分数法曲率量上界下共形浸入的极限
Limits of conformal immersions under a bound on a fractional normal curvature quantity
摘要：
我们考虑了从球$B \subset \mathbb{R}^2$到$\mathbb{R}^3$的弱收敛$W^{1,2}$ -映射$Φ_k$的极限，它们是共形浸入.假设一个法曲率项是小的，即如果正常的地图$u$，我们有一些$s \in (\frac{1}{2},1)$ $$ \int_{B} \int_{B} \left | \frac{u_k(x) \wedge u_k(y)}{|x-y|^{s}} \right|^{\frac{2}{s}}\, \frac{dx\, dy}{|x-y|^{2}} < \varepsilon $$，然后我们表明，我们可以通过到极限，并获得一个几乎处处浸入$Φ$或$Φ$崩溃，是常数。这与T. Toro和S. Müller and V. Sverak，and F. Hélein，他在更强的假设下得到了类似的陈述，即第二基本型是有界的（但也是更强的结果：局部双Lipschitz参数化）。分数法曲率的假设是模糊的曲率能量，如缩放不变的限制，表面的切点能量考虑由Strzelecki，冯德莫塞尔等人。我们希望，最终在这项工作中的分析可以用来定义弱浸入与这些种的能量界限。
英文摘要：
We consider limits of weakly converging $W^{1,2}$-maps $Φ_k$ from a ball $B \subset \mathbb{R}^2$ into $\mathbb{R}^3$ which are conformal immersions. Under the assumption that a normal curvature term is small, namely if for the normal map $u$ we have for some $s \in (\frac{1}{2},1)$ $$ \int_{B} \int_{B} \left | \frac{u_k(x) \wedge u_k(y)}{|x-y|^{s}} \right|^{\frac{2}{s}}\, \frac{dx\, dy}{|x-y|^{2}} < \varepsilon $$ then we show that we can either pass to the limit and obtain an almost everywhere immersion $Φ$ or $Φ$ collapses and is constant. This is in the spirit of the results by T. Toro, and S. Müller and V. Sverak, and F. Hélein, who obtained similar statements under the stronger assumptions that the second fundamental form is bounded (but also stronger result: a locally bi-Lipschitz parametrization). The fractional normal curvature assumption is vaguely reminiscent of curvature energies such as the scaling-invariant limits of tangent-point energies for surfaces as considered by Strzelecki, von der Mosel et al. and we hope that eventually the analysis in this work can be used to define weak immersions with these kind of energy bounds.
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