On the Existence of Closed Biconservative Surfaces in Space Forms
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866929666166620160 |
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| author | Montaldo, Stefano Pampano, Alvaro |
| author_facet | Montaldo, Stefano Pampano, Alvaro |
| contents | Biconservative surfaces of Riemannian 3-space forms $N^3(ρ)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3κ_1+κ_2=0$ between their principal curvatures $κ_1$ and $κ_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round 3-sphere, $S^3(ρ)$. However, none of these closed surfaces is embedded in $S^3(ρ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_03233 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the Existence of Closed Biconservative Surfaces in Space Forms Montaldo, Stefano Pampano, Alvaro Differential Geometry Biconservative surfaces of Riemannian 3-space forms $N^3(ρ)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3κ_1+κ_2=0$ between their principal curvatures $κ_1$ and $κ_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round 3-sphere, $S^3(ρ)$. However, none of these closed surfaces is embedded in $S^3(ρ)$. |
| title | On the Existence of Closed Biconservative Surfaces in Space Forms |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2009.03233 |