{"id":1498,"date":"2013-07-26T05:39:52","date_gmt":"2013-07-26T10:39:52","guid":{"rendered":"http:\/\/sites.williams.edu\/Morgan\/?p=1498"},"modified":"2018-03-21T06:07:25","modified_gmt":"2018-03-21T11:07:25","slug":"isoperimetric-profile-continuous","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2013\/07\/26\/isoperimetric-profile-continuous\/","title":{"rendered":"Isoperimetric Profile Continuous?"},"content":{"rendered":"

Note added 5 June\u00a02016<\/em>. A noncompact counterexample is given by Nardulli and Pansu, arxiv.org<\/a>. On the positive side, see the comment below by Milman and Flores\/Nardulli [FN].<\/p>\n

Note added 21 March 2018.<\/em> A 2D (noncompact) counterexample is given by Papasoglu and Swenson [PS], via expander graphs.<\/p>\n

Given a smooth Riemannian manifold, the isoperimetric profile I(V<\/em>) gives the infimum perimeter of smooth regions of volume V<\/em>.<\/p>\n

Proposition 1.<\/strong> In a compact smooth Riemannian manifold of dimension at least two, the isoperimetric profile is continuous. <\/em><\/p>\n

Note that the isoperimetric profile on the circle is discontinuous at 0.<\/p>\n

Proof.<\/em> Uppersemicontinuity is easy, because in dimension at least two a region with volume V<\/em>0<\/sub> 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<\/sub><\/em>\u00a0approaching V<\/em>0<\/sub>\u00a0and perimeters Pi<\/sub><\/em>\u00a0approaching the limit inferior P<\/em>0<\/sub>\u00a0of the infimum perimeters. By the Compactness Theorem of Geometric Measure Theory and lowersemicontinuity of perimeter, there is a limit integral current with volume V<\/em>0<\/sub>\u00a0and perimeter at most P<\/em>0<\/sub>. It can be smoothed to a smooth region with volume V<\/em>0<\/sub>\u00a0and perimeter at most about P<\/em>0<\/sub>.<\/p>\n

Remark<\/em>. 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<\/sub><\/em>\u00a0approaching V<\/em>0<\/sub>\u00a0from above with perimeters Pi<\/sub><\/em>\u00a0approaching P<\/em>0<\/sub>\u00a0< I(V<\/em>0<\/sub>). 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:<\/p>\n

Proposition 2<\/strong>\u00a0(Adams, Morgan, Nardulli, 2013). There is a smooth manifold with density in every dimension for which the isoperimetric profile is not continuous<\/em>.<\/p>\n

Proof<\/em>. Let the manifold consist of spheres Sn<\/sub><\/em>\u00a0of density n<\/em>+1 (n<\/em>\u22653), unweighted volume 1\/n<\/em>, and hence volume 1 + 1\/n<\/em>, sequentially connected by nice short thin tubes from Sn<\/sub><\/em>\u00a0to Sn<\/sub><\/em>+1\u00a0<\/sub>of volume 1\/20(2n<\/sup><\/em>), replacing small balls, both small balls contained in a single small ball Bn<\/sub><\/em>\u00a0of volume 1\/20(2n<\/sup><\/em>), each tube separated in the middle by a sphere of area less than 1\/n<\/em>. Note that the manifold has regions (spheres Sn<\/sub><\/em>\u00a0and 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<\/sub><\/em>\u00a0outside the small balls Bn<\/sub><\/em>, because the rest of the manifold has volume less than 1\/10, most efficiently in S<\/em>3<\/sub>, yielding a positive lower bounded on the perimeter to enclose volume 1. In Sn<\/sub><\/em>\u00a0– Bn<\/sub><\/em>, you cannot do better than enclosing volume in a spherical cap up against Bn<\/sub><\/em>\u00a0with free boundary, a ball, or complement. Efficiency requires the volume in each Sn<\/sub><\/em>\u00a0–\u00a0Bn<\/sub><\/em>\u00a0to 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 S<\/em>3<\/sub>. If all are small, it’s better to combine them in S<\/em>3<\/sub>. This yields a positive lower bound on the perimeter. Therefore the isoperimetric profile is not continuous.<\/p>\n

Acknowledgements.<\/em> Morgan and Nardulli thank the organizers\u00a0Luis Florit and Wolfgag Ziller\u00a0for the opportunities for discussion at Encounters in Geometry, Cabo Fr\u00edo, R\u00edo de Janeiro, Brasil, June 3-7, 2013, supported by\u00a0CNPq, FAPERJ, FAPESP, CAPES, IMPA, and the National Science Foundation.<\/p>\n

[FN]\u00a0Abraham Mu\u00f1oz Flores, Stefano Nardulli,\u00a0Local H\u00f6lder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry,\u00a0Geometria Dedicata (2016?),\u00a0arxiv.org\/abs\/1606.05020<\/a><\/p>\n

[PS] Panos Papasoglu and Eric Swenson, A surface with discontinuous isoperimetric profile,\u00a0https:\/\/arxiv.org\/abs\/1803.07375<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Note added 5 June\u00a02016. 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) […]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-1498","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1498","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=1498"}],"version-history":[{"count":14,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1498\/revisions"}],"predecessor-version":[{"id":2713,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1498\/revisions\/2713"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=1498"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=1498"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=1498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}