Collapsing and Dirac-Type Operators
Collapsing and Dirac-Type Operators
复制标题
折叠算子和狄拉克算子
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
J. Lott
中科院分区：
文献类型：
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作者：
J. Lott
本文研究了一个几何Dirac型算子在有界直径和有界截面曲率的坍缩下的谱极限。在光滑极限空间B的情形下，我们证明了谱的极限是由B上某个一阶微分算子的谱给出的，它可以用超联络构造。在一般极限空间X中，我们用G-流形X/上的横截椭圆算子来表示极限算子，其中X = X//G.作为应用，我们给出了Dirac型算子平方的第k个特征值在直径和截面曲率方面不具有一致上界的流形的一个特征.我们还给出了有限体积流形上具有负截面曲率的Dirac型算子的本质谱的一个公式。
We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-manifold X/ with X = X//G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.