Lipschitz Bounds and Uniform Convergence for Sequences of Bounded Rough Riemannian Metrics
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915838455447552 |
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| author | Allen, Brian Falcao, Bernardo Pacheco, Harry Sanchez, Bryan |
| author_facet | Allen, Brian Falcao, Bernardo Pacheco, Harry Sanchez, Bryan |
| contents | Here we study what we call bounded rough Riemannian metrics $(M,g)$, which are positive definite, symmetric tensors on each tangent space, $T_pM$, which are bounded and measurable as functions in coordinates. This is enough structure to study the length space given by taking the infimum of the length of all piecewise smooth curves connecting points $p,q \in M$. The goal is to find the weakest conditions one can place on $g$ which can guarantee Lipschitz or uniform bounds from above and below. For each condition, an example is given showing that the condition cannot be weakened any further which also explores the geometric intuition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05669 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lipschitz Bounds and Uniform Convergence for Sequences of Bounded Rough Riemannian Metrics Allen, Brian Falcao, Bernardo Pacheco, Harry Sanchez, Bryan Differential Geometry Metric Geometry Here we study what we call bounded rough Riemannian metrics $(M,g)$, which are positive definite, symmetric tensors on each tangent space, $T_pM$, which are bounded and measurable as functions in coordinates. This is enough structure to study the length space given by taking the infimum of the length of all piecewise smooth curves connecting points $p,q \in M$. The goal is to find the weakest conditions one can place on $g$ which can guarantee Lipschitz or uniform bounds from above and below. For each condition, an example is given showing that the condition cannot be weakened any further which also explores the geometric intuition. |
| title | Lipschitz Bounds and Uniform Convergence for Sequences of Bounded Rough Riemannian Metrics |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2603.05669 |