爆破分析及其几何应用
|
摘要
爆破分析是偏微分方程和几何分析研究中的一个重要课题,其主要包括能量等式和no neck性质。本文的主要研究对象是调和映射及调和映射流的一些推广,例如Dirac-调和映射、双调和映射以及双调和映射流。本文证明了Dirac-调和映射和双调和映射的no neck性质。此外,本文还构造出双调和映射流在有限时间爆破的一些例子。具体而言,本文的主要结果列举如下。
I.Dirac-调和映射的neck分析
定理0.1.设(M,9)为紧的黎曼曲面,∑M是M上的spin从,(N,h)为任一紧的黎曼流形,{Φk,ψk}是从(M,g)到(N,h)的一列能量有界的Dirac-调和映射,设{Φk,ψk}弱收敛到Dirac-调和映射{Φ,ψ},P1,…,PI是爆破点集合,那么存在子列(不妨仍记为{Φk,ψk})和有限个Dirac-调和映射球{σli,ξli},i=1,...,I;l=1,...,Lt,使得Φ(D)∪ii1∪Lil=1(σli(S2))是连通的。
Ⅱ.双调和映射的neck分析
定理0.2.设B1是R4中的单位球,(N,h)是黎曼流形,ui是从B1到N的一列双调和映射满足其中A是常数。假设存在一列正数λi→0使得对于任意R4中的紧集K,有若ω是唯一的bubble,u∞是ui的弱极限,则我们有
注记0.3.为了叙述上简单起见,在上述定理中我们做了一个不必要的假设,即在爆破过程中只有一个bubble。事实上,根据Ding和Tian[15]的归纳假设我们知道定理对于多个bubble的情形同样是成立的。
利用上述定理的证明方法我们还可以给出双调和映射的能量等式和可去奇点的新证明。即我们有以下两个推论：
推论0.4.在上定理的假设条件下,我们有
推论0.5.设u是定义在B1\{0}上的光滑双调和映射,如果
则u可以光滑的延拓到B1。
Ⅲ.双调和映射流的有限时间爆破
定理0.6.设uit是从(B1,9)到(N,h)的一列逼近双调和映射,即ut满足其中9是球面度量。设A>0和p≥4/3为两常数,且假设存在一列正数λi→0使得对于任意R4中的紧集K,都有强收敛若ω是唯一的bubble,u∞是ui的弱极限,则我们有
定理0.7.设M'是一维数大于4的闭的流形且π4(M')≠(?),记Tm是m维环面,M=M'#Tm(连通和)。对于M上的任何度量9,我们都可以找到初始映射u00：S4→M使得从u0出发的双调和映射流在有限时间爆破。
Blow-up analysis is an important topic in the study of partial differential equations and geometric analysis, which usually include energy identity and neck analysis. In the present paper, we are concerned on the popularization of harmonic maps and harmonic map flow, such as Dirac-harmonic maps, biharmonic maps and biharmonic map flow. I will prove in this paper that there is no neck during the blow-up process of Dirac-harmonic maps and biharmonic maps. On the other hand, I will construct some initial maps and the target manifold with some topology condition to prove the biharmonic map flow must blowup in finite time. In detail, the main results of the paper are as follows.
Ⅰ.Neck analysis of Dirac-harmonic maps
Theorem0.1. Let (M, g) be a compact Riemannian surface and SM be the spin bun-dle over M.(N, h) be another compact Riemannian manifold. For a sequence of smooth Dirac-harmonic maps {(φκ,ψκ)} with uniform bounded energy E(φκ,ψκ）≤∧ Ⅱ.Neck analysis of biharmonic maps
Theorem0.2. Suppose B4(?) R4is a unite sphere and (N, h) is a compact Riemannian manifold. Let Ui be a sequence of biharmonic maps from B4to N satisfying for some A>O. Assume that there is a sequence positive λi→O such that ui(λix)→w on any compact set K K (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then,
Remark0.3. For convenient, we make a unnecessary assumption in the above theorem, that is there is only one bubble in the blow-up process. But, according to Ding and Tian[15I's induction method, we know the above theorem also holds for two or more bubbles.
