关于希尔伯特流形上完备性的共形灵活性与极小化测地线
On Conformal Flexibility of Completeness and Minimizing Geodesics on Hilbert Manifolds
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中文总结 AI 辅助
研究希尔伯特流形上完备性相关问题，通过共形变形得到度量和测地线完备且有长度极小化测地线的代表，同时建立了度量完备性的局部灵活性现象，即能在保持测地线完备性的同时破坏度量完备性。
中文摘要 AI 辅助
本文是一个更广泛计划的一部分，该计划研究有限维黎曼几何的哪些特征在无限维中仍然存在。霍普夫 - 里诺定理即使在希尔伯特流形上也不成立：度量完备性和测地线完备性不一定一致，且这两个性质都不能保证给定端点之间存在长度极小化测地线。尽管如此，第一个主要结果表明，光滑可分希尔伯特流形上每个光滑强黎曼度量的共形类都包含一个光滑强代表，它是度量和测地线完备的，且同一连通分量中的任意两点由长度极小化测地线连接。相比之下，第二个主要结果建立了一个本质上是无限维的度量完备性的局部灵活性现象。给定任何规定的希尔伯特范数球，一个度量完备的强度量允许一个共形变形，该变形在球外等于一个，保持测地线完备性，并破坏度量完备性。
英文摘要
This article is part of a broader programme investigating which features of finite-dimensional Riemannian geometry persist in infinite dimensions. The Hopf--Rinow theorem fails even for Hilbert manifolds: metric and geodesic completeness need not agree, and neither property guarantees a length-minimizing geodesic between prescribed endpoints. Despite this failure, our first main result shows that the conformal class of every smooth strong Riemannian metric on a smooth separable Hilbert manifold contains a smooth strong representative that is metrically and geodesically complete and such that every two points in the same connected component are joined by a length-minimizing geodesic. By contrast, our second main result establishes a local flexibility phenomenon for metric completeness that is inherently infinite-dimensional. Given any prescribed Hilbert-norm ball, a metrically complete strong metric admits a conformal deformation which is equal to one outside that ball, preserves geodesic completeness, and destroys metric completeness.
