球中具有平行平均曲率向量的曲面的挤压刚性
Pinching rigidity of surfaces with parallel mean curvature vector in spheres
AI总结：
研究浸入单位球中具有平行平均曲率向量和正高斯曲率的闭曲面，建立三个西蒙斯型积分恒等式，得到前两个尖锐端点间隙等结果，并结合分类定理刻画端点情形。
AI中文摘要：
受球中极小曲面的西蒙猜想启发，我们研究浸入单位球中的具有平行平均曲率向量和正高斯曲率的闭曲面。设\(h\)为第二基本形式，\(\mathbf{H}\)为平均曲率向量场，令\(\tilde h = h - \mathbf{H}g\)且\(\tilde S = |\tilde h|^2 = |h|^2 - 2H^2\)。我们为\(\tilde S\)建立了三个西蒙斯型积分恒等式，扩展了极小情形下的第一、第二和第三间隙恒等式。作为应用，我们得到了前两个尖锐端点间隙以及第三区间的一些刚性和振荡估计。通过将这些恒等式与卡拉比和丘成桐的分类定理相结合，我们进一步刻画了端点情形。
英文摘要：
Inspired by the Simon conjecture for minimal surfaces in spheres, we study closed surfaces with parallel mean curvature vector and positive Gaussian curvature immersed in unit spheres. Let $h$ be the second fundamental form, let $\mathbf{H}$ be the mean curvature vector field, and set $\tilde h=h-\mathbf{H}g$ and $\tilde S=|\tilde h|^2=|h|^2-2H^2$, where $H=|\mathbf{H}|$ and $g$ is the induced Riemannian metric on the surface $M$. We establish three Simons-type integral identities for $\tilde S$, which extend the first, second and third gap identities in the minimal case. As applications, we obtain the first two sharp endpoint gaps and several rigidity and oscillation estimates in the third interval. We further characterize the endpoint cases by combining these identities with the classification theorems of Calabi and Yau.