By the proof of above theorem, I can give a new method to prove the energy identity and removable singularity of biharmonic maps. So, we can get the following two corollary.
Corollary0.4. Under the same assumption of above theorem, we have
Corollary0.5. Let u be a smooth biharmonic map on B1\{0}. If then u can be extended to a smooth biharmonic map on B1.
Ⅲ.Finite time blow-up of biharmonic map flow
Theorem0.6. Let ui be a sequence of approximate biharmonic maps from B4to (N, h) satisfying with for some∧>0and p≥4/3. Assume that there is a positive sequence λi→0such that on any compact set K C (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then,
Theorem0.7. Suppose that M'is any closed manifold of dimension m>4with nontrivial π4(M') and let M'##Tm be the connected sum of M' with the torus of the same dimension. For any Riemannian metric g on M, we can find (infinitely many) initial map u0:S4→M such that the biharmonic map flow starting from u0develops finite time singularity.
I.Dirac-调和映射的neck分析
定理0.1.设(M,9)为紧的黎曼曲面,∑M是M上的spin从,(N,h)为任一紧的黎曼流形,{Φk,ψk}是从(M,g)到(N,h)的一列能量有界的Dirac-调和映射,设{Φk,ψk}弱收敛到Dirac-调和映射{Φ,ψ},P1,…,PI是爆破点集合,那么存在子列(不妨仍记为{Φk,ψk})和有限个Dirac-调和映射球{σli,ξli},i=1,...,I;l=1,...,Lt,使得Φ(D)∪ii1∪Lil=1(σli(S2))是连通的。
Ⅱ.双调和映射的neck分析
定理0.2.设B1是R4中的单位球,(N,h)是黎曼流形,ui是从B1到N的一列双调和映射满足其中A是常数。假设存在一列正数λi→0使得对于任意R4中的紧集K,有若ω是唯一的bubble,u∞是ui的弱极限,则我们有
注记0.3.为了叙述上简单起见,在上述定理中我们做了一个不必要的假设,即在爆破过程中只有一个bubble。事实上,根据Ding和Tian[15]的归纳假设我们知道定理对于多个bubble的情形同样是成立的。
利用上述定理的证明方法我们还可以给出双调和映射的能量等式和可去奇点的新证明。即我们有以下两个推论：
推论0.4.在上定理的假设条件下,我们有
推论0.5.设u是定义在B1\{0}上的光滑双调和映射,如果
则u可以光滑的延拓到B1。
Ⅲ.双调和映射流的有限时间爆破
定理0.6.设uit是从(B1,9)到(N,h)的一列逼近双调和映射,即ut满足其中9是球面度量。设A>0和p≥4/3为两常数,且假设存在一列正数λi→0使得对于任意R4中的紧集K,都有强收敛若ω是唯一的bubble,u∞是ui的弱极限,则我们有
定理0.7.设M'是一维数大于4的闭的流形且π4(M')≠(?),记Tm是m维环面,M=M'#Tm(连通和)。对于M上的任何度量9,我们都可以找到初始映射u00：S4→M使得从u0出发的双调和映射流在有限时间爆破。
Blow-up analysis is an important topic in the study of partial differential equations and geometric analysis, which usually include energy identity and neck analysis. In the present paper, we are concerned on the popularization of harmonic maps and harmonic map flow, such as Dirac-harmonic maps, biharmonic maps and biharmonic map flow. I will prove in this paper that there is no neck during the blow-up process of Dirac-harmonic maps and biharmonic maps. On the other hand, I will construct some initial maps and the target manifold with some topology condition to prove the biharmonic map flow must blowup in finite time. In detail, the main results of the paper are as follows.
