Geometric variational problems and its visualization

考虑了M乘R上的双调和映射的存在性，它是由M上的调和映射通过M上度量的共形变换约化而成的。证明了当M的维数为3或4时，存在一个度量和这样的双调和映射，但当M的维数大于5时，不存在这样的双调和映射.在Laplacian第一特征值的最小化/最大化条件下，计算了具有两个导热系数的材料的最佳形状。本文还考虑了单壁碳纳米管的长度指数。

I consider the existence of biharmonic map on M times R, which is reduced from a harmonic map on M by conformal change of metric on M. I prove that if dimension of M is 3 or 4, there exists a metric and such biharmonic map, but if dimension of M is greater than 5, there are no such biharmonic maps.I also consider optimization problem of heat diffusion. I calculate the optimize shape of materials with two values of heat conductivity under the minimization/maximaization of the first eigenvalue of Laplacian. I also consider the LENGTH INDEX of single wall carbon nanotube.