本書是一部研究非線性色散方程，特別是幾何發展方程的專著。波映射是在黎曼流形(M, g)上取值的最簡單的波方程，其拉格朗日算子同標量方程中的基本一樣，僅有的不同是長度的測量與度量g有關。通過Noether定理，拉格朗日對稱表明了波映射的守恆律，如能量守恆。在坐標系中，波映射有半線性系統波方程給出。在過去的20年中，一些表述這個系統的局部和全局適定性問題的重要方法出現了。由於弱色散效應，波映射定義在低維Minkowski空間，如Rt,x1+2上，呈現出特別的技術難題。這一類波函數有格外重要臨界能量特性，事實上即能量尺度和方程極其相似。本書將在雙曲平面中實現集中緊性方法的應用，這一實現的最大挑戰是，將產生更多有關解的詳細信息。目次：導論和概述；S[k]和N[k]空間；Hodge分解和空結構；S和N空間有關的雙線性估計；三線性估計；五線性和更高階非線性；一些基本擾動結論；BMO，Ap和權重交換子估計；Bahouri-Gerard集中緊性方法；主定理證明；附錄。讀者對性：數學專業、數值分析、非線性方程和幾何發展方程專業的廣大學者。Joachim Krieger（J.克里格,瑞士）是國際知名學者，在數學和物理學界享有盛譽。本書凝聚了作者多年科研和教學成果，適用於科研工作者、高校教師和研究生。

Introduction and overview1.1 The main result and its history1.2 Wave maps to H21.3 The small data theory1.4 The Bahouri—Gerard concentration compactness method1.5 The Kenig—Merle agument1.6 An overview of the book2 The spaces S[k] and N[k]2.1 Preliminaries2.2 The null—frame spaces2.3 The energy estimate2.4 A stronger S[k]—norm， and time localizations2.5 Solving the inhomogeneous wave equation in the Coulomb gauge3 Hodge decomposition and null—structures4 Bilinear estimates involving S and N spaces4.1 Basic L2—bounds4.2 An algebra estimate for S[k]4.3 Bilinear estimates involving both S[k1] and N[k2] waves4.4 Null—form bounds in the high—high case4.5 Null—form bounds in the low—high and high—low cases5 Trilinear estimates5.1 Reduction to the hyperbolic case5.2 Trilinear estimates for hyperbolic S—waves5.3 Improved trilinear estimates with angular alignment6 Quintilinear and higher nonlinearities6.1 Error terms of order higher than five7 Some basic perturbative results7.1 A blow—up criterion7.2 Control of wave maps via a fixed L2—profile8 BMO， Ap， and weighted commutator estimates9 The Bahouri—Gerard concentration compactness method9.1 The precise setup for the Bahouri—Gerard method9.2 Step 1： Frequency decomposition of initial data9.3 Step 2： Frequency localized approximations to the data9.4 Step 3： Evolving the lowest—frequency nonatomic part9.5 Completion of some proofs9.6 Step 4： Adding the first large component9.7 Step 5： Invoking the induction hypothesis9.8 Completion of proofs9.9 Step 6 of the Bahouri—Gerard process； adding all atoms10 The proof of the main theorem10.1 Some preliminary properties of the limiting profiles10.2 Rigidity I： Harmonic maps and reduction to the self—similar case10.3 Rigidity II： The self—similar case11 Appendix11.1 Completing a proof11.2 Completion of proofs11.3 Completion of a proof11.4 Completion of a proof11.5 Competion of proofsReferencesIndex