座落在平面凸區域上的最小網絡之組態

In this project, we are interested in the configurations of curve networks, whose elastic energy are at equilibria achieved by gradient flows. Simpler cases to start the study include tripod-like networks and tree-type networks. Our approach is to view networks as unions of curves joint at junction points by setting up certain boundary conditions at junction/boundary points, and then consider geometric gradient flow of networks. In fact , various of ``boundary conditions" at the junction points provide different type of difficulty for the geometric gradient flow. Among various boundary conditions, the most difficult one is the so-called clamped boundary condition, namely the end point is fixed and the tangent indicatrix at any end point is a prescribed (constant) unit vector. The main result in this case is stated in our report.