arXiv math/06125262006-12-18math.APmath.DG

Willmore表面的分析方面

Analysis aspects of Willmore surfaces

Riviere Tristan

摘要：

在${\R}^m$中，我们发现了一个新的Willmore泛函的Euler-Lagrange方程。新的Willmore方程形式为发散形式，且非线性项由Jacobian构成。除此之外，如果$\bH$表示曲面的平均曲率向量，则这种新形式写作${\mathcal L}\bH=0$，其中${\mathcal L}$是定义良好的局部可逆自伴算子。这3个事实在Willmore曲面的分析中有许多结果。第一个结果是，长期存在的开放问题，以赋予意义的Willmore欧拉-拉格朗日方程的浸入只有$L^2$有界的第二基本形式，现在解决。然后，我们建立了弱$W^{2,p}-$ Willmore曲面的正则性为任何$p$的高斯映射是连续的：$p>2$。这是基于弱Willmore曲面的$ε-$正则性结果的证明。然后，我们建立了能量小于$8π-δ$的Willmore曲面的弱紧性结果，对于每个$δ>0$ .这个定理是基于一个点可移性的结果，我们证明了在${\R}^m$的Wilmore曲面。这个结果推广到任意余维，E.Kuwert和R.Schaetzle在${\R}^3$中建立了曲面的结果。最后，我们从这一点的可移动性的结果推导出强紧性，模莫比乌斯群作用，Willmore环面低于能级$8π-δ$在3和4维。第三个维度的问题在以前的工作中已经解决了。

英文摘要：

We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in ${\R}^m$. This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of jacobians. Additionally to that, if $\bH$ denotes the mean curvature vector of the surface, this new form writes ${\mathcal L}\bH=0$ where ${\mathcal L}$ is a well defined locally invertible self-adjoint operator. These 3 facts have numerous consequences in the analysis of Willmore surfaces. One first consequence is that the long standing open problem to give a meaning to the Willmore Euler-Lagrange equation for immersions having only $L^2$ bounded second fundamental form is now solved. We then establish the regularity of weak $W^{2,p}-$Willmore surfaces for any $p$ for which the Gauss map is continuous : $p>2$. This is based on the proof of an $ε-$regularity result for weak Willmore surfaces. We establish then a weak compactness result for Willmore surfaces of energy less than $8π-δ$ for every $δ>0$. This theorem is based on a point removability result we prove for Wilmore surfaces in ${\R}^m$. This result extends to arbitrary codimension a result that E.Kuwert and R.Schaetzle established for surfaces in ${\R}^3$. Finally, we deduce from this point removability result the strong compactness, modulo the Möbius group action, of Willmore tori below the energy level $8π-δ$ in dimensions 3 and 4. The dimension 3 case was already solved in a previous work.

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