本课题主要研究三类与高维Willmore猜想密切相关的子流形的几何和分析：.1、利用Moebius第二基本形式关于Moebius度量的特征多项式定义牛顿变换和二阶微分算子，计算广义Willmore泛函用Moebius不变量刻画的第一变分公式和第二变分公式，从而用Moebius几何的方法考察W_r-极小子流形，构造W_r-极小子流形的新例子。.2、把曲面的共形约束变分推广到子流形的共形约束变分，考察Willmore泛函的共形约束变分，计算约束Willmore子流形所应满足的方程。同时考虑Willmore泛函在共形约束变分下的第二变分公式。.3、考察具有消失共形不变量A ̃的子流形的局部几何和整体几何，探讨其分类问题和拼挤现象。

This project mainly study geometry and analysis about three types of submanifolds closely related to the higher-dimensional Willmore conjecture:.1. Use the eigenpolynomial of the Moebius second fundamental form with respect to the Moebius metric to define Newton transforms and the second-order differential operators. Calculate the first variation formulas and the second variation formulas of generalized Willmore functionals using Moebius invariants. Then use the Moebius geometry method to study W_r-minimal submanifolds and construct new examples..2. Generalize the conformal constrained variation of surfaces to the conformal constrained variation of submanifolds. Consider the conformal constraint variation of Willmore functional. Calculate the equations to be satisfied for constrained Willmore submanifolds and consider the second variational formula of Willmore functional under conformal constraint variation..3. Investigate the local geometry and the global geometry of submanifolds with vanishing conformal invariant A ̃ . Discuss the classification problem and the spelling phenomenon.