Ⅰ.Neck analysis of Dirac-harmonic maps
Theorem0.1. Let (M, g) be a compact Riemannian surface and SM be the spin bun-dle over M.(N, h) be another compact Riemannian manifold. For a sequence of smooth Dirac-harmonic maps {(φκ,ψκ)} with uniform bounded energy E(φκ,ψκ）≤∧
Theorem0.2. Suppose B4(?) R4is a unite sphere and (N, h) is a compact Riemannian manifold. Let Ui be a sequence of biharmonic maps from B4to N satisfying for some A>O. Assume that there is a sequence positive λi→O such that ui(λix)→w on any compact set K K (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then,
Remark0.3. For convenient, we make a unnecessary assumption in the above theorem, that is there is only one bubble in the blow-up process. But, according to Ding and Tian[15I's induction method, we know the above theorem also holds for two or more bubbles.
By the proof of above theorem, I can give a new method to prove the energy identity and removable singularity of biharmonic maps. So, we can get the following two corollary.
Corollary0.4. Under the same assumption of above theorem, we have
Corollary0.5. Let u be a smooth biharmonic map on B1\{0}. If then u can be extended to a smooth biharmonic map on B1.
Ⅲ.Finite time blow-up of biharmonic map flow
Theorem0.6. Let ui be a sequence of approximate biharmonic maps from B4to (N, h) satisfying with for some∧>0and p≥4/3. Assume that there is a positive sequence λi→0such that on any compact set K C (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then,
Theorem0.7. Suppose that M'is any closed manifold of dimension m>4with nontrivial π4(M') and let M'##Tm be the connected sum of M' with the torus of the same dimension. For any Riemannian metric g on M, we can find (infinitely many) initial map u0:S4→M such that the biharmonic map flow starting from u0develops finite time singularity.
引文
[1]Robert A Adams and John JF Fournier, Sobolev spaces, vol.140, Academic press, 2003.
[2]Christine Breiner and Tobias Lamm, Compactness results for sequences of ap-proximate biharmonic maps, arXiv preprint arXiv:1312.5787 (2013).
[3]Kung-Ching Chang, Wei Yue Ding, Rugang Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, Journal of Differential Geometry 36 (1992), no.2,507-515.
[4]Sun-Yung A Chang, Lihe Wang, and Paul C Yang, A regularity theory of bihar-monic maps, Communications on Pure and Applied Mathematics 52 (1999), no. 9,1113-1137.
[5]J. Chen and Y. Li, Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows, ArXiv e-prints (2013).
[6]Qun Chen, Jurgen Jost, Jiayu Li, and Guofang Wang, Regularity theorems and en-ergy identities for dirac-harmonic maps, Mathematische Zeitschrift 251 (2005), no.1,61-84.
[7]______, Dirac-harmonic maps, Mathematische Zeitschrift 254 (2006), no.2, 409-432.
[8]Qun Chen, Jiirgen Jost, and Guofang Wang, Liouville theorems for Dirac-harmonic maps, Journal of Mathematical Physics 48 (2007), no.11,113517,1-13.
[9]______, The maximum principle and the dirichlet problem for dirac-harmonic maps, Calculus of Variations and Partial Differential Equations 47 (2013), no. 1-2,87-116.
[10]Qun Chen, Jiirgen Jost, Guofang Wang, Miaomiao Zhu, The boundary value problem for dirac-harmonic maps, Journal of the European Mathematical Society 15 (2011), no.3,997-1031.
[11]Yunmei Chen and Wei-Yue Ding, Blow-up and global existence for heat flows of harmonic maps, Inventiones mathematicae 99 (1990), no.1,567-578.
[12]Yunmei Chen and Michael Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Mathematische Zeitschrift 201 (1989), no.1, 83-103.
[13]Pierre Degne, Quantum fields and strings:a course for mathematicians, vol.2, 1999.
[14]Weiyue Ding, Jiayu Li, and Wei Li, Nonstationary weak limit of a stationary harmonic map sequence, Comm. Pure Appl. Math.56 (2003), no.2,270-277. MR 1934622 (2003h:58020)
[15]Weiyue Ding and Gang Tian, Energy identity for a class of approximate har-monic maps from surfaces, Comm. Anal. Geom.3 (1995), no.3-4,543-554. MR 1371209 (97e:58055)
[16]James Eells and Luc Lemaire, A report on harmonic maps, Bulletin of the London mathematical society 10 (1978), no.1,1-68.
