2025/06/05 by Orriols, Gerard, Rivière, Tristan
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2506.05141
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 \mathbbS3, taking values in the Grassmannian of 2-planes in ℝ4. This lower bound is proved to be optimal for any genus g ∈ ℕ but attained only when g = 0. For g ≠ 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 \mathbbS3 in relation to the Willmore conjecture.