论文标题
具有纤维边界指标的流形的光谱几何形状I：低能解析
Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent
论文作者
论文摘要
我们研究了装有纤维边界指标的歧管上的霍奇拉普拉斯人的低能解析。我们确定分解为纤维边界（aka $ ϕ $ - ）的确切渐近行为，当分解参数趋于零时，伪差算子。这概括了Guillarmou和Sher的先前工作，后者考虑了渐近的圆锥指标，这与纤维是点时的特殊情况相对应。在非平凡纤维的情况下，新功能是，在纤维中谐波和其正交补体上的形式的子空间上，分辨率具有不同的渐近行为。为了解决这个问题，我们引入了一个适当的“分裂”伪分化的微积分，并在Grieser和Hunsicker的工作基础上进行并扩展了工作。我们的工作为讨论$ ϕ $ manifolds的频谱不变性的讨论奠定了基础。
We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $ϕ$-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on $ϕ$-manifolds.
