Let $$(M_i, g_i)_{i \in \mathbb {N}}$$ ( M i , g i ) i ∈ N be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold ( B , h ) in the Gromov–Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator $$\mathcal {D}^B$$ D B on B . We give an explicit description of $$\mathcal {D}^B$$ D B and characterize the special case where $$\mathcal {D}^B$$ D B equals the Dirac operator on B .

中文翻译：

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坍缩到光滑极限空间下的狄拉克算子

令 $$(M_i, g_i)_{i \in \mathbb {N}}$$ ( M i , gi ) i ∈ N 是一系列具有均匀有界曲率和直径的自旋流形，收敛到低维黎曼Gromov-Hausdorff 拓扑中的流形 ( B , h )。然后，恰好 Dirac 算子的频谱收敛到 B 上某个一阶椭圆微分算子 $$\mathcal {D}^B$$ DB 的频谱。我们给出了 $$\mathcal {D}^B$$ DB 的明确描述，并刻画了 $$\mathcal {D}^B$$ DB 等于 B 上的狄拉克算子的特殊情况。