The $L^2$-norm of the Euler class for Foliations on closed irreducible Riemannian 3-Manifolds
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913899452825600 |
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| author | Bolotov, Dmitry V. |
| author_facet | Bolotov, Dmitry V. |
| contents | An upper bound for the $L^2$- norm of the Euler class $e(\cal F)$ of an arbitrary transversally orientable foliation $\cal F$ of codimension one, defined on a three-dimensional closed irreducible orientable Riemannian 3-manifold $M^3$ is given in terms of constants bounding the volume, the radius of injectivity, the sectional curvature of $M^3$ and the modulus of mean curvature of the leaves. As a consequence we get that only finitely many cohomolo\-gical classes of the group $H^2(M^3)$ that can be realized by the Euler class $e(\cal F)$ of a two-dimensional transversely oriented foliation $\cal F $ whose leaves have the modulus of mean curvature which is bounded above by the fixed constant $H_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_06807 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The $L^2$-norm of the Euler class for Foliations on closed irreducible Riemannian 3-Manifolds Bolotov, Dmitry V. Geometric Topology An upper bound for the $L^2$- norm of the Euler class $e(\cal F)$ of an arbitrary transversally orientable foliation $\cal F$ of codimension one, defined on a three-dimensional closed irreducible orientable Riemannian 3-manifold $M^3$ is given in terms of constants bounding the volume, the radius of injectivity, the sectional curvature of $M^3$ and the modulus of mean curvature of the leaves. As a consequence we get that only finitely many cohomolo\-gical classes of the group $H^2(M^3)$ that can be realized by the Euler class $e(\cal F)$ of a two-dimensional transversely oriented foliation $\cal F $ whose leaves have the modulus of mean curvature which is bounded above by the fixed constant $H_0$. |
| title | The $L^2$-norm of the Euler class for Foliations on closed irreducible Riemannian 3-Manifolds |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2212.06807 |