Minimizing the Gauss map area of surfaces in $\mathbb{S}^3$
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| Format: | Preprint |
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2025
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| _version_ | 1866916781173506048 |
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| author | Orriols, Gerard Rivière, Tristan |
| author_facet | Orriols, Gerard Rivière, Tristan |
| contents | We establish the lower bound of $4π(1+g)$ for the area of the Gauss map of any immersion of a closed oriented surface of genus $g$ into $\mathbb{S}^3$, taking values in the Grassmannian of $2$-planes in $\mathbb{R}^4$. This lower bound is proved to be optimal for any genus $g \in \mathbb{N}$ but attained only when $g = 0$. For $g \neq 0$ we describe the behavior of any minimizing sequence of embeddings: we prove that, modulo extraction of a subsequence, the surfaces converge in the Hausdorff distance to a round sphere $S$, and the integral cycles carried by the Gauss maps split into $g+1$ spheres, each of area $4π$; one of them corresponds to the cycle carried by the Gauss map of $S$, while the other $g$ arise from the concentration of negative Gauss curvature at $g$ points of $S$.
The results of this paper are used by the second author to define a nontrivial homological 4-dimensional min-max scheme for the area of Gauss maps of immersions into $\mathbb{S}^3$ in relation to the Willmore conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimizing the Gauss map area of surfaces in $\mathbb{S}^3$ Orriols, Gerard Rivière, Tristan Differential Geometry We establish the lower bound of $4π(1+g)$ for the area of the Gauss map of any immersion of a closed oriented surface of genus $g$ into $\mathbb{S}^3$, taking values in the Grassmannian of $2$-planes in $\mathbb{R}^4$. This lower bound is proved to be optimal for any genus $g \in \mathbb{N}$ but attained only when $g = 0$. For $g \neq 0$ we describe the behavior of any minimizing sequence of embeddings: we prove that, modulo extraction of a subsequence, the surfaces converge in the Hausdorff distance to a round sphere $S$, and the integral cycles carried by the Gauss maps split into $g+1$ spheres, each of area $4π$; one of them corresponds to the cycle carried by the Gauss map of $S$, while the other $g$ arise from the concentration of negative Gauss curvature at $g$ points of $S$. The results of this paper are used by the second author to define a nontrivial homological 4-dimensional min-max scheme for the area of Gauss maps of immersions into $\mathbb{S}^3$ in relation to the Willmore conjecture. |
| title | Minimizing the Gauss map area of surfaces in $\mathbb{S}^3$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.05141 |