Chen Daguang (Tsinghua University) Fan Huijun (Peking University) Feng Huitao (Nankai University) Fu Jixiang (Fudan University) Futaki Akito (University of Tokyo) Guo Zhen (Yunnan Normal University) Kunikawa Keita (Nagoya University) Li Haizhong (Tsinghua University) Li Jiayu (University of Science and Technology of China Liu Xiaobo (Peking University) Mabuchi Toshiki (Osaka University) Saito Shunsuke (University of Tokyo) Tamaru Hiroshi (Hiroshima University) Tian Gang (Peking University and Princeton University) Xu Guangbo (Princeton University) Yamaguchi Takao (Kyoto University) Yan Wenjiao (Beijing Normal University) Zhang Weiping (Nankai University) Zhu Xiaohua (Peking University)
Organizing Committee Cheng Qing-Ming (Fukuoka University), Sano Yuji (Fukuoka University)
March&23& & 6VOW FO\ LcXW "9cYc YO H\WdS aWbg 03 ) 13)) OPcQVW G aVWYW "BaOYO H\WdS aWbg 8fb S OZ N OVZS dS aW \a T bVS LOc GWO\ 7 \OZRa \ 6 \XSQbc S 13 ) m +)3+) 7W\\S bW S +)3,) + 3,) 7WaQcaaW \ \ 7WTTS S\bWOZ :S Sb g March&24& 6VOW GWO\ :O\U o5SWXW\U H\WdS aWbg O\R C W\QSb \ H\WdS aWbgp )1q.) )2q-) MVO\U JSW W\U "AO\YOW H\WdS aWbg C aWbWdS aQOZO Qc dObc S \ O\WT ZRa O\R T ZWObW \a 6VOW WgO YO ESWY "G V Yc H\WdS aWbg )q)) )q.) GO O c W aVW " W aVW O H\WdS aWbg STb W\dO WO\b Sb WQa O\R acP O\WT ZR US Sb g q ) +q)) W OWhV \U "GaW\UVcO H\WdS aWbg k H\W cS\Saa T QZ aSR aSZT aW WZO a ZcbW \a b lY Qc dObc S TZ e c\QV bW S 6VOW 7W\U DW\U "9cRO\ H\WdS aWbg -3.) .3-) 9c WfWO\U "9cRO\ H\WdS aWbg 4 ac dSg \ POZO\QSR Sb WQa /3)) /3.) c\WYOeO SWbO "AOU gO H\WdS aWbg F S \ \SfWabS\QS SacZba T b O\aZObW\U a ZWb \a 03 ) 13)) W WOgc "H\WdS aWbg T GSQV\ Z Ug O\R FQWS\QSa T 6VW\O 8\S Ug WRS\bWbg T O aS cS\QS T FOQYa HVZS\PSQY O a b O a VS S 13 ) m +)3+) 7W\\S bW S +)3,) + 3,) 7WaQcaaW \ \ a SQWOZ b WQa March&25& 6VOW OPcQVW G aVWYW "BaOYO H\WdS aWbg )1q.) )2q-) 9O\ cWXc\o5SWXW\U H\WdS aWbgp 9cYOgO QObSU g T OcRO :W\hPc U RSZ )q)) )q.) FOWb FVc\acYS "H\WdS aWbg T G Yg FbOPWZWbg T O\bW QO\ \WQOZZg POZO\QSR Sb WQa q ) +q)) Kc :cO\UP "C W\QSb \ H\WdS aWbg :OcUSR ZW\SO aWU O RSZ O\R ORWOPObWQ ZW Wb c\QV bW S 6VOW Wc KWO P "5SWXW\U H\WdS aWbg /3)) /3.) LO\ JS\XWO "5SWXW\U A OZ H\WdS aWbg A OZ aQOZO Qc dObc S W\S cOZWbg \ bVS T QOZ acP