[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-290422-105":59,"doc-detail-290422-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","on-the-breakdown-criterion-in-general-relativity-sergiu-klainerman-and-igor-rodnianski","ON THE BREAKDOWN CRITERION IN GENERAL - Relativity - Sergiu Klainerman and Igor Rodnianski","","Geometric criterion for breakdown of an Einstein vacuum spacetime foliated by constant mean curvature (maximal) hypersurfaces is developed. The spacetime can be extended as long as the second fundamental form and the first derivatives of the logarithm of the foliation lapse remain uniformly bounded, without restrictions on the size of initial data. The main theorem characterizes the first breakdown time using a sharp lim sup condition involving L1 norms of k(t) and log n(t).",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/on-the-breakdown-criterion-in-general-relativity-sergiu-klainerman-and-igor-rodnianski/290422/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/on-the-breakdown-criterion-in-general-relativity-sergiu-klainerman-and-igor-rodnianski/290422.png","ImageObject",300,407,{"name":92,"@type":93},"Mia  ","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-09-17",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What geometric quantities determine when breakdown occurs in the foliated Einstein vacuum spacetime?","Question",{"text":112,"@type":113},"Breakdown is characterized by the behavior of the second fundamental form k(t) and the lapse n(t) of the foliation, specifically via L1 norms and their lim sup at the first breakdown time.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Under what condition can the spacetime be extended beyond a time t* \u003C 0?",{"text":117,"@type":113},"The extension holds beyond any t* for which the supremum of the L1 norms of k(t) and r log n(t) stays strictly below 1 on the interval up to t*.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the result compare with earlier breakdown criteria such as Anderson’s and Beale–Kato–Majda’s?",{"text":121,"@type":113},"Anderson’s criterion ties breakdown to L1 boundedness of the spacetime curvature R, while the present work uses k and log n, requiring one less degree of differentiability. It is also compared to the Beale–Kato–Majda framework, which uses blow-up integrals of quantities like vorticity.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},290422,1789641659,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144,"preview_page_count":4},687207024478,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","ON THE BREAKDOWN CRITERION IN GENERAL  \nRELATIVITY  \nSERGIU KLAINERMAN AND IGOR RODNIANSKI  \nAbstract. We give a geometric criterion for the breakdown of an Einstein vacuum space-time foliated by a constant mean curvature, or maximal, foliation. More precisely we show that the foliated space-time can be extended as long as the the second fundamental form and the 􀀌rst derivatives of the logarithm of the lapse of the foliation remain uniformly bounded. We make  \nno restrictions on the size of the initial data.  \n1. Introduction  \nThis paper is concerned with the problem of a geometric criterion for breakdown of solutions (M ; g) of the vacuum Einstein equations.  \nR􀀋􀀌 (g) = 0 : (1)  \nTo describe the problem we assume that a part of space-time M􀀃 􀀚 M is foliated by the level hypersurface of a time function t, monotonically increasing towards future, with lapse n and second fundamental form k de􀀌ned by,  \nk (X; Y ) = 􀀀 g (DX T; Y ); n = 􀀀􀀀 g (Dt; Dt)􀀁 􀀀1=2 (2)  \nwhere T is the future unit normal to 􀀆 t , D is the space-time covariant derivative associated with g, and X; Y are tangent to 􀀆 t and . Let 􀀆 0 be a 􀀌xed leaf of the tfoliation, corresponding to t = t0 . We shall refer to 􀀆 0 as initial slice. We assume that the space-time region M􀀃 is globally hyperbolic, i.e. every causal curve from a point p 2 M􀀃 intersects 􀀆 0 at precisely one point. We also assume that the initial slice veri􀀌es the following assumption.  \nA 1 . There exists a 􀀌nite covering of 􀀆 0 by a 􀀌nite number of charts U such that for any 􀀌xed chart, the induced metric g veri􀀌es  \n􀀁  1 j􀀘j 2 􀀔 gij (x)􀀘i􀀘j 􀀔 􀀁0 j􀀘j 2 ; 8x 2 U (3)  \nwith 􀀁 0 a 􀀌xed positive number.  \nWe consider the following two situations:  \n1991 Mathematics Subject Classi􀀌cation. 35J10  \nThe 􀀌rst author is partially supported by NSF grant DMS-0070696 . The second author is partially supported by NSF grant DMS-0702270 .  \n2 SERGIU KLAINERMAN AND IGOR RODNIANSKI  \n(1) The surfaces 􀀆 t are asymptotically 􀀍at and maximal.  \ntrk = 0 :  \n(2) The surfaces 􀀆 t are compact, of Yamabe type 􀀀1, and of constant, negative mean curvature. They form what is called a (CMC) foliation .  \ntrk = t; t \u003C 0  \nThough our methods apply equally well to both situations we shall only consider here the latter case, which is somewhat easier to treat due to the compactness of the level surfaces 􀀆 t. We shall thus assume in what follows that the region M􀀃 is equal to [ t2[t0 ;t􀀃 ) 􀀆 t , with t􀀃 \u003C 0. We can also assume that the initial hypersurface 􀀆 0 corresponds to t0 = 􀀀1.  \nRemark. In the second case the CMC conjecture asserts that it should be possible to extend the foliation, in a smooth manner, to all values of trk = t \u003C 0, see [And] and references therein.  \nGiven p 2 M􀀃 we can de􀀌ne a point-wise norm j􀀅(p)j of any space-time tensor 􀀅 via decomposition  \nX = 􀀀X0 T + X  ; X 2 TM;  X 2 T 􀀆 t  \nWe denote by k􀀅(t)kLp the Lp norm of 􀀅 on 􀀆 t. More precisely,  \nk􀀅(t)kLp = Z􀀆 t j􀀅jpdvg  \nwith dvg the volume element of the metric g of 􀀆 t. The main result of this paper is the following theorem.  \nTheorem 1.1 (Main theorem) . Let (M ; g) be a globally hyperbolic development of 􀀆 0 foliated by the CMC level hypersurfaces of a time function t \u003C 0, such that 􀀆 0 corresponds to the level surface t = t0 . Assume that 􀀆 0 veri􀀌es A1 . Then the 􀀌rst time T􀀃 \u003C 0, with respect to the t-foliation, of a breakdown is characterized by the condition  \nlim sup 􀀀kk(t)kL1 + kr log n(t)kL1 􀀁 = 1 (4)  \nt!T􀀃􀀀  \nMore precisely the space-time together with the foliation 􀀆 t can be extended beyond any value t􀀃 \u003C 0 for which,  \nsup kk(t)kL1 + kr log n(t)kL1 = 􀀁 0 \u003C 1 (5)  \nt2 [t0 ;t􀀃 )  \nCondition (5) can be reformulated in terms of the deformation tensor of the future unit normal T, 􀀙 = (T)􀀙 = LT g. By a simple calculation, expressed relative to an orthonormal frame e0 = T; e1 ; e2 ; e3 , we 􀀌nd,  \n􀀙00 = 0; 􀀙0i = n 􀀀1rin; 􀀙ij = 􀀀2kij : (6)  \nConsistent with the statement of the main theorem we assume that T is an approximate Killing vector􀀌eld in t","cbCaif1ZiHM0997J","https://ap.wps.com/l/cbCaif1ZiHM0997J","pdf",361970,38,"English","# Abstract\n# Introduction\n# Main theorem (Theorem 1.1)\n# Extension criteria and deformation tensor\n# Connections to earlier breakdown criteria","[{\"question\":\"What geometric quantities determine when breakdown occurs in the foliated Einstein vacuum spacetime?\",\"answer\":\"Breakdown is characterized by the behavior of the second fundamental form k(t) and the lapse n(t) of the foliation, specifically via L1 norms and their lim sup at the first breakdown time.\"},{\"question\":\"Under what condition can the spacetime be extended beyond a time t* \\u003c 0?\",\"answer\":\"The extension holds beyond any t* for which the supremum of the L1 norms of k(t) and r log n(t) stays strictly below 1 on the interval up to t*.\"},{\"question\":\"How does the result compare with earlier breakdown criteria such as Anderson’s and Beale–Kato–Majda’s?\",\"answer\":\"Anderson’s criterion ties breakdown to L1 boundedness of the spacetime curvature R, while the present work uses k and log n, requiring one less degree of differentiability. It is also compared to the Beale–Kato–Majda framework, which uses blow-up integrals of quantities like vorticity.\"}]","ON THE BREAKDOWN CRITERION IN GENERAL - Relativity - Sergiu Klainerman and Igor Rodnianski | PDF",96]