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

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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.