O\WT ZRa T Wa O O Sb WQ Vg S ac TOQSa 03 ) 13)) :c MVS\oLc\\O\ A OZ H\WdS aWbgp 6VO OQbS WhObW \ T O QZOaa T bg WQOZ JWZZ S acP O\WT ZRa 13 ) m +)3+) ESQS bW \ +)3,) + 3,) 7WaQcaaW \ \ a SQWOZ b WQa & & & & & March&26& 6VOW 9O\ cWXc\ "5SWXW\U H\WdS aWbg )1q.) )2q-) Wc KWO P "5SWXW\U H\WdS aWbg 6 \\SQbW\U bVS \baSdWQV JWbbS\ O\R RUS bOc Tc\QbW \a Pg bVS IW Oa S Ob a )q)) )q.) 9S\U cWbO "AO\YOW H\WdS aWbg B\ O 6 \XSQbc S T F'F' 6VS \ q ) +q)) LO OUcQVW GOYO " g b H\WdS aWbg 6 O Wa \ O\UZSa O\R d Zc S c\QV bW S 6VOW W OWhV \U "GaW\UVcO H\WdS aWbg -3.) .3-) MVc KWO VcO "5SWXW\U H\WdS aWbg C S \Saa T 9 Tc\QbW \OZ /3)) /3.) 6VS\ 7OUcO\U "GaW\UVcO H\WdS aWbg GVS SabW ObSa T bVS UO a T Q \aSQcbWdS SWUS\dOZcS T O ZOQWO\ 03 ) 13)) 9cbOYW 4YWb "H\WdS aWbg T G Yg I Zc S W\W WhW\U W\QW ZS T Q \T OZZg OSVZS 8W\abSW\ OfeSZZ Sb WQa 13 ) m +)3+) 7W\\S bW S +)3,) + 3,) 7WaQcaaW \ \ a SQWOZ b WQa March&27& 6VOW MVO\U JSW W\U "AO\YOW H\WdS aWbg )13,) )23+) GWO\ :O\U o5SWXW\U H\WdS aWbg O\R C W\QSb \ H\WdS aWbgp
4Pab OQba GVS SabW ObSa T bVS UO a T Q \aSQcbWdS SWUS\dOZcS T O ZOQWO\ 6VS\ 7OUcO\U "GaW\UVcO H\WdS aWbg \ bVWa bOZY eS eWZZ bOZY OP cb bVS SabW ObSa T bVS UO a T Q \aSQcbWdS SWUS\dOZcS T O ZOQWO\' 9 bVS SWUS\dOZcS PZS T bVS 7W WQVZSb O ZOQWO\ \ bVS P c\RSR R OW\ W\ 8cQZWRSO\ a OQS E\ eS PbOW\ bVS SabW ObSa T bVS c S P c\Ra T bVS UO T Q \aSQcbWdS SWUS\dOZcSa eVWQV O S bVS PSab aaWPZS W\ bVS SO\W\U T bVS RS a T SWUS\dOZcSa' GVWa Wa bVS X W\b e Y eWbV C TSaa \UQO\U LO\U O\R 7 ' GO MVS\U' 9cYOgO QObSU g T OcRO :W\hPc U RSZ 9O\ cWXc\o5SWXW\U H\WdS aWbgp \ bVWa S b eWZZ UWdS O WU ca ObVS ObWQOZ Q \ab cQbW \ T bVS 9cYOgO QObSU g T OcROc :W\hPc U RSZ eVWQV eS S RSaQ WPSR P WSTZg Pg SQS\b e Y T :OW bb S JWbbS\ O\R O O\ d \baSdWQV F WPSZ O\ \ bVS OZUSP O T bVS W\T OSR' GVWa Wa O RcZW PZS OP cb JWbbS\ S cObW \ eWbV STaQVSbh P c\RO g Q \RWbW \' GVWa Wa O X W\b e Y eWbV 7W\Ugc LO\U O\R JS\TS\U WO\U' B\ O 6 \XSQbc S T F'F' 6VS \ cWbO 9S\U "AO\YOW H\WdS aWbg \ bVWa bOZY eS eWZZ SaS\b O T T O\ ZR Q \XSQbc S T C T' F'F' 6VS \ \ OTTW\S O\WT ZRa' 4 ac dSg \ POZO\QSR Sb WQa 9c WfWO\U "9cRO\ H\WdS aWbg \ bVWa bOZY eWZZ T Qca \ O TSe Oa SQba T POZO\QSR Sb WQa \ O Q OQb VS WbWO\ Sb WQa' I Zc S W\W WhW\U W\QW ZS T Q \T