论文标题
尖锐的特征值估计和相关刚性定理
Sharp eigenvalue estimates and related rigidity theorems
论文作者
论文摘要
在本文中，已经获得了不同类型的第一个非零特征值的尖锐边界。此外，当达到这些界限时，可以表征相关的刚性。首先，首先，通过应用[10,13]中证明的主教型体积比较，并在[26]中对laplacian的第一个非零steklov特征值进行[26]中的第一个非零steklov特征值进行了比较，在[26]中，与辐射曲率上的上限相比，在[26]中进行了歧管，在radial curventer corplocts中进行了范围，在适当的上限范围内，我们可以在第一个范围内进行。这些歧管上的大地球可以通过在模型空间（即球形对称的歧管）中的地半径相同的地面球的上方界定。此外，只有当这两个测量球彼此等值时，才能实现第一个非零温特策尔特征值的上限。该结论可以看作是[9,26]中特征值比较的扩展。其次，我们证明了漂流Laplacian的一般Reilly公式，然后使用该公式为第一个非零的steklov特征值提供了急剧的下限，以紧凑的平滑度量度量空间具有边界和凸电势。此外，只有针对规定半径的欧几里得球才能实现这种下限。
In this paper, sharp bounds for the first nonzero eigenvalues of different type have been obtained. Moreover, when those bounds are achieved, related rigidities can be characterized. More precisely, first, by applying the Bishop-type volume comparison proven in [10,13] and the Escobar-type eigenvalue comparisons for the first nonzero Steklov eigenvalue of the Laplacian proven in [26], for manifolds with radial sectional curvature upper bound, under suitable preconditions, we can show that the first nonzero Wentzell eigenvalue of the geodesic ball on these manifolds can be bounded from above by that of the geodesic ball with the same radius in the model space (i.e., spherically symmetric manifolds) determined by the curvature bound. Besides, this upper bound for the first nonzero Wentzell eigenvalue can be achieved if and only if these two geodesic balls are isometric with each other. This conclusion can be seen as an extension of eigenvalue comparisons in [9,26]. Second, we prove a general Reilly formula for the drifting Laplacian, and then use the formula to give a sharp lower bound for the first nonzero Steklov eigenvalue of the drifting Laplacian on compact smooth metric measure spaces with boundary and convex potential function. Besides, this lower bound can be achieved only for the Euclidean ball of the prescribed radius.
