Construction of the fundamental solution for equations of parabolic type and its applications to geometric problems

1.岩崎千人在《用伪微分算子的符号演算证明高斯-邦纳-陈定理》的论文中给出了局部形式的高斯-邦纳-陈定理的一个解析证明，该方法只计算抛物型方程的基本解的主项，并构造了抛物型方程.根据1998年的计划，将上述方法应用于有边流形，得到了Laplacian的谱与流形曲率的关系。这个结果是Gunther-Schimingg关于无边界流形的结果的推广。这一结果在德国的会议上得到了证实。利用构造退化抛物方程基本解的方法，得到了Kaehler流形上Riemann-Roch Thorem的局部形式.这一成果在1998年大坂会议和1998年波茨坦研讨会上都作了介绍。

1. An analytical proof of the local version of the Gauss-Banner-Chern Theorem has been obtained by Chisato Iwasaki in a paper titled A proof of the Gauss Bonnet-Chern Theorem by the symbolic calculus of pseudo-differential operators, only calculating the main term of the fundamental solution for equation of parabolic type, which was constructed by the new method.2. Applying the above method to manifolds with boundary, we obtained the relation of spectrum of Laplacian and curvatures of manifolds, according to the plan of 1998. This result is one of extension of that of Gunther-Schiminng on manifolds without boundary. This results was tailked in the conference in Germany.3. Local version of Riemann-Roch Thorem for Kaehler manifolds is obtained, by applying the method of construction of the fundamental solution for the degenerate parabolic equation. This result is presented in both conference at Osaka in 1998 and workshop at Potsdom in 1998.