OZZg OSVZS 8W\abSW\ OfeSZZ Sb WQa 9cbOYW 4YWb "H\WdS aWbg T G Yg JS RWaQcaa \ bVS d Zc S W\W WhObW \ W\QW ZS T Q \T OZZg OSVZS 8W\abSW\ OfeSZZ Sb WQa W\ bVS aW WZO a W Wb Oa bVS OSVZS EWQQW a ZWb \a O\R FOaOYW 8W\abSW\ Sb WQa' GVWa bOZY Wa POaS \ O X W\b e Y eWbV OXW S B\ ' 6VO OQbS WhObW \ T O QZOaa T bg WQOZ JWZZ S acP O\WT ZRa :c MVS\oLc\\O\ A OZ H\WdS aWbgp \ bVWa bOZY JS SaS\b O\ W\bSU OZ W\S cOZWbg \ O Q OQb JWZZ S Vg S ac TOQS' \ O bWQcZO bVS Q \T OZ QZOaaSa T JWZZ S b W O S QVO OQbS WhSR Pg caW\U O Q \T OZ W\dO WO\b Tc\QbW \' \ ORRWbW \OZ eS W\b RcQS O QZOaa T a SQWOZ JWZZ S acP O\WT ZRa' FW\QS bVWa QZOaa T JWZZ S acP O\WT ZRa W\QZcRSa Y\ e\ W\bS SabW\U SfO ZSa eS QOZZ Wb bg WQOZ JWZZ S acP O\WT ZRa' JS SaS\b WQV Q \Q SbS SfO ZSa T bVWa QZOaa JWZZ S acP O\WT ZRa' F S \ \SfWabS\QS SacZba T b O\aZObW\U a ZWb \a c\WYOeO SWbO "AOU gO H\WdS aWbg G O\aZObW\U a ZWb \a "SbS \OZ a ZcbW \a O S bg S aW\UcZO Wbg RSZa T SO\ Qc dObc S TZ e' \ g bOZY eWZZ aV e a S 5S \abSW\ bg S SacZba T b O\aZObW\U a ZWb \a W\ VWUVS Q RW S\aW \a' k H\W cS\Saa T QZ aSR aSZT aW WZO a ZcbW \a b lY Qc dObc S TZ e W OWhV \U "GaW\UVcO H\WdS aWbg \ SQS\b be O S a 6V W 7OaYOa cZ a O\R 5 S\RZS 6V W 7OaYOa cZ a VOdS dSR bVOb O\g \ RW S\aW \OZ ab WQbZg Q \dSf QZ aSR aSZT aW WZO a ZcbW \a b k \$ l\ Qc dObc S TZ e W\ E eWbV kn (\$+ cab PS O\ SZZW a WR c\R a VS S' 5g ORO bW\U bVSW bSab Tc\QbW \a O\R Sf Z W\U S bWSa T bVS Y bV SZS S\bO g ag Sb WQ Tc\QbW \a lY W\bS\aWdSZg eS aV e bVOb T O\g TWfSR Y eWbV n Y k n \ O\g ab WQbZg Q \dSf QZ aSR aSZT aW WZO a ZcbW \a b lY Qc dObc S TZ e W\ E\$ cab PS O c\R a VS S'GVWa Wa X W\b e Y eWbV cW O O\R FVO\hS :O ' 8\S Ug WRS\bWbg T O aS cS\QS T FOQYa HVZS\PSQY O a b O a VS S W WOgc "H\WdS aWbg T GSQV\ Z Ug O\R FQWS\QSa T 6VW\O Energy identity for a sequence of Sacks-Uhlenbeck maps to a sphere
Li, Jiayu
Abstract
For a map u from a Riemann surface M to a Riemannian manifold and α > 1, the α energy functional is defined as
2α Eα(u)= |∇u| dx. !M
We call uα asequenceofSacks-Uhlenbeckmapsifuα is a critical point of Eα and sup Eα(uα) < ∞. α>1 In this talk, when the target manifold is a standard sphere SK ,weprove the energy identity for a sequence of Sacks-Uhlenbeck maps during blowing up.
