Mahler型定理与次黎曼紧Heisenberg流形的非坍缩极限
Mahler type theorem and non-collapsed limits of sub-Riemannian compact Heisenberg manifolds
AI总结：
本文证明带次黎曼度量的紧Heisenberg流形在直径与Popp测度界下具有非坍缩GH子列极限，且极限仍为同维紧Heisenberg流形。
摘要：
本文研究了具有次黎曼度量的一列紧Heisenberg流形的非坍缩Gromov–Hausdorff极限。在严格次黎曼情形下，我们证明：若一列流形具有直径上界和Popp测度下界，则它存在在Gromov–Hausdorff拓扑中收敛的子列，并且其极限与一个相同维数的紧Heisenberg流形等距。对于黎曼情形，在附加Ricci曲率下界的假设下，也证明了相同结论。
英文摘要：
In this paper, we study a non-collapsed Gromov--Hausdorff limit of a sequence of compact Heisenberg manifolds with sub-Riemannian metrics. In the case of strictly sub-Riemannian case, we show that if a sequence has an upper bound of the diameter and a lower bound of Popp's measure, then it has a convergent subsequence in the Gromov--Hausdorff topology, and the limit is isometric to a compact Heisenberg manifold of the same dimension. The same conclusion is also shown for Riemannian case with the additional assumption on Ricci curvature lower bounds.
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