## Isoperimetric Profile Continuous?

*Note added 5 June 2016*. A noncompact counterexample is given by Nardulli and Pansu, arxiv.org. On the positive side, see the comment below by Milman and Flores/Nardulli [FN].

*Note added 21 March 2018.* A 2D (noncompact) counterexample is given by Papasoglu and Swenson [PS], via expander graphs.

Given a smooth Riemannian manifold, the isoperimetric profile I(*V*) gives the infimum perimeter of smooth regions of volume *V*.

**Proposition 1.** *In a compact smooth Riemannian manifold of dimension at least two, the isoperimetric profile is continuous. *

Note that the isoperimetric profile on the circle is discontinuous at 0.

*Proof.* Uppersemicontinuity is easy, because in dimension at least two a region with volume *V*_{0} can be slightly altered to have nearby volumes and not much more perimeter. It is lowersemicontinuity that requires compactness. Take a sequence of regions with volumes *V _{i}* approaching

*V*

_{0}and perimeters

*P*approaching the limit inferior

_{i}*P*

_{0}of the infimum perimeters. By the Compactness Theorem of Geometric Measure Theory and lowersemicontinuity of perimeter, there is a limit integral current with volume

*V*

_{0}and perimeter at most

*P*

_{0}. It can be smoothed to a smooth region with volume

*V*

_{0}and perimeter at most about

*P*

_{0}.

*Remark*. It is apparently an open question whether the isoperimetric profile need be continuous in a noncompact smooth Riemannian manifold. In a counterexample there would be regions of volume *V _{i}* approaching

*V*

_{0}from above with perimeters

*P*approaching

_{i}*P*

_{0}< I(

*V*

_{0}). We tried in vain to create such a counterexample using pieces of increasing negative curvature after Buser-Sarnak, Katz-Schaps-Vishne, and Schmutz. Such a counterexample is possible in the larger category of manifolds with density:

**Proposition 2** (Adams, Morgan, Nardulli, 2013). *There is a smooth manifold with density in every dimension for which the isoperimetric profile is not continuous*.

*Proof*. Let the manifold consist of spheres *S _{n}* of density

*n*+1 (

*n*≥3), unweighted volume 1/

*n*, and hence volume 1 + 1/

*n*, sequentially connected by nice short thin tubes from

*S*to

_{n}*S*

_{n}_{+1 }of volume 1/20(2

*), replacing small balls, both small balls contained in a single small ball*

^{n}*B*of volume 1/20(2

_{n}*), each tube separated in the middle by a sphere of area less than 1/*

^{n}*n*. Note that the manifold has regions (spheres

*S*and half of the adjacent tubes) of volume converging to 1 and perimeter converging to 0. Consider any region of volume 1. Volume at least 9/10 and at most 1 is contained in the spheres

_{n}*S*outside the small balls B

_{n}*, because the rest of the manifold has volume less than 1/10, most efficiently in*

_{n}*S*

_{3}, yielding a positive lower bounded on the perimeter to enclose volume 1. In

*S*–

_{n}*B*, you cannot do better than enclosing volume in a spherical cap up against

_{n}*B*with free boundary, a ball, or complement. Efficiency requires the volume in each

_{n}*S*–

_{n}*B*to be small or large, and at most one is large. If one is large, it is better to move the rest of the volume there, and best if it’s in

_{n}*S*

_{3}. If all are small, it’s better to combine them in

*S*

_{3}. This yields a positive lower bound on the perimeter. Therefore the isoperimetric profile is not continuous.

*Acknowledgements.* Morgan and Nardulli thank the organizers Luis Florit and Wolfgag Ziller for the opportunities for discussion at Encounters in Geometry, Cabo Frío, Río de Janeiro, Brasil, June 3-7, 2013, supported by CNPq, FAPERJ, FAPESP, CAPES, IMPA, and the National Science Foundation.

[FN] Abraham Muñoz Flores, Stefano Nardulli, Local Hölder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry, Geometria Dedicata (2016?), arxiv.org/abs/1606.05020

[PS] Panos Papasoglu and Eric Swenson, A surface with discontinuous isoperimetric profile, https://arxiv.org/abs/1803.07375.

## Emanuel Milman:

Hi Frank,

long time! I hope all is well.

I just came across this old post, and wanted to remind the readers of an old argument of Buser (“A note on the isoperimetric constant”, ASENS 1982 – Lemma 3.4), for proving the continuity, and in fact the Holder continuity with exponent (n-1)/ n , of the isoperimetric profile. I adapted his argument (in fact, an adaption of Gallot’s adaption of Buser’s argument) in Lemma 6.9 of my paper http://arxiv.org/pdf/0712.4092v5.pdf to the weighted Riemannian setting. The result is as follows:

Let (M,g) denote an n-dimensional (n≥2) smooth complete oriented connected Riemannian manifold (no compactness assumed) and let d denote the induced geodesic distance. Let \mu denote an absolutely continuous probability measure with respect to vol_M, such that its density is bounded from above on every ball (but not necessarily from below, nor do we assume it is continuous). Then the isoperimetric profile I=I(M,d,μ) is absolutely continuous on [0,1], and in fact is locally of H ̈older exponent (n−1)/n.

I guess the key point is that the measure I considered was always a probability (or finite) measure. So it is not the compactness that is essential for the continuity of the profile, but finiteness of the measure (+ local boundedness of the density). Incidentally, I used the Minkowski definition of weighted perimeter, but by regularity theory, this a-poteriori coincides with any other definition used, and in any case is irrelevant for the proof of the above statement.

There are some other useful statements in Section 6 of my paper above, like addressing when is I symmetric about the point 1/2, i.e. I(t) = I(1-t) ? This hold as soon as I is lower-semi-continuous (which is of course typically the case).

All the best,

4 June 2016, 6:59 amEmanuel.