论文标题
带有典型边界条件的扰动接触式Instantons的几何分析
Geometric analysis of perturbed contact instantons with Legendrian boundary conditions
论文作者
论文摘要
在本文中，我们提供以下非线性椭圆系统的分析基础，称为\ emph {hamiltonian perterted contact instanton方程}，$$（du -x_h \ otimesγ）^{π（0,1）= 0，\ quad d（e^{g_ {h，u}} u^*（λ+ h \ otimesγ）\ circ j）= 0 $$与触点triad $（m，λ，j）$相关联，并在传奇人物边界条件下与汉密尔顿$ h $及其边界价值问题联系在一起。 （1）我们确定了针对扰动接触式汉密尔顿轨迹的作用功能的正确选择，该轨迹为系统提供了梯度结构并得出其第一个变化公式。 （2）我们确定了对方程有限能量解决方案的气泡分析的正确选择。 （3）我们为解决方案开发椭圆规则理论，称为\ emph {扰动联系instantons}：我们首先建立了由哈密顿微积分约束的全局$ w^{2,2} $联系人的三合会连接三合会$（m，λ，j）$。然后，我们通过$ d_hu $的$π$ - 组件与$ d_hu $的reeb-component之间的$ c^{k，α} $ - 通过$π$ -COMPONEN之间的交替启动式参数进行估算。一路上，我们还建立了$ w^{1,4} $的边界规则定理 - 在弱的legendrian边界条件下，扰动接触式instanton方程的弱解决方案。 （4）基于这种规律性理论，我们证明在有限能量的假设下，渐近$ c^\ infty $收敛会导致刺穿。
In the present article, we provide analytic foundation of the following nonlinear elliptic system, called the \emph{Hamiltonian-perturbed contact instanton equation}, $$ (du - X_H \otimes γ)^{π(0,1)} = 0, \quad d(e^{g_{H, u}}u^*(λ+ H \otimes γ)\circ j) = 0 $$ associated to a contact triad $(M,λ,J)$ and contact Hamiltonian $H$ and its boundary value problem under the Legendrian boundary condition. (1) We identify the correct choice of the action functional for perturbed contact Hamiltonian trajectories which provides a gradient structure for the system and derive its first variation formula. (2) We identify the correct choice of the energy for the bubbling analysis for the finite energy solutions for the equation. (3) We develop elliptic regularity theory for the solution, called \emph{perturbed contact instantons}: We first establish a global $W^{2,2}$ bound by the Hamiltonian calculus and the harmonic theory of the vector-valued one form $d_Hu : = du - X_H(u)\otimes γ$ and its relevant Weitzenböck formulae utilizing the contact triad connection of the contact triad $(M,λ, J)$. Then we establish $C^{k,α}$-estimates by an alternating boot-strap argument between the $π$-component of $d_Hu$ and the Reeb-component of $d_Hu$. Along the way, we also establish the boundary regularity theorem of $W^{1,4}$-weak solutions of perturbed contact instanton equation under the weak Legendrian boundary condition. (4) Based on this regularity theory, we prove an asymptotic $C^\infty$ convergence result at a puncture under the hypothesis of finite energy.