[17]Lawrence C Evans, Partial regularity for stationary harmonic maps into spheres, Archive for rational mechanics and analysis 116 (1991), no.2,101-113.
[18]Andreas Gastel, The extrinsic polyharmonic map heat flow in the critical dimen-sion, Advances in Geometry 6 (2006), no.4,501-521.
[19]David Gilbarg and Neil S Trudinger, Elptic partial differential equations of sec-ond order, vol.224, Springer Verlag,2001.
[20]Frederic Helein, Regularte des applications faiblement harmoniques entre une surface et une variete riemannienne, C. R. Acad. Sci. Paris Ser. I Math.312 (1991), no.8,591-596. MR 1101039 (92e:58055)
[21]______, Regularity of weakly harmonic maps from a surface into a manifold with symmetries, manuscripta mathematica 70 (1991), no.1,203-218.
[22]______, Harmonic maps, conservation laws and moving frames,Cambridge Uni- versity Press, vol.150,2002.
[23]Peter Hornung and Roger Moser, Energy identity for intrinsically biharmonic maps in four dimensions, Analysis & PDE 5 (2012), no.1,61-80.
[24]Jiirgen Jost, Two-dimensional geometric variational problems, Pure and Apped Mathematics (New York), John Wiley & Sons Ltd., Chichester,1991, A Wiley-Interscience Pubcation. MR 1100926 (92h:58045)
[25]Jurgen Jost, Xiaohuan Mo, and Miaomiao Zhu, Some explicit constructions of dirac-harmonic maps, Journal of Geometry and Physics 59 (2009), no.11,1512-1527.
[26]Tobias Lamm, Heat flow for extrinsic biharmonic maps with small initial energy, Ann. Global Anal. Geom.,26 (2004), no.4 369-384.
[27]Tobias Lamm and Tristan Riviere, Conservation laws for fourth order systems in four dimensions, Communications in Partial Differential Equations 33 (2008), no. 2,245-262.
[28]Paul Laurain and Tristan Riviere, Energy quantization for biharmonic maps, Ad-vances in Calculus of Variations 6 (2013), no.2,191-216.
[29]H Blaine Lawson and Marie-Louise Michelsohn, Spin geometry, vol.38,1989.
[30]Jiayu Li and Xiangrong Zhu, Non existence of quasi-harmonic spheres, Calc. Var. Partial Differential Equations 37 (2010), no.3-4,441-460. MR 2592981 (2011a:58029)
[31]——, Small energy compactness for approximate harmomic mappings, Com-mun. Contemp. Math.13 (2011), no.5,741-763. MR 2847227
[32]——, Energy identity for the maps from a surface with tension field bounded in lp, Pacific Journal of Mathematics 260 (2012), no.1,181-195.
[33]Yuxiang Li and Youde Wang, A weak energy identity and the length of necks for a sequence of sacks-uhlenbeck a-harmonic maps, Advances in Mathematics 225 (2010), no.3,1134-1184.
[34]——, Energy identities for a-harmonic sequences are not true, arXiv preprint arⅩⅳ:1303.2862 (2013).
[35]Fang-Hua Lin, Gradient estimates and blow-up analysis for stationary harmonic maps, Annals of Mathematics 149 (1999),785-829.
[36]Fanghua Lin and Changyou Wang, Energy identity of harmonic map flows from surfaces at finite singular time, Calculus of Variations and Partial Differential Equations 6 (1998), no.4,369-380.
[37]______, Harmonic and quasi-harmonic spheres, Communications in Analysis and Geometry 7 (1999), no.2,397-429.
