Lu's conjecture for minimal surfaces in codimension two

2026/07/23 by Jianquan Ge, Fagui Li, Yunheng Zhang
Mathematics · #math.DG #msc:53C20 #msc:53C24 #msc:53C42

paper · pdf

13 pages. All comments are welcome

arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

Let M²→\mathbbS⁴ be a closed minimal immersion, let S be the squared norm of its second fundamental form, and let λ₁≥λ₂≥0 be the eigenvalues of Lu's fundamental matrix. We classify all such immersions for which S+λ₂ is constant. We prove that the constant can only be 0 or 2. In the first case the image is a totally geodesic 2-sphere; in the second case it is either a Clifford torus in a totally geodesic \mathbbS³ or the Veronese surface in \mathbbS⁴. In particular, there is no closed minimal surface in \mathbbS⁴ with constant S+λ₂>2. Consequently, Lu's second-gap conjecture holds for minimal surfaces in codimension two. Together with the hypersurface result of Peng--Terng and the counterexamples of Li--Zhao in every codimension m≥3, this completes the codimension picture for minimal surfaces.

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