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

1.岩崎Chisato在题为“用伪微分算子的符号演算证明Gauss-Banner-Chern定理的证明”一文中，仅计算用新方法构造的抛物型方程基本解的主项，得到了Gauss-Banner-Chern定理的局部形式的解析证明。将上述方法应用于带边界的流形，得到了拉普拉斯谱与流形曲率之间的关系。这是Gunther-Schiminng的结果在无边界流形上的推广之一。这一结果在德国的会议上被否决了。利用构造退化抛物型方程基本解的方法，得到了Kaehler流形上的Riemann-Roch定理的局部形式。这一结果在1998年在大阪举行的会议和1998年在波茨顿举行的研讨会上都有介绍。

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.