论文标题
加权$ \ infty $ -willmore Spheres
Weighted $\infty$-Willmore Spheres
论文作者
论文摘要
在两键$σ$上，我们认为最小化合适的沉浸式$ f \，\ colonσ\ rightarrow \ rightarrow \ rightarrow \ rightarrow \ rymbb {r}^3 $平均曲率$ h $的加权$ l^\ infty $ norm n $ h $，由处方的环境函数给定的权衡，由处方的环境函数给定的表面积$，固定的表面积约束。我们表明，在一个低能的假设下，阻止拓扑问题引起的解决方案，以及更一般的一组``pseudo-minimiser''表面必须满足以$ p \ rightArrow \ rightarrow \ rightarrow \ right \ for the euler-lagrange equation for euler-lagrange equarne for the Euler-lagrange equarne the Euler-lagrange套件的二阶系统，以解决euler-lagrange interage $ l^P $ l^P $ l^p $ l^p $ l^p pde。该系统提供了有关表面几何行为的一些信息，特别意味着它们的平均曲率最多承担了三个值：$ h \ in \ {\ pm \ pm \ vert \ vert \ vert \ vertξh\ vert \ vert \ vert \ vert_ {l^\ infty} \} $从pde System和$ h nonod inod-nod inod and nonod设置（如果是$ h = 0 $ h = 0 $）。
On the two-sphere $Σ$, we consider the problem of minimising among suitable immersions $f \,\colon Σ\rightarrow \mathbb{R}^3$ the weighted $L^\infty$ norm of the mean curvature $H$, with weighting given by a prescribed ambient function $ξ$, subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of ``pseudo-minimiser'' surfaces must satisfy a second-order PDE system obtained as the limit as $p \rightarrow \infty$ of the Euler-Lagrange equations for the approximating $L^p$ problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: $H \in \{ \pm \vert \vert ξH \vert \vert_{L^\infty} \}$ away from the nodal set of the PDE system, and $H = 0$ on the nodal set (if it is non-empty).
