{"id":1173,"date":"2024-11-03T08:59:48","date_gmt":"2024-11-03T16:59:48","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/hades\/?p=1173"},"modified":"2024-11-21T17:24:47","modified_gmt":"2024-11-22T01:24:47","slug":"stability-of-catenoid-for-the-hyperbolic-vanishing-mean-curvature-equation-in-4-spatial-dimensions","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2024\/11\/03\/stability-of-catenoid-for-the-hyperbolic-vanishing-mean-curvature-equation-in-4-spatial-dimensions\/","title":{"rendered":"Stability of catenoid for the hyperbolic vanishing mean curvature equation in 4 spatial dimensions"},"content":{"rendered":"<p>The HADES seminar on Tuesday, <strong>November 5th<\/strong>, will be at\u00a0<strong>2<\/strong><strong>:00pm<\/strong>\u00a0in\u00a0<strong>Room 740.<\/strong><\/p>\n<p><strong>Speaker:\u00a0<\/strong><a href=\"https:\/\/math.berkeley.edu\/~ning_tang\/\">Ning Tang<\/a><\/p>\n<p><strong>Abstract: <\/strong>The hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space is a quasilinear wave equation that serves as the hyperbolic counterpart of the minimal surface equation in Euclidean space. This talk will concern the modulated nonlinear asymptotic stability of the $1+4$ dimensional hyperbolic catenoid, viewed as a stationary solution to the HVMC equation. This stability result is under a &#8220;codimension-one&#8221; assumption on initial perturbation, modulo suitable translation and boost (i.e. modulation), without any symmetry assumptions. In comparison to the $n \\geq 5$ case addressed by L\u00fchrmann-Oh-Shahshahani, proving catenoid stability in $n = 4$ dimensions shares additional difficulties with its $3$ dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties for the $n = 3$ case, the strong Huygens principle, as well as a miracle cancellation in the source term, plays an important role in the work of Oh-Shahshahani to obtain strong late time tails. Without these special structural advantages in $n = 4$ dimensions, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy with higher $r^p$-weights so that an improved pointwise decay can be established.<\/p>\n<p>In this talk, I will first give an outline of the proof and then focus on a model case to illustrate our approach to achieving an improved decay in $4$ dimensions.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday, November 5th, will be at\u00a02:00pm\u00a0in\u00a0Room 740. Speaker:\u00a0Ning Tang Abstract: The hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space is a quasilinear wave equation that serves as the hyperbolic counterpart of the minimal surface equation in Euclidean space. This talk will concern the modulated nonlinear asymptotic stability of the $1+4$ [&hellip;]<\/p>\n","protected":false},"author":95,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20],"tags":[],"class_list":["post-1173","post","type-post","status-publish","format-standard","hentry","category-fall-2024"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/users\/95"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=1173"}],"version-history":[{"count":2,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1173\/revisions"}],"predecessor-version":[{"id":1175,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1173\/revisions\/1175"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=1173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=1173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=1173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}