1
6 \\SQbW\U bVS \baSdWQV JWbbS\ O\R RUS bOc Tc\QbW \a Pg bVS IW Oa S Ob a Wc KWO P "5SWXW\U H\WdS aWbg \baSdWQV JWbbS\ bOc Tc\QbW \ O\R bVS RUS bOc Tc\QbW \ O S US\S ObW\U Tc\QbW \a T be bg Sa T W\bS aSQbW \ \c PS a \ RcZW a OQSa T abOPZS Qc dSa' 5 bV T bVS O S bOc Tc\QbW \a T bVS C VWS O QVg' \ bVWa bOZY eWZZ RSaQ WPS V e b Q \\SQb bVSaS be bOc Tc\QbW \a Pg RWTTS S\bWOZ S Ob a PSZ \UW\U b bVS NeWRSVObi: "NW\Tbg j U c ' \RSSR bVSaS be bOc Tc\QbW \a QO\ PS Q \\SQbSR caW\U IW Oa S Ob a' GVWa dSa O Q \XSQbc S abSR Pg 4ZSfO\R d' GVWa Wa O X W\b e Y eWbV :SVO JO\U' 8fb S OZ N OVZS dS aW \a T bVS LOc GWO\ 7 \OZRa \ 6 \XSQbc S OPcQVW G aVWYW "BaOYO H\WdS aWbg \ N OVZS 8W\abSW\ QOaSa bVS LOc GWO\ 7 \OZRa \ Q \XSQbc S \ ZO WhSR OZUSP OWQ O\WT ZRa eOa a ZdSR OTTW ObWdSZg Pg GWO\ O\R 6VS\ 7 \OZRa \ Fc\' eSdS T US\S OZ ZO WhObW \a S US\S OZZg T Sfb S OZ N OVZS QOaSa bVS US\S OZWhSR dS aW \a T bVS Q \XSQbc S O S abWZZ S\' \ bVWa bOZY ZZ RWaQcaa acQV dS aW \a T bVS Q \XSQbc S T QcaW\U \ bVS SfWabS\QS PZS ' FbOPWZWbg T O\bW QO\ \WQOZZg POZO\QSR Sb WQa FOWb FVc\acYS "H\WdS aWbg T G Yg JS abcRg bVS Oag b bWQ PSVOdW T cO\bWhSR 7W\U Tc\QbW \OZa OZ \U 5S U O\ US RSaWQ Oga O\R dS bVOb bVS aZ S Ob W\TW\Wbg QO\ PS Sf SaaSR W\ bS a T 7 \OZRa \ 9cbOYW W\dO WO\ba O\R 6V e eSWUVba' 5OaSR \ bVS aZ S T cZO eS W\b RcQS O \Se OZUSP US Sb WQ abOPWZWbg \ 9O\ O\WT ZRa O\R aV e bVOb bVS SfWabS\QS T O\bW QO\ \WQOZZg POZO\QSR Sb WQa W ZWSa c abOPWZWbg' GVS SZObW \ PSbeSS\ c abOPWZWbg O\R bVS a Wa OZa RWaQcaaSR' 4a O\ bVS O ZWQObW \ T bVS aZ S T cZO eS USb bVS Z eS P c\R SabW ObS \ bVS 6OZOPW ZWYS Tc\QbW \OZa \ 9O\ O\WT ZRa' GVWa Wa O X W\b e Y eWbV E' GOYOVOaVW' STb W\dO WO\b Sb WQa O\R acP O\WT ZR US Sb g GO O c W aVW " W aVW O H\WdS aWbg 9 O ZSTb W\dO WO\b Sb WQ \ O UWdS\ WS U c eS QO\ Q \ab cQb O acP O\WT ZR eVS S bVS O PWS\b a OQS Wa bVS a OQS T OZZ ZSTb W\dO WO\b Sb WQa \ bVOb WS U c ' JS Sf SQb bVOb \WQS ZSTb W\dO WO\b Sb WQa O S Q Sa \RW\U b \WQS acP O\WT ZRa' \ bVWa bOZY eS W\b RcQS c T O Se Y O\R S\bW \ a S SacZba SZObSR b acQV Q Sa \RS\QS' :OcUSR ZW\SO aWU O RSZ O\R ORWOPObWQ ZW Wb Kc :cO\UP "C W\QSb \ H\WdS aWbg GVS UOcUSR ZW\SO aWU O RSZ ": F W\b RcQSR Pg JWbbS\ VOa PSS\ dS g W\TZcS\bWOZ W\ ab W\U bVS g' b c\WTWSa bVS : d JWbbS\ bVS g O\R bVS O\ROc :W\hPc U bVS g O\R