{"id":549,"date":"2011-04-04T09:15:20","date_gmt":"2011-04-04T13:15:20","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=549"},"modified":"2011-04-04T09:15:20","modified_gmt":"2011-04-04T13:15:20","slug":"monotoncity-from-lelong-to-federer-fleming","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2011\/04\/04\/monotoncity-from-lelong-to-federer-fleming\/","title":{"rendered":"Monotoncity from Lelong to Federer-Fleming"},"content":{"rendered":"<p>The most important lemma of regularity for an <em>m<\/em>-dimensional area-minimizing surface <em>S<\/em> in <strong>R<\/strong><sup><em>n<\/em><\/sup> is the monotonicity of the mass ratio about a point <em>p<\/em> of <em>S<\/em>, i.e., the ratio of the area inside an <em>n<\/em>-ball about <em>p<\/em> of radius <em>r<\/em> to the volume of an <em>m<\/em>-ball of radius <em>r<\/em> [FF, Thm. 9.13 and proof], [M, 9.3]. Here in Lahore, Pakistan, at the <a href=\"http:\/\/www.sms.edu.pk\/\">Abdus Salam School of Mathematical Sciences<\/a>, Rein Zeinstra told me that such monotonicity generalizes a similar result by Lelong [L] for complex analytic varieties (which are area minimizing). Similarly, the compactness theorem of geometric measure theory generalizes a later such result for complex analytic varieties.\u00a0See\u00a0<a href=\"http:\/\/www.ams.org\/mathscinet\/search\/publdoc.html?pg1=IID&amp;s1=292468&amp;vfpref=html&amp;r=8&amp;mx-pid=206337\">a Math Review<\/a> by Stoll.<\/p>\n<p>[FF] Herbert Federer and Wendell H. Fleming, Normal and integral currents, <em>Ann. of Math.<\/em> 72 (1960), 458-520.<\/p>\n<p>[L] Pierre Lelong,\u00a0Propri\u00e9t\u00e9s m\u00e9triques des vari\u00e9t\u00e9s  analytiques complexes  d\u00e9finies par une \u00e9quation,  <em>Ann. Sci. \u00c9cole Norm. Sup. <\/em><strong>67 <\/strong>(1950), 393\u2013419.<\/p>\n<p>[M] Frank Morgan, Geometric Measure Theory: a Beginner&#8217;s Guide, Academic Press, 4th ed., 2009.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The most important lemma of regularity for an m-dimensional area-minimizing surface S in Rn is the monotonicity of the mass ratio about a point p of S, i.e., the ratio of the area inside an n-ball about p of radius r to the volume of an m-ball of radius r [FF, Thm. 9.13 and proof], [&hellip;]<\/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-549","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/549","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=549"}],"version-history":[{"count":0,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/549\/revisions"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=549"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=549"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=549"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}