论文标题
最小边界最小表面的粘度方法
The viscosity method for min-max free boundary minimal surfaces
论文作者
论文摘要
我们将Rivière引入的粘度方法适应自由边界案例。也就是说，给定一个面向紧凑的表面$σ$，可能具有边界，封闭的环境riemannian歧管$（\ Mathcal {m}^m，g）$和一个封闭的嵌入式submanifold $ \ mathcal {n}^n}^n \ subset \ subset \ subset \ nathcal \ mathcal {m mathcal {m mathcal {m mathcal pristiations $ criptions $ criptions $ criptions $ to and uncompitiation $ saps $ saps $ saps $ saps ultion as ass usps $（几乎是; \ begin {align*}＆e_σ（φ）：= \ operatorName {aind}（φ）+σ\ perperatorname {length}（φ| _ {\partialς}）+σ^4 \int_σ| {\int_σ| {\ mathrm {i \！i}}}^φ|^4 \，\ operatorname {vol}_φ\ end end {align*}上的浸入$φ：σ\ to \ nathcal {m nathcal {m} $带约束$ $ $ $ $（\partialς）和合适的熵条件。 结果，给定任何集合的$ \ nathcal {f} $的紧凑型浸入空间的子集$（σ，\partialς）\ to（\ nathcal {m}，\ nathcal {n}）$，假设$ \ \ \ \ \ \ \ mathcal {f} $ stable spable spable smape of Isign Inign aLign a lign*sim nirnign* ＆β：= \ inf_ {a \ in \ mathcal {f}} \ max_ {φ\ in a} \ operatorname {aind}（φ）\ end \ end {align {align*}的总和是有限的许多分支最小的沉浸液的总和$φ_ {（i）}：σ_ {（i）} \ to \ nathcal {m} $，带有$ \partial_νφ_ {（i）} \ perp t \ perp t \ perp t \ nathcal {n} $沿$ \partialς_ {（i）的$ i（$ i）$ n $ n $ n $ n $ n $ n $ can但不能具有更复杂的拓扑结构。 我们采用一个观点，该观点广泛利用$e_σ$的差异不变性，并在此过程中简化了原始工作中的几个参数。某些零件概括为封闭的高维域，为此，我们在极限内获得了可更整流的固定varifold。
We adapt the viscosity method introduced by Rivière to the free boundary case. Namely, given a compact oriented surface $Σ$, possibly with boundary, a closed ambient Riemannian manifold $(\mathcal{M}^m,g)$ and a closed embedded submanifold $\mathcal{N}^n\subset\mathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Φ$ for the functional \begin{align*} &E_σ(Φ):=\operatorname{area}(Φ)+σ\operatorname{length}(Φ|_{\partialΣ})+σ^4\int_Σ|{\mathrm {I\!I}}^Φ|^4\,\operatorname{vol}_Φ\end{align*} on immersions $Φ:Σ\to\mathcal{M}$ with the constraint $Φ(\partialΣ)\subseteq\mathcal{N}$, as $σ\to 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $\mathcal{F}$ of compact subsets of the space of smooth immersions $(Σ,\partialΣ)\to(\mathcal{M},\mathcal{N})$, assuming $\mathcal{F}$ to be stable under isotopies of this space we show that the min-max value \begin{align*} &β:=\inf_{A\in\mathcal{F}}\max_{Φ\in A}\operatorname{area}(Φ) \end{align*} is the sum of the areas of finitely many branched minimal immersions $Φ_{(i)}:Σ_{(i)}\to\mathcal{M}$ with $\partial_νΦ_{(i)}\perp T\mathcal{N}$ along $\partialΣ_{(i)}$, whose (connected) domains $Σ_{(i)}$ can be different from $Σ$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_σ$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.
