<p>In this paper, we study the compactness properties of solutions to the hyperbolic geometric flow. We establish a Cheeger–Gromov type compactness theorem for sequences of complete pointed solutions with uniformly bounded geometry and a uniform lower bound on the injectivity radius at the initial time. Under these assumptions, we show that any such sequence admits a subsequence which converges smoothly, in the pointed sense, to a complete solution of the hyperbolic geometric flow on the same time interval. Our approach relies on uniform curvature bounds, uniform metric equivalence, and Cheeger–Gromov compactness arguments. The results provide a natural framework for the analysis of geometric limits and singularity formation in the hyperbolic geometric flow.</p>

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A Compactness Theorem for the Hyperbolic Geometric Flow

  • Zohreh Aral

摘要

In this paper, we study the compactness properties of solutions to the hyperbolic geometric flow. We establish a Cheeger–Gromov type compactness theorem for sequences of complete pointed solutions with uniformly bounded geometry and a uniform lower bound on the injectivity radius at the initial time. Under these assumptions, we show that any such sequence admits a subsequence which converges smoothly, in the pointed sense, to a complete solution of the hyperbolic geometric flow on the same time interval. Our approach relies on uniform curvature bounds, uniform metric equivalence, and Cheeger–Gromov compactness arguments. The results provide a natural framework for the analysis of geometric limits and singularity formation in the hyperbolic geometric flow.