2025/09/26 by Cheng, Da Rong, Karigiannis, Spiro, Madnick, Jesse
#53C15 #53C38 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.22155
A classical theorem of Micallef says that if F \colon (Σ, g) → ℝ4 is a stable minimal immersion of an oriented 2-dimensional complete Riemannian manifold (that is parabolic) into ℝ4, it is necessarily holomorphic with respect to some parallel orthogonal complex structure on ℝ4. We generalize this theorem by replacing ℝ4 with ℝ2 + 2k for any codimension 2k, under the additional hypothesis that the normal bundle N Σ is equipped with a complex structure that is compatible with the induced metric and parallel with respect to the induced connection. This is a necessary assumption for such a theorem to hold, and it is automatically satisfied in the classical case k=1. We also briefly discuss possible further generalizations of such a result to other calibrations and to Smith maps.