F-Willmore曲面的间隙现象

对于空间形式中的2维曲面,定义了F-Willmore泛函,此泛函包括经典的Willmore泛函作为特殊情形.F-Willmore泛函的临界点称为F-Willmore曲面.推导了第1变分公式并由此构造了F-Willmore曲面的典型例子.利用自伴算子作用于特殊的实验函数,得到了Simons类积分不等式,讨论了F-Willmore曲面的间隙现象,定出了间隙端点对应的特殊曲面.

Gap Phenomenon of F-Willmore Surfaces

For a $2$-dimensional surface in space forms, an $F$-Willmore functional is constructed, which includes and generalizes the well-known classic Willmore functional of surface. The critical point of $F$-Willmore is called $F$-Willmore surface, for which the variational equation and the Simons' typeintegral inequality are obtained. Moreover, for some particular functions $F$, the author constructs examples of $F$-Willmore surface and gives a characterization of $F$-Willmore tori and Veronese surface by use of Simons' type integral inequality.