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 V0 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 Vi approaching V0 and perimeters Pi approaching the limit inferior P0 of the infimum perimeters. By the Compactness Theorem of Geometric Measure Theory and lowersemicontinuity of perimeter, there is a limit integral current with volume V0 and perimeter at most P0. It can be smoothed to a smooth region with volume V0 and perimeter at most about P0.
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 Vi approaching V0 from above with perimeters Pi approaching P0 < I(V0). 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 Sn of density n+1 (n≥3), unweighted volume 1/n, and hence volume 1 + 1/n, sequentially connected by nice short thin tubes from Sn to Sn+1 of volume 1/20(2n), replacing small balls, both small balls contained in a single small ball Bn of volume 1/20(2n), each tube separated in the middle by a sphere of area less than 1/n. Note that the manifold has regions (spheres Sn 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 Sn outside the small balls Bn, because the rest of the manifold has volume less than 1/10, most efficiently in S3, yielding a positive lower bounded on the perimeter to enclose volume 1. In Sn – Bn, you cannot do better than enclosing volume in a spherical cap up against Bn with free boundary, a ball, or complement. Efficiency requires the volume in each Sn – Bn 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 S3. If all are small, it’s better to combine them in S3. 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.
Welcome to my blog. I also have a blog at the
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.