dWRSa O UZ POZ WQbc S T W ag Sb g' \ O aS WSa T O S a eS VOdS ORS STT ba W\ Q \ab cQbW\U O ObVS ObWQOZ bVS g T : F caW\U ag ZSQbWQ US Sb WQ SbV Ra' \ bVWa bOZY eWZZ SaS\b c SQS\b Q \ab cQbW \ W\ bVS a QOZZSR US Sb WQ VOaS T : F ' \ O bWQcZO eWZZ Sf ZOW\ V e b RSTW\S O : d JWbbS\ bg S W\dO WO\b T : F ' T bW S S Wba eWZZ Sf ZOW\ bVS SZObW \ PSbeSS\ bVS : F W\dO WO\b O\R : d JWbbS\ W\dO WO\b dWO ORWOPObWQ ZW Wb' GVWa Wa O W\b e Y eWbV C T' GWO\' 6 O Wa \ O\UZSa O\R d Zc S LO OUcQVW GOYO " g b H\WdS aWbg JS W\b RcQS O \Se US Sb WQ W\dO WO\b T a OQSa eWbV Qc dObc S P c\RSR T PSZ e Pg SO\a T Q O Wa \ O\UZSa' GV cUV bVWa Wa O RWabO\QS W\dO WO\b W\ \Obc S Wb bc \a cb bVOb Wb Wa QZ aSZg SZObSR eWbV bVS d Zc S' 9 W\abO\QS eS QO\ Q ZSbSZg RSbS W\S WT bVS a OQS Wa Q ZZO aSR Pg caW\U bVWa W\dO WO\b' GVWa Wa O X W\b e Y eWbV 4gOb WbacWaVW' A OZ aQOZO Qc dObc S W\S cOZWbg \ bVS T QOZ acP O\WT ZRa T Wa O O Sb WQ Vg S ac TOQSa LO\ JS\XWO "5SWXW\U A OZ H\WdS aWbg 4\ Wa O O Sb WQ Vg S ac TOQS W\ c\Wb a VS Sa VOa be T QOZ acP O\WT ZRa' 6 \RWbW \ 4 ZOga O Q cQWOZ ZS W\ bVS QZOaaWTWQObW \ bVS g T Wa O O Sb WQ
Vg S ac TOQSa' GVWa O S RSbS W\Sa 64 bVS aSb T W\ba eWbV 6 \RWbW \ 4 W\
T QOZ acP O\WT ZRa' b bc \a cb bVOb bVS W\ba W\ 64 SOQV O\ c S P c\R T bVS \ OZ aQOZO Qc dObc S "aVO S bVO\ bVOb W\ 77II W\S cOZWbg ' JS OZa
RSbS W\S bVS aSba 6C " W\ba eWbV O OZZSZ aSQ \R Tc\RO S\bOZ T O\R 68 " W\ba eWbV 8W\abSW\ Q \RWbW \ eVWQV OQVWSdS be Z eS P c\Ra T bVS \ OZ aQOZO Qc dObc S' GVWa bOZY Wa POaSR \ X W\b e Y eWbV 'D':S O\R M'M'GO\U' C aWbWdS aQOZO Qc dObc S \ O\WT ZRa O\R T ZWObW \a MVO\U JSW W\U "AO\YOW H\WdS aWbg GVS TO ca WQV\S eWQh dO\WaVW\U bVS S abObSa bVOb bVS W hSP cQV 4 VOb US\ca T O QZ aSR EWS O\\WO\ a W\ O\WT ZR T aWbWdS aQOZO Qc dObc S S cOZa b hS ' JS RWaQcaa dO W ca US\S OZWhObW \a T bVWa QZOaaWQOZ SacZb' C S \Saa T 9 Tc\QbW \OZ MVc KWO VcO "5SWXW\U H\WdS aWbg 9 Tc\QbW \OZ ZOga O\ W bO\b ZS W\ bVS abcRg T abOPWZWbg T OSVZS O\WT ZRa' \ 220 GWO\ UOdS O\ O OQV b dS bVS S \Saa T 9 Tc\QbW \OZ Pg SabOPZWaVW\U O a bV ZS O T EWQQW TZ e' \ bVWa bOZY eS UWdS O\ bVS T T GWO\ a SacZb Pg caW\U bVS SUcZO Wbg T Q ZSf \US 4 S S S cObW \' Bc \Se O OQV QO\ PS OZa US\S OZWhSR b abcRg beWabSR 9 Tc\QbW \OZ Oaa QWObSR b aW\UcZO OSVZS 8W\abSW\ Sb WQa' GVWa Wa O X W\b e Y eWbV GWO\'