AI 中文总结
研究余维数为二的极小曲面的卢猜想，通过证明若$S+\lambda_2$为常数且大于2则$S+\lambda_2\geq3$，得出该猜想在此情形下成立，结合其他结果完善了极小曲面卢第二间隙猜想的余维数情况。
AI 中文摘要
设$M^2\to\Sph^4$为一个闭极小浸入，$S$为其二阶基本形式的平方范数，$\lambda_1\geq\lambda_2\geq0$为卢基本矩阵的特征值。我们证明若$S+\lambda_2$为常数且大于2，则$S+\lambda_2\geq3$。因此卢的第二间隙猜想对余维数为二的极小曲面成立。结合彭-特恩的超曲面结果以及李-赵在每个余维数$m\geq3$下的反例，我们的定理完善了极小曲面的卢第二间隙猜想的余维数情况。
英文摘要
Let $M^2\to\mathbb{S}^4$ be a closed minimal immersion, let $S$ be the squared norm of its second fundamental form, and let $λ_1\geqλ_2\geq0$ be the eigenvalues of Lu's fundamental matrix. We classify all such immersions for which $S+λ_2$ 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 $\mathbb{S}^3$ or the Veronese surface in $\mathbb{S}^4$. In particular, there is no closed minimal surface in $\mathbb{S}^4$ with constant $S+λ_2>2$. 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 $m\geq3$, this completes the codimension picture for minimal surfaces.
Comments13 pages. All comments are welcome
