发表机构
South China Normal University; Beijing Normal University; Beijing Institute of Technology（华南师范大学; 北京师范大学; 北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过谱比较方法证明了单位球面中闭极小曲面的Simon猜想，并推广到平行平均曲率向量曲面及高维旋转对称情形。
AI 中文摘要
本文证明了单位球面中浸入的连通闭极小曲面的Simon猜想中的所有缺口。我们的证明使用Lin、Wang和Xu的线丛阶梯以及Dolbeault计算，在高斯曲率$K_g\geq1$和$K_g\leq1$下获得二维球面上的双侧有序谱比较，并在等式情形下得到刚性。在连续的Calabi尺度上应用这些比较，迫使高斯曲率取两个端点值之一。我们还建立了具有平行平均曲率向量的闭曲面的相应刚性定理。最后，我们在Ricci下界和截面曲率上界条件下证明了旋转对称的高维类比。
英文摘要
In this paper, we prove all gaps in the Simon conjecture for connected closed minimal surfaces immersed in unit spheres. Our proof uses the line-bundle ladder of Lin, Wang, and Xu and a Dolbeault computation to obtain two-sided ordered spectral comparisons on the two-sphere under Gaussian curvature $K_g\geq1$ and $K_g\leq1$, with rigidity in the equality case. Applying these comparisons at consecutive Calabi scales forces the Gaussian curvature to be one of the two endpoint values. We also establish the corresponding rigidity theorem for closed surfaces with parallel mean curvature vector. Finally, we prove a rotationally symmetric higher-dimensional analogue under a Ricci lower bound and a sectional-curvature upper bound.
Comments28 pages. All comments are welcome
