{"id":541,"date":"2011-03-29T13:21:05","date_gmt":"2011-03-29T17:21:05","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=541"},"modified":"2011-03-29T13:21:05","modified_gmt":"2011-03-29T17:21:05","slug":"complex-curves-minimize-curvature","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2011\/03\/29\/complex-curves-minimize-curvature\/","title":{"rendered":"Complex Curves Minimize Curvature"},"content":{"rendered":"<p>In Section 3 of his article on &#8220;<a href=\"http:\/\/www.ams.org\/journals\/bull\/2011-48-02\/S0273-0979-2010-01319-7\/home.html\">Crystals, Proteins, Stability and Isoperimetry<\/a>&#8221; in the April 2011 <em>Bulletin of the American Mathematical Society<\/em>, Misha Gromov suggests that a complex subvariety <em>S<\/em> of a K\u00e4hler manifold <em>X<\/em> minimizes a curvature energy defined as the volume of the unit tangent bundle <em>S<\/em>\u2032 in the Grassmannian bundle <em>X<\/em>\u2032 of <em>X<\/em>. Here we note that this holds for complex curves in the following sense:<\/p>\n<p><strong>Proposition.<\/strong> <em>Let S be a complex curve in C<sup>n<\/sup>. Then compact portions of S minimize curvature energy among surfaces (rectifiable currents) with the same boundary and the same tangent planes (almost everywhere) along the boundary.<\/em><\/p>\n<p><em>Proof.<\/em> The Grassmannian bundle is a K\u00e4hler manifold, and <em>S\u2032<\/em> is a complex curve, hence volume minimizing for given homology class and boundary, which corresponds to boundary and tangent planes along the boundary for <em>S<\/em>.<\/p>\n<p>For example {<em>x<\/em><sup>2<\/sup>=<em>y<\/em><sup>3<\/sup>} in <em>C<\/em><sup>2<\/sup>, with that interesting singularity at the origin. I don&#8217;t know what condition on a more general K\u00e4hler manifold\u00a0<em>X<\/em> guarantees that <em>X<\/em>\u2032 is a K\u00e4hler manifold.<\/p>\n<p>Incidentally, there is a recent arXiv post on &#8220;<a href=\"http:\/\/arxiv.org\/abs\/1103.0167\">Willmore minimizers with prescribed isoperimetric ratio<\/a>&#8221; by Johannes Schygulla.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Section 3 of his article on &#8220;Crystals, Proteins, Stability and Isoperimetry&#8221; in the April 2011 Bulletin of the American Mathematical Society, Misha Gromov suggests that a complex subvariety S of a K\u00e4hler manifold X minimizes a curvature energy defined as the volume of the unit tangent bundle S\u2032 in the Grassmannian bundle X\u2032 of [&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-541","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/541","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=541"}],"version-history":[{"count":0,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/541\/revisions"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=541"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=541"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=541"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}