arXiv 0911.37672009-11math.DG

曲面上的有限指标算子

Finite index operators on surfaces

Jose M. Espinar

备注：: Section 4 has changed completely

摘要：

我们考虑微分算子$L$作用于形式为$$L = Δ+ V -a K ,$$的黎曼曲面$Σ$上的函数，其中$Δ$是$Σ$的拉普拉斯算子，$K$是高斯曲率，$a$是正常数，$V \in C^{\infty}(Σ)$。这样的算子$L$作为$Σ$的稳定性算子出现在具有常平均曲率的黎曼三流形中（对于$V$和$a$的特定选择）。我们假设$L$对$Σ$上支持的函数是非正作用的。如果势$V:= c + P $（其中$c$为非负常数）验证可积性条件（即$P \in L^1(Σ)$和$P$为非正）或关于点$p_0 \in Σ$的衰减条件（即$|P(q)|\leq M/d(p_0,q)$（其中$d$为$Σ$中的距离函数）），则我们控制$Σ$的拓扑和共形类型。此外，我们建立了一个{ \it距离引理}。我们应用这样的结果，以完成定向稳定的$H-$表面沉浸在一个杀淹没。

英文摘要：

We consider differential operators $L$ acting on functions on a Riemannian surface, $Σ$, of the form $$L = Δ+ V -a K ,$$where $Δ$ is the Laplacian of $Σ$, $K$ is the Gaussian curvature, $a$ is a positive constant and $V \in C^{\infty}(Σ)$. Such operators $L$ arise as the stability operator of $Σ$ immersed in a Riemannian three-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume $L$ is nonpositive acting on functions compactly supported on $Σ$. If the potential, $V:= c + P $ with $c$ a nonnegative constant, verifies either an integrability condition, i.e. $P \in L^1(Σ)$ and $P$ is non positive, or a decay condition with respect to a point $p_0 \in Σ$, i.e. $|P(q)|\leq M/d(p_0,q)$ (where $d$ is the distance function in $Σ$), we control the topology and conformal type of $Σ$. Moreover, we establish a {\it Distance Lemma}. We apply such results to complete oriented stable $H-$surfaces immersed in a Killing submersion.

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