Solitons to Mean Curvature Flow in the hyperbolic 3-space
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| author | de Lima, R. F. Ramos, A. K. Santos, J. P. dos |
| author_facet | de Lima, R. F. Ramos, A. K. Santos, J. P. dos |
| contents | We consider {translators} (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic $3$-space $\mathbb H^3$, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in $\mathbb H^3$, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in $\mathbb H^3$ and apply it to prove that properly immersed translators to MCF in $\mathbb H^3$ are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the $x_3$-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in $\mathbb H^3$ which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in $\mathbb H^3$ and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in $\mathbb R^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14136 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solitons to Mean Curvature Flow in the hyperbolic 3-space de Lima, R. F. Ramos, A. K. Santos, J. P. dos Differential Geometry 53E10 We consider {translators} (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic $3$-space $\mathbb H^3$, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in $\mathbb H^3$, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in $\mathbb H^3$ and apply it to prove that properly immersed translators to MCF in $\mathbb H^3$ are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the $x_3$-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in $\mathbb H^3$ which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in $\mathbb H^3$ and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in $\mathbb R^3$. |
| title | Solitons to Mean Curvature Flow in the hyperbolic 3-space |
| topic | Differential Geometry 53E10 |
| url | https://arxiv.org/abs/2307.14136 |