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arXiv 0901.1246math.DGmath.SP
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关于$\bar{X}$ -有界极小子流形的谱

On the spectrum of $\bar{X}$-bounded minimal submanifolds

Isabel M. C. Salavessa

备注：
v.2 13 pages. We improve some theorems, correct some misprints and add a new theorem (theorem 5) in section 2 that shows complete bounded minimal immersed submanifolds have unbounded second fundamental form, for suitable curvature conditions of the ambient space
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摘要：

证明了在第第二基本型的$(\bar{X}^{\top}, \bar{X}^{\bot})$ -分支无穷远点的有界性条件下，具有强凸向量场$\bar{X}$的黎曼流形上的完备极小$\bar{X}$ -有界和$\bar{X}$ -适当浸入子流形的本质谱为零.同样的结论也适用于任何完全极小$h$ -有界和$h$ -适当浸入子流形，它位于支持非负严格凸函数$h$的黎曼流形$\oM$的开集。这扩展了最近的结果贝萨，豪尔赫和蒙特内哥罗的马丁-莫拉莱斯极小曲面的频谱。我们的证明使用作为主要工具的扩展Barta的定理在\cite{BM}

英文摘要：

We prove, under a certain boundedness condition at infinity on the $(\bar{X}^{\top}, \bar{X}^{\bot})$-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal $\bar{X}$-bounded and $\bar{X}$-properly immersed submanifold on a Riemannian manifold endowed with a strongly convex vector field $\bar{X}$. The same conclusion also holds for any complete minimal $h$-bounded and $h$-properly immersed submanifold that lies in a open set of a Riemannian manifold $\oM$ supporting a nonnegative strictly convex function $h$. This extends a recent result of Bessa, Jorge and Montenegro on the spectrum of Martin-Morales minimal surfaces. Our proof uses as main tool an extension of Barta's theorem given in \cite{BM}

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