Sard properties for polynomial maps in infinite dimension
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909998232109056 |
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| author | Lerario, Antonio Rizzi, Luca Tiberio, Daniele |
| author_facet | Lerario, Antonio Rizzi, Luca Tiberio, Daniele |
| contents | Sard's theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true -- even under the assumption that the map is ``polynomial'' -- and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard's theorem in this setting. Our motivation comes from sub-Riemannian geometry and, as an application of our results, we prove the sub-Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece-wise real-analytic controls with large enough radius of convergence, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02296 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sard properties for polynomial maps in infinite dimension Lerario, Antonio Rizzi, Luca Tiberio, Daniele Differential Geometry Functional Analysis Metric Geometry Optimization and Control 53C17, 14P10 Sard's theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true -- even under the assumption that the map is ``polynomial'' -- and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard's theorem in this setting. Our motivation comes from sub-Riemannian geometry and, as an application of our results, we prove the sub-Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece-wise real-analytic controls with large enough radius of convergence, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls. |
| title | Sard properties for polynomial maps in infinite dimension |
| topic | Differential Geometry Functional Analysis Metric Geometry Optimization and Control 53C17, 14P10 |
| url | https://arxiv.org/abs/2407.02296 |