Energy identity for approximate harmonic maps from surfaces to general targets

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• 作者：Wendong Wang^a ; ^{wendong@dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Dongyi Wei^b ; ^{jnwdyi@163.com" class="auth_mail" title="E-mail the corresponding author} ; Zhifei Zhang^c ; ^{zfzhang@math.pku.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Energy identity ; Harmonic map ; Hopf differential
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：776-803
• 全文大小：435 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002212361630297X&_mathId=si1.gif&_user=111111111&_pii=S002212361630297X&_rdoc=1&_issn=00221236&md5=590e4612ee3feea83c643954d273ca1d" title="Click to view the MathML source">u_nclass="mathContainer hidden">class="mathCode">$u_n$ be a sequence of mappings from a closed Riemannian surface M to a general Riemannian manifold N  . If class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002212361630297X&_mathId=si1.gif&_user=111111111&_pii=S002212361630297X&_rdoc=1&_issn=00221236&md5=590e4612ee3feea83c643954d273ca1d" title="Click to view the MathML source">u_nclass="mathContainer hidden">class="mathCode">$u_n$ satisfies where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002212361630297X&_mathId=si3.gif&_user=111111111&_pii=S002212361630297X&_rdoc=1&_issn=00221236&md5=2d6e42294fe20bfc00dd3a010b98c7e2" title="Click to view the MathML source">τ(u_n)class="mathContainer hidden">class="mathCode">$\tau (u_n)$ is the tension field of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002212361630297X&_mathId=si1.gif&_user=111111111&_pii=S002212361630297X&_rdoc=1&_issn=00221236&md5=590e4612ee3feea83c643954d273ca1d" title="Click to view the MathML source">u_nclass="mathContainer hidden">class="mathCode">$u_n$, then the so called energy identity and neckless property hold during blowing up. This result is sharp by Parker's example, where the tension fields of the mappings from Riemannian surface are bounded in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002212361630297X&_mathId=si4.gif&_user=111111111&_pii=S002212361630297X&_rdoc=1&_issn=00221236&md5=e428ebf4778fcc92e3cbee8aa2f62a9e" title="Click to view the MathML source">L¹(M)class="mathContainer hidden">class="mathCode">$L^1(M)$ but the energy identity fails.