Minimizing the Gauss map area of surfaces in $\mathbb{S}^3$

Abstract

We establish the lower bound of $4\pi(1+g)$ 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 $4\pi$; 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.