2007 Fiscal Year Final Research Report Summary
The fundamental solution for the heat equation on manifolds and its applications
| Project/Area Number |
17540168
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | University of Hyogo |
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Principal Investigator |
IWASAKI Chisato University of Hyogo, Graduate School of Material science, Professor (30028261)
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| Co-Investigator(Kenkyū-buntansha) |
UMEDA Tomio University of Hyogo, 大学院・物質理学研究科, Professor (20160319)
HOSHIRO Toshihiko University of Hyogo, 大学院・物質理学研究科, Professor (40211544)
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| Project Period (FY) |
2005 – 2007
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| Keywords | partial differential equation / parabolic equation / fundamental solution / pseudo-differential operator / index theorem / curvature |
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| Research Abstract |
Gunther and Schimming had studied a relation between curvature of Riemannian manifolds and a summation of the trace of the kernel for the heat equation acting to the de Rham complex on manifolds. This summation is one kind of generalization of the alternating summation. So, their result is a generalization of a local version of Gauuss-Bonnet-Chern theorem. By a new method, we tried to generalize this result for both to Dolbeault complex on Kaehler manifolds and to de Rham complex on Riemann manifolds with boundary. In this research, it is important to construct asymptotic behavior of the fundamental solution for the heat equation. After this construction we must discuss a relation of the main part of the asymptotic behavior and curvature of manifolds.
In case manifolds have no boundary, it is enough to construct the fundamental solution for the Cauchy problem. But for manifolds with boundary, we must study how quantities, which depend on the boundary manifolds, appear in an asymptotic b
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ehavior of the fundamental solution. Also when we construct the fundamental solution, we must introduce a new calculation for symbolic calculus because some cancellation occurs in summing a generalization of the alternating sum. Incase Dolbeault complex, we must study more precise behavior of an asymptotic solution of the fundamental solution.
C. Iwasaki presented a generalization of a local index theorem for Dolbeault complex on Kaehler manifolds, which obtained by the above method, at the conference in Italy in August 2005, in Germany on August 2006, in Taiwan in January 2007 and at several conferences in Japan. This result has already been published. C. Iwasaki presented a generalization of a local index theorem for de Rham complex on Riemann manifolds with boundary, which obtained by the above method, at ISAAC Congress in Ankara at August 2007. More over this result is presented is the RIMS Conference at October 2007, and papers about this result are to appear.
We started to study both the construction of the fundamental solution and spectrum of the Fokker-Planck equation. Less
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Research Products
(42 results)
