Analytic surgery and the eta invariant
Analytic surgery and the eta invariant
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分析手术和 eta 不变量
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
R. Melrose
中科院分区：
文献类型：
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作者：
R. Mazzeo;R. Melrose
设（M，h）是奇维紧致自旋流形，其中H是具有二次定义函数x2 ∈C∞（M）的嵌入超曲面。设是与度量jhen相关的狄拉克算子，其中∈>0是一个参数。极限度量g 0是紧流形上的一个精确b-度量
上划线M $$
通过沿H切割M并紧化为一个有边界的流形而得到，即它给出M/H个有横截面的渐近圆柱形端点
$$部分上划线M $$
H的双重覆盖。在此双重覆盖上的诱导Dirac算子可逆的假设下，我们证明了其中是[Me 1]中引入的eta不变量的“B”版本，r1（∈）和r2（∈）是光滑的，在∈=0处为零，是局部几何数据的积分，并且其中
$$ ilde eta（in）$$
是的小特征值的有限维eta不变量或签名。如果是可逆的，
$$ ilde eta（in）equiv 0$$
即使不可逆，这也在n/n中成立。事实上，在Hermitian Clifford模的广义Dirac算子的背景下，讨论会更自然地进行，这些结果通过分析一致远离谱且接近于零的预解式族得到了证明。这导致了对小特征值行为的精确描述。相应的“热演算”也被构造。它包含，因此相当精确地描述了，一致为∈→0的热核。这种微积分与McDonald（[Mc]）的外科伪微分算子微积分有关，但又有所不同。
AbstractLet (M, h) be an odd-dimensional compact spin manifold in whichH is an embedded hypersurface with quadratic defining functionx2∈C∞ (M). Let be the Dirac operator associated to the metric jhen where ∈>0 is a parameter. The limiting metric,g0, is an exact b-metric on the compact manifold with boundary
$$overline M $$
obtained by cuttingM alongH and compactifying as a manifold with boundary, i.e. it givesM/H asymptotically cylindrical ends with cross-section
$$partial overline M $$
, a double cover ofH. Under the assumption that the induced Dirac operator on this double cover is invertible we show that where is the ‘b’ version of the eta invariant introduced in [Me1],r1(∈) andr2(∈) are smooth, vanish at ∈=0 and are integrals of local geometric data, and where
$$ ilde eta ( in )$$
is the finite dimensional eta invariant, or signature, for the small eigenvalues of. If is invertible then
$$ ilde eta ( in ) equiv 0$$
and Even if is not invertible this holds in ℝ/ℤ. In fact the discussion takes place more naturally in the context of the generalized Dirac operators associated to Hermitian Clifford modules.These results are proved by analyzing the resolvent family of, uniformly away from the spectrum and near zero. This leads to a precise description of the behaviour of the small eigenvalues. The corresponding ‘heat calculus’ is also constructed. It contains, and hence describes rather precisely, the heat kernel for uniformly as ∈→0. This calculus is related to, but different from, the surgery pseudodifferential operator calculus of McDonald ([Mc]).