Geometric properties on the threshold of the Sobolev type inequalities and generalization to metric spaces

在非紧黎曼流形和无限图上，全局几何结构、几何不等式和长时间热核行为之间的关系已被许多研究者研究得很好。在本研究中，我们考虑连通和，得到了瓶颈性结构与等周不等式、Poincare不等式和长时间热核行为之间的精确关系。

On non-compact Riemannian manifolds and infinite graphs, therelationship among global geometric structure, geometric inequalities and long time heatkernel behavior is well investigated by many researchers. In this research, we considerconnected sums and we obtain a precise relationship between a structure of bottlenecknessand isoperimetric inequality, Poincare inequality and long time heat kernel behavior.