[38]______, Harmonic and quasi-harmonic spheres. Ⅱ, Comm. Anal. Geom.10 (2002), no.2,341-375. MR 1900755 (2003d:58029)
[39]______, Harmonic and quasi-harmonic spheres. Ⅲ. Rectifiability of the parabolic defect measure and generalized varifold flows, Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002), no.2,209-259. MR 1902744 (2003d:58028)
[40]______, The analysis of harmonic maps and their heat flows, World Scientific, 2008.
[41]Lei Liu, No neck for dirac-harmonic maps, Calculus of Variations and Partial Differential Equations, DOI 10.1007/s00526-013-0702-9. (to appear)
[42]Lei Liu and Hao Yin, Neck analysis for biharmonic maps, arXiv preprint arXiv:1312.4600 (2013).
[43]______, On the finite time blow-up of biharmonic map flow in dimension four, arⅩⅳ1401.6274 (2014).
[44]Yong Luo, Energy identity and removable singularities of maps from a Riemann surface with tension field unbounded in L2, Pacific J. Math.256 (2012), no.2, 365-380. MR 2944981
[45]Roger Moser, A variational problem pertaining to biharmonic maps, Communi-cations in Partial Differential Equations 33 (2008), no.9,1654-1689.
[46]______, Weak solutions of a biharmonic map heat flow, Advances in Calculus of Variations 2 (2009), no.1,73-92.
[47]Thomas Parker et al., Bubble tree convergence for harmonic maps, J. Diff. Geom 44 (1996), no.3,595-633.
[48]Thomas H Parker and Jon G Wolfson, Pseudo-holomorphic maps and bubble trees, The Journal of Geometric Analysis 3 (1993), no.1,63-98.
[49]Jie Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres, Communications in Analysis and Geometry 3 (1995),297-316.
[50]Jie Qing and Gang Tian, Bubbling of the heat flows for harmonic maps from surfaces, Communications on pure and apped mathematics 50 (1997), no.4,295-310.
[51]J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no.1,1-24. MR 604040 (82f:58035)
[52]Christoph Scheven, Dimension reduction for the singular set of biharmonic maps, Advances in Calculus of Variations 1 (2008), no.1,53-91.
[53]______, An optimal partial regularity result for minimizers of an intrinsically defined second-order functional, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.26, Elsevier,2009, pp.1585-1605.
[54]Richard Schoen and Karen Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom 17 (1982), no.2,307-335.
[55]______, Regularity of minimizing harmonic maps into the sphere, Inventiones mathematicae 78 (1984), no.1,89-100.
[56]Michael Struwe, On the evolution of harmonic mappings of riemannian surfaces, Commentarii Mathematici Helvetici 60 (1985), no.1,558-581.
[57]_______, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom 28 (1988), no.3,485-502.
[58]_______, Partial regularity for biharmonic maps, revisited, Calculus of Variations and Partial Differential Equations 33 (2008), no.2,249-262.
[59]Peter Topping, Winding behaviour of finite-time singularities of the harmonic map heat flow, Mathematische Zeitschrift 247 (2004), no.2,279-302.
[60]Changyou Wang, Bubble phenomena of certain palais-smale sequences from sur-faces to general targets, Houston J. Math 22 (1996), no.3,559-590.
[61]______,Biharmonic maps from R4 into a riemannian manifold, Mathematische Zeitschrift 247 (2004), no.1,65-87.
[62]______, Remarks on biharmonic maps into spheres, Calculus of Variations and Partial Differential Equations 21 (2004), no.3,221-242.
[63]______, Stationary biharmonic maps from Rm into a riemannian manifold, Com-munications on pure and applied mathematics 57 (2004), no.4,419-444.
[64]______, Heat flow of biharmonic maps in dimensions four and its application, Pure and Applied Mathematics Quarterly 3 (2007), no.2.
[65]Changyou Wang and Deliang Xu, Regularity of dirac-harmonic maps, Interna-tional Mathematics Research Notices 2009 (2009), no.20,3759-3792.
[66]Changyou Wang and Shenzhou Zheng, Energy identity for a class of approximate biharmonic maps into sphere in dimension four, arXiv preprint arXiv:1110.6536 (2011).
