could be bridged.*

**Here is the abstract:** We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature $latex M^n, n\ge 2$, is bounded below by the volume of the unit sphere in Euclidean space $latex R^n$. This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in M, via Kleiner’s variational approach, and thus settles the Cartan-Hadamard conjecture. The proof employs a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for smooth approximation of the signed distance function. Immediate applications include sharp extensions of the Faber-Krahn and Sobolev inequalities to manifolds of nonpositive curvature.

The proof is quite involved (the paper is 80 pages long) and is based on extending Kleiner’s approch for the 3- and 4- dimensional case. Or Hershkovich raised on FB an even more general question related to isospectral relations based on higher homology.

We also note a 2013 paper by Benoît Kloeckner and Greg Kuperberg The Cartan-Hadamard conjecture and the little prince

A page from Kloeckner-Kuperberg’s paper, and related FB posts by Greg (below).

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*Related*

What are the answers to the questions in Greg Kuperberg’s Facebook posts?

Let me mention that a discrete version of the planar case and a problem on the spacial space are presented in the 2018 paper An isoperimetric inequality for planar triangulations by Angel, Benjamini, and Horesh https://arxiv.org/abs/1604.05863.

To Peter Kagey’s question I suppose that Benoît and Greg’s paper would be a good source.

There appears to be at least one problem in the argument that the authors have ceded in an update to the arXiv preprint: https://arxiv.org/abs/1908.09814.

We have posted an updated version of the paper at

link

which fixes that issue.

Dear Mohammad, many thanks for the update and congratulations on the result.