{"id":261,"date":"2021-04-20T07:16:29","date_gmt":"2021-04-20T07:16:29","guid":{"rendered":"https:\/\/geometry.ulb.ac.be\/bowl\/?page_id=261"},"modified":"2021-05-17T13:08:47","modified_gmt":"2021-05-17T13:08:47","slug":"costante-bellettini-ucl","status":"publish","type":"page","link":"https:\/\/geometry.ulb.ac.be\/bowl\/costante-bellettini-ucl\/","title":{"rendered":"Costante Bellettini (UCL)"},"content":{"rendered":"\n<p>Costante Bellettini will talk on 25th May at 1.45pm UK time, 2.45pm Belgian time. Costante&#8217;s title is <em>&#8220;Existence of hypersurfaces with prescribed mean-curvature&#8221;<\/em> and his abstract is below.<\/p>\n\n\n\n<!--more-->\n\n\n\n<h4 class=\"wp-block-heading\">Existence of hypersurfaces with prescribed mean-curvature<\/h4>\n\n\n\n<p><em>Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. We prove that there exists a closed hypersurface M whose mean curvature attains the values prescribed by g (joint work with Neshan Wickramasekera, Cambridge). Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is C^2 immersed and two-sided (it admits a global unit normal); the scalar mean curvature at x is g(x) with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when g is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.The construction of M is carried out largely by means of PDE principles: (i) a minmax for an Allen&#8211;Cahn (or Modica-Mortola) energy, involving a parameter that, when sent to 0, leads to an interface from which the desired PMC hypersurface is extracted; (ii) quasi-linear elliptic PDE and\u00a0geometric-measure-theory\u00a0arguments, to obtain regularity conclusions for said interface; (iii) parabolic semi-linear PDE (together with specific features of the Allen-Cahn framework), to tackle cancellation phenomena that can happen when\u00a0sending to 0 the Allen-Cahn parameter.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Costante Bellettini will talk on 25th May at 1.45pm UK time, 2.45pm Belgian time. Costante&#8217;s title is &#8220;Existence of hypersurfaces with prescribed mean-curvature&#8221; and his abstract is below.<\/p>\n","protected":false},"author":1,"featured_media":62,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"class_list":{"0":"post-261","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"entry"},"featured_image_src":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-content\/uploads\/sites\/8\/2020\/08\/Bowl-chrysanthemum-by-Yamaken-600x400.jpg","featured_image_src_square":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-content\/uploads\/sites\/8\/2020\/08\/Bowl-chrysanthemum-by-Yamaken-600x600.jpg","_links":{"self":[{"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/pages\/261","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/comments?post=261"}],"version-history":[{"count":4,"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/pages\/261\/revisions"}],"predecessor-version":[{"id":281,"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/pages\/261\/revisions\/281"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/media\/62"}],"wp:attachment":[{"href":"https:\/\/geometry.ulb.ac.be\/bowl\/wp-json\/wp\/v2\/media?parent=261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}