论文标题
当n = 3时，Hélein关于保形因子有限性的猜想的证明
A Proof of Hélein's Conjecture on Boundedness of Conformal Factors when n=3
论文作者
论文摘要
为了将单元光盘的平滑映射到定向的格拉曼尼亚歧管$ \ mathbb g_ {n，2} $，Hélein（2002）猜想，猜测库仑框架的全球存在与有界的保形因子的整体存在，提供了$ | \ boldsymbol a |^2 $ s的squend felliment fellymalter felly $ use的整体构图。此后，已经表明，$ | \ boldsymbol a |^2 $的积分上的最佳界限保证了此结果：$γ_3=8π$和$γ_n=4π$ for $ n \ geq 4 $。对于等温浸入，该假设等同于说主要曲率平方之和的积分不小于$γ_n$。这里的目的是证明当$ n = 3 $时，相同的结论在较弱的假设下得出。特别是，当$ | \ boldsymbol a | $是方形融合，并且$ | k | $，$ k $，高斯曲率的积分不足时，它具有等温浸入率时。由于$ 2 | k | \ leq | \ boldsymbol a |^2 $这意味着等温浸入的已知结果，但是当$ | \ boldsymbol a |^2 $很大时，$ | k | $可能很小。在较弱的假设下的结果是由Enneper的表面和立体投影显示的。然后将这种方法扩展到纯粹的分析，以调查第二个基本形式的长度是正方形集成的情况。
For smooth mappings of the unit disc into the oriented Grassmannian manifold $\mathbb G_{n,2}$, Hélein (2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of $|\boldsymbol A|^2$, the squared-length of the second fundamental form, is less than $γ_n=8π$. It has since been shown that the optimal bounds on the integral of $|\boldsymbol A|^2$ that guarantee this result are: $γ_3 = 8π$ and $γ_n = 4π$ for $n \geq 4$. For isothermal immersions, this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than $γ_n$. The goal here is to prove that when $n=3$ the same conclusion holds under weaker hypotheses. In particular, it holds for isothermal immersions when $|\boldsymbol A|$ is square-integrable and the integral of $|K|$, $K$ the Gauss curvature, is less than $4π$. Since $2|K| \leq |\boldsymbol A|^2$ this implies the known result for isothermal immersions, but $|K|$ may be small when $|\boldsymbol A|^2$ is large. That the result under the weaker hypothesis is sharp is shown by Enneper's surface and stereographic projections. The method, which is purely analytic, is then extended to investigate the case when the length of the second fundamental form is square-integrable.
