论文标题
规定的平均曲率最小 - 最大理论在某些非紧密歧管中
Prescribed Mean Curvature Min-Max Theory in Some Non-Compact Manifolds
论文作者
论文摘要
本文开发了一种用于在某些非紧凑型歧管中应用单参数规定的平均曲率最小理论的技术。我们提供两个主要应用程序。首先，修复一个尺寸$ 3 \ le n+1 \ le 7 $，并考虑一个平滑函数$ h \ colon \ colon \ mathbb {r}^{n+1} \ to \ mathbb {r} $，这是渐进的近乎无限的正常数。我们表明，在$ h $上的某些其他假设下，存在$ \ mathbb {r}^{n+1} $的封闭式hypersurface $σ$，并带有$ h $规定的平均曲率。其次，让$（m^3，g）$为渐近平面的3个manifold，并修复常数$ c> 0 $。我们表明，在$ m $上的额外假设下，可以在$ m $中找到固定平均曲率$ c $的封闭表面$σ$。
This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension $3\le n+1 \le 7$ and consider a smooth function $h\colon \mathbb{R}^{n+1}\to \mathbb{R}$ which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on $h$, there exists a closed hypersurface $Σ$ in $\mathbb{R}^{n+1}$ with mean curvature prescribed by $h$. Second, let $(M^3,g)$ be an asymptotically flat 3-manifold and fix a constant $c > 0$. We show that, under an additional assumption on $M$, it is possible to find a closed surface $Σ$ of constant mean curvature $c$ in $M$.
