论文标题
由球形多边形形成的表面的等等不等式
An isoperimetric inequality for surfaces formed from spherical polygons
论文作者
论文摘要
我们为一个封闭表面家族的等速度不等式提供了新的证据，该家族的表面曲率相同等于表面光滑的一方面。这些表面是由凸形的球形多边形形成的，多边形的每个顶点导致表面上的非平滑点。例如，由球形灯光形成的表面是革命的表面，具有两个非平滑尖端。结合一个直接的近似论点，这种不等式首先由Bérard，Besson和Gallot证明，在那里他们提供了Lévy-Gromov等等不平等的概括。不平等意味着针对球体的地理凸子集的等等不平等，并且使用Faber-Krahn定理，这也暗示着在封闭表面上给定区域的第一个dirichlet特征值上的下限。通过近似值，我们将其转换为在球体的地球凸子集中包含的域的第一个dirichlet-neumann特征值上的下限。
We give a new proof of an isoperimetric inequality for a family of closed surfaces, which have Gaussian curvature identically equal to one wherever the surface is smooth. These surfaces are formed from a convex, spherical polygon, with each vertex of the polygon leading to a non-smooth point on the surface. For example, the surface formed from a spherical lune is a surface of revolution, with two non-smooth tips. Combined with a straightforward approximation argument, this inequality was first proved by Bérard, Besson, and Gallot, where they provide a generalization of the Lévy-Gromov isoperimetric inequality. The inequality implies an isoperimetric inequality for geodesically convex subsets of the sphere, and, using a Faber-Krahn theorem, it also implies a lower bound on the first Dirichlet eigenvalue of a region of a given area on the closed surfaces. Via approximation, we convert this into a lower bound on the first Dirichlet-Neumann eigenvalue of domains contained in geodesically convex subsets of the sphere.
