子流形共形不变泛函的构造、变分与间隙现象的研究

Motivated by the completely solved of 2 dimensional Willmore Conjecture, the project would research the construction and variation problems of new types conformal invariant functional which mainly concern the covariant derivative tensors of second fundamental form in sub-manifolds. First, the conformal transformation rules of second fundamental form tensors and their covariant derivatives would be formulated; Second,through the trace-zero and other linear algebra techniques,the combinatorial construction methods of conformal invariant tensors would be designed; Third,the 1st and 2nd variational formulas of constructed functional would be calculated,and the examples of critical points would be constructed using the algebraic and ordinary differential equations; Forth, through concrete computation, the stable analysis and lower bound of functional would be estimated; Finally,through self-adjoint operators and special test functions, the gap phenomenon of critical points would be established.The project should flourish the research of Willmore conjecture related problems and build a bridge between the special functional and the topology and geometrical properties of sub-manifolds.

本项目以2维Willmore猜想的解决为契机，研究子流形共形不变泛函的构造与变分问题。通过发掘第二基本型等曲率张量的协变导数的共形变化规律与组合构造方法来设计新型的共形不变泛函；通过变分计算研究泛函临界点子流形的构造、稳定性分析与间隙现象；通过典型例子的估计对泛函的最优下界做出一系列的初步猜想。本项目的研究将丰富特殊子流形的类型，丰富子流形间隙现象类型，丰富Willmore猜想的类型，加深对子流形上的特殊泛函与子流形的几何、拓扑性质的关系的认识。

本项目以2维Willmore猜想的解决为契机，主要研究子流形共形不变泛函的构造方法、变分计算、例子构造、稳定性分析、间隙现象以及泛函的下界估计等问题。通过研究获得了如下结果：一是通过子流形基本方程得到了迹零第二基本型的变化规律，并以此为基础发展了通用的组合方法构造了多类型的子流形共形不变泛函；二是推导了子流形变分计算的通用公式，为了降低变分计算的难度，发展了作为中间过渡的牛顿张量构造方法并推导了其性质；三是计算了多类型共形不变泛函的变分公式，利用变分公式研究了泛函临界点的例子构造和稳定性分析；四是研究了特殊的共形不变泛函以及低阶曲率泛函的间隙现象，发展了几类点态与全局间隙定理；五是初步探索了共形不变泛函的下界估计问题。本项目的研究丰富了特殊子流形的类型，丰富了子流形间隙现象类型，丰富了Willmore猜想的类型，对于加深对子流形上的特殊泛函与子流形的几何、拓扑性质的关系的认识有一定意义，同时在人工智能中机器学习领域的流形学习子领域有一定的应用价值，在军事上可用于分析电磁空间波形追逃博弈的终点流形分析。