[67]______, Energy identity of approximate biharmonic maps to riemannian mani-folds and its application, Journal of Functional Analysis 263 (2012), no.4,960-987.
[68]Liang Zhao, Energy identities for dirac-harmonic maps, Calculus of Variations and Partial Differential Equations 28 (2007), no.1,121-138.
[69]Miaomiao Zhu, Dirac-harmonic maps from degenerating spin surfaces i:the neveu-schwarz case, Calculus of Variations and Partial Differential Equations 35 (2009), no.2,169-189.
[70]______, Regularity for weakly dirac-harmonic maps to hypersurfaces, Annals of Global Analysis and Geometry 35 (2009), no.4,405-412.
[71]Xiangrong Zhu, No neck for approximate harmonic maps to the sphere, Nonnear Analysis:Theory, Methods & Appcations (2012).
[2]Christine Breiner and Tobias Lamm, Compactness results for sequences of ap-proximate biharmonic maps, arXiv preprint arXiv:1312.5787 (2013).
[3]Kung-Ching Chang, Wei Yue Ding, Rugang Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, Journal of Differential Geometry 36 (1992), no.2,507-515.
[4]Sun-Yung A Chang, Lihe Wang, and Paul C Yang, A regularity theory of bihar-monic maps, Communications on Pure and Applied Mathematics 52 (1999), no. 9,1113-1137.
[5]J. Chen and Y. Li, Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows, ArXiv e-prints (2013).
[6]Qun Chen, Jurgen Jost, Jiayu Li, and Guofang Wang, Regularity theorems and en-ergy identities for dirac-harmonic maps, Mathematische Zeitschrift 251 (2005), no.1,61-84.
[7]______, Dirac-harmonic maps, Mathematische Zeitschrift 254 (2006), no.2, 409-432.
[8]Qun Chen, Jiirgen Jost, and Guofang Wang, Liouville theorems for Dirac-harmonic maps, Journal of Mathematical Physics 48 (2007), no.11,113517,1-13.
[9]______, The maximum principle and the dirichlet problem for dirac-harmonic maps, Calculus of Variations and Partial Differential Equations 47 (2013), no. 1-2,87-116.
[10]Qun Chen, Jiirgen Jost, Guofang Wang, Miaomiao Zhu, The boundary value problem for dirac-harmonic maps, Journal of the European Mathematical Society 15 (2011), no.3,997-1031.
[11]Yunmei Chen and Wei-Yue Ding, Blow-up and global existence for heat flows of harmonic maps, Inventiones mathematicae 99 (1990), no.1,567-578.
[12]Yunmei Chen and Michael Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Mathematische Zeitschrift 201 (1989), no.1, 83-103.
[13]Pierre Degne, Quantum fields and strings:a course for mathematicians, vol.2, 1999.
[14]Weiyue Ding, Jiayu Li, and Wei Li, Nonstationary weak limit of a stationary harmonic map sequence, Comm. Pure Appl. Math.56 (2003), no.2,270-277. MR 1934622 (2003h:58020)
[15]Weiyue Ding and Gang Tian, Energy identity for a class of approximate har-monic maps from surfaces, Comm. Anal. Geom.3 (1995), no.3-4,543-554. MR 1371209 (97e:58055)
[16]James Eells and Luc Lemaire, A report on harmonic maps, Bulletin of the London mathematical society 10 (1978), no.1,1-68.
[17]Lawrence C Evans, Partial regularity for stationary harmonic maps into spheres, Archive for rational mechanics and analysis 116 (1991), no.2,101-113.
[18]Andreas Gastel, The extrinsic polyharmonic map heat flow in the critical dimen-sion, Advances in Geometry 6 (2006), no.4,501-521.
[19]David Gilbarg and Neil S Trudinger, Elptic partial differential equations of sec-ond order, vol.224, Springer Verlag,2001.
