Entropy Stability for products of negatively curved symmetric spaces
| Autor: | Jang, Hyun Chul |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $(M,g_0)$ be a closed oriented $n$-manifold that is locally isometric to a product $(X^{n_1}_1,g_1)\times\cdots (X_k^{n_k},g_k)$, where each $n_i\ge 3,$ and each factor $(X_i^{n_i},g_i)$ is a negatively curved symmetric space. We study the stability of minimal entropy rigidity for such manifolds. Specifically, we consider whether an entropy-minimizing sequence $(M,g_i)$ converges to the model space in the measured Gromov-Hausdorff sense after removing negligible subsets. Previously, Song [Son23] established this type of stability for negatively curved symmetric spaces, where both the $n$-volume of the removed subsets and the $(n-1)$-volume of their boundaries converge to zero. We construct a counterexample demonstrating that this stronger stability notion does not generally hold in the product case; in particular, the condition that the $(n-1)$-volume of the boundary of removed subsets converges to zero cannot be imposed. Nonetheless, we prove that an entropy-minimizing sequence $(M,g_i)$ converges to the model space after removing subsets whose $n$-volume converges to zero in the measured Gromov-Hausdorff topology. This result provides a weaker form of stability compared to the negatively symmetric case. A key ingredient in establishing this stability is our proof of the intrinsic uniqueness of the spherical Plateau solution for products of negatively curved symmetric spaces, which is of independent interest. Comment: 26 pages; Comments welcome! |
| Databáze: | arXiv |
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