[20]Frederic Helein, Regularte des applications faiblement harmoniques entre une surface et une variete riemannienne, C. R. Acad. Sci. Paris Ser. I Math.312 (1991), no.8,591-596. MR 1101039 (92e:58055)
[21]______, Regularity of weakly harmonic maps from a surface into a manifold with symmetries, manuscripta mathematica 70 (1991), no.1,203-218.
[22]______, Harmonic maps, conservation laws and moving frames,Cambridge Uni- versity Press, vol.150,2002.
[23]Peter Hornung and Roger Moser, Energy identity for intrinsically biharmonic maps in four dimensions, Analysis & PDE 5 (2012), no.1,61-80.
[24]Jiirgen Jost, Two-dimensional geometric variational problems, Pure and Apped Mathematics (New York), John Wiley & Sons Ltd., Chichester,1991, A Wiley-Interscience Pubcation. MR 1100926 (92h:58045)
[25]Jurgen Jost, Xiaohuan Mo, and Miaomiao Zhu, Some explicit constructions of dirac-harmonic maps, Journal of Geometry and Physics 59 (2009), no.11,1512-1527.
[26]Tobias Lamm, Heat flow for extrinsic biharmonic maps with small initial energy, Ann. Global Anal. Geom.,26 (2004), no.4 369-384.
[27]Tobias Lamm and Tristan Riviere, Conservation laws for fourth order systems in four dimensions, Communications in Partial Differential Equations 33 (2008), no. 2,245-262.
[28]Paul Laurain and Tristan Riviere, Energy quantization for biharmonic maps, Ad-vances in Calculus of Variations 6 (2013), no.2,191-216.
[29]H Blaine Lawson and Marie-Louise Michelsohn, Spin geometry, vol.38,1989.
[30]Jiayu Li and Xiangrong Zhu, Non existence of quasi-harmonic spheres, Calc. Var. Partial Differential Equations 37 (2010), no.3-4,441-460. MR 2592981 (2011a:58029)
[31]——, Small energy compactness for approximate harmomic mappings, Com-mun. Contemp. Math.13 (2011), no.5,741-763. MR 2847227
[32]——, Energy identity for the maps from a surface with tension field bounded in lp, Pacific Journal of Mathematics 260 (2012), no.1,181-195.
[33]Yuxiang Li and Youde Wang, A weak energy identity and the length of necks for a sequence of sacks-uhlenbeck a-harmonic maps, Advances in Mathematics 225 (2010), no.3,1134-1184.
[34]——, Energy identities for a-harmonic sequences are not true, arXiv preprint arⅩⅳ:1303.2862 (2013).
[35]Fang-Hua Lin, Gradient estimates and blow-up analysis for stationary harmonic maps, Annals of Mathematics 149 (1999),785-829.
[36]Fanghua Lin and Changyou Wang, Energy identity of harmonic map flows from surfaces at finite singular time, Calculus of Variations and Partial Differential Equations 6 (1998), no.4,369-380.
[37]______, Harmonic and quasi-harmonic spheres, Communications in Analysis and Geometry 7 (1999), no.2,397-429.
[38]______, Harmonic and quasi-harmonic spheres. Ⅱ, Comm. Anal. Geom.10 (2002), no.2,341-375. MR 1900755 (2003d:58029)
[39]______, Harmonic and quasi-harmonic spheres. Ⅲ. Rectifiability of the parabolic defect measure and generalized varifold flows, Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002), no.2,209-259. MR 1902744 (2003d:58028)
[40]______, The analysis of harmonic maps and their heat flows, World Scientific, 2008.
[41]Lei Liu, No neck for dirac-harmonic maps, Calculus of Variations and Partial Differential Equations, DOI 10.1007/s00526-013-0702-9. (to appear)
[42]Lei Liu and Hao Yin, Neck analysis for biharmonic maps, arXiv preprint arXiv:1312.4600 (2013).
[43]______, On the finite time blow-up of biharmonic map flow in dimension four, arⅩⅳ1401.6274 (2014).
[44]Yong Luo, Energy identity and removable singularities of maps from a Riemann surface with tension field unbounded in L2, Pacific J. Math.256 (2012), no.2, 365-380. MR 2944981
[45]Roger Moser, A variational problem pertaining to biharmonic maps, Communi-cations in Partial Differential Equations 33 (2008), no.9,1654-1689.
[46]______, Weak solutions of a biharmonic map heat flow, Advances in Calculus of Variations 2 (2009), no.1,73-92.
[47]Thomas Parker et al., Bubble tree convergence for harmonic maps, J. Diff. Geom 44 (1996), no.3,595-633.
[48]Thomas H Parker and Jon G Wolfson, Pseudo-holomorphic maps and bubble trees, The Journal of Geometric Analysis 3 (1993), no.1,63-98.
[49]Jie Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres, Communications in Analysis and Geometry 3 (1995),297-316.
[50]Jie Qing and Gang Tian, Bubbling of the heat flows for harmonic maps from surfaces, Communications on pure and apped mathematics 50 (1997), no.4,295-310.
[51]J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no.1,1-24. MR 604040 (82f:58035)
[52]Christoph Scheven, Dimension reduction for the singular set of biharmonic maps, Advances in Calculus of Variations 1 (2008), no.1,53-91.
[53]______, An optimal partial regularity result for minimizers of an intrinsically defined second-order functional, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.26, Elsevier,2009, pp.1585-1605.
[54]Richard Schoen and Karen Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom 17 (1982), no.2,307-335.
[55]______, Regularity of minimizing harmonic maps into the sphere, Inventiones mathematicae 78 (1984), no.1,89-100.
[56]Michael Struwe, On the evolution of harmonic mappings of riemannian surfaces, Commentarii Mathematici Helvetici 60 (1985), no.1,558-581.
[57]_______, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom 28 (1988), no.3,485-502.
[58]_______, Partial regularity for biharmonic maps, revisited, Calculus of Variations and Partial Differential Equations 33 (2008), no.2,249-262.
[59]Peter Topping, Winding behaviour of finite-time singularities of the harmonic map heat flow, Mathematische Zeitschrift 247 (2004), no.2,279-302.
[60]Changyou Wang, Bubble phenomena of certain palais-smale sequences from sur-faces to general targets, Houston J. Math 22 (1996), no.3,559-590.
[61]______,Biharmonic maps from R4 into a riemannian manifold, Mathematische Zeitschrift 247 (2004), no.1,65-87.
[62]______, Remarks on biharmonic maps into spheres, Calculus of Variations and Partial Differential Equations 21 (2004), no.3,221-242.
[63]______, Stationary biharmonic maps from Rm into a riemannian manifold, Com-munications on pure and applied mathematics 57 (2004), no.4,419-444.
[64]______, Heat flow of biharmonic maps in dimensions four and its application, Pure and Applied Mathematics Quarterly 3 (2007), no.2.
[65]Changyou Wang and Deliang Xu, Regularity of dirac-harmonic maps, Interna-tional Mathematics Research Notices 2009 (2009), no.20,3759-3792.
[66]Changyou Wang and Shenzhou Zheng, Energy identity for a class of approximate biharmonic maps into sphere in dimension four, arXiv preprint arXiv:1110.6536 (2011).
[67]______, Energy identity of approximate biharmonic maps to riemannian mani-folds and its application, Journal of Functional Analysis 263 (2012), no.4,960-987.
[68]Liang Zhao, Energy identities for dirac-harmonic maps, Calculus of Variations and Partial Differential Equations 28 (2007), no.1,121-138.
[69]Miaomiao Zhu, Dirac-harmonic maps from degenerating spin surfaces i:the neveu-schwarz case, Calculus of Variations and Partial Differential Equations 35 (2009), no.2,169-189.
[70]______, Regularity for weakly dirac-harmonic maps to hypersurfaces, Annals of Global Analysis and Geometry 35 (2009), no.4,405-412.
[71]Xiangrong Zhu, No neck for approximate harmonic maps to the sphere, Nonnear Analysis:Theory, Methods & Appcations (2012).