The determination of the nonlinear term by the bifurcating curve of a solution of a nonlinear boundary valne problem

• Project/Area Number: 05640157
• Research Category: Grant-in-Aid for General Scientific Research (C)
• Allocation Type: Single-year Grants
• Research Field: 解析学
• Research Institution: Tokyo University of Fisheries
• Principal Investigator: TSUBOI Kenji (1994) Tokyo University of Fisheries, Department of Natural Sciences, Assistent Professor, 水産学部, 助教授 (50180047); 上村 豊 (1993) 東京水産大学, 水産学部, 助教授 (50134854)
• Co-Investigator(Kenkyū-buntansha): 坪井 堅二 東京水産大学, 水産学部, 助教授 (50180047)
• Project Period (FY): 1993 – 1994
• Project Status: Completed (Fiscal Year 1994)
• Budget Amount: ¥1,500,000 (Direct Cost: ¥1,500,000); Fiscal Year 1994: ¥700,000 (Direct Cost: ¥700,000); Fiscal Year 1993: ¥800,000 (Direct Cost: ¥800,000)
• Keywords: elliptic operator / Atiyah-Singer index / fixedpoint formula / Futaki invariant / Eistein-taehlermetric / determinant / Witten holonomy / Futaki-Morita invariant polynomial / 微分作用素 / アインシュタイン計量 / 微分作用素のdeterminant / Wittenのホロノミー公式 / 非線形境界値問題 / 逆問題 / 分岐曲線 / 非線形項

Research Abstract

Using the Atiyah-Singer index and the Atiyah-Bott-Lefschetz-Singer fixed point formula, we obtained an explicit calculation formula for the lifted Futaki invariant. (The Futaki invariant is a lie algebra homomorphism and the lifted Futaki invariant is a lifting of the Futaki invariant to a Lie group homomorphism.) Using the calculation formula above, we obtained the following results.
(1) The lifted Futaki invariant of a complex 2-dimensional kaehler surface with positive first Chern class and with a generic complex structure vanishes if and only if the surface admits an Einstein-Kaehler metric.
(2) The lifted Futaki invariant for a certain general automorphism of a complete intersection vanishes. (Note that every complete intersection is expected to admit an Einstein-Kaehler metric.)
We moreover defined the determinant of elliptic operators, obtained an explicit calculation formula for the determinant and proved the special case of the Witten holonomy formula as an application of the calculation formula.
Furthermoro, using the determinant of elliptic operators, we defined a Lie group homomorphism which is a lifting of the Futaki-Morita invariant polynomial (which is a generalization of the Futaki invariant) and obtained the explicit calculation formula for the Lie group homomorphism.

Research Products

• [Publications] Kenji Tsuboi: "The lifted Futaki invoriants and the Spin^C-Dirac operators" Osaka J.of Math.(to oppear).
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] Kenji Tsuboi: "On the determinant and the holonomt of eguivariaht elliptic operators" Proc.of Amer.Math.Soc.(to appear).
  • Description: 「研究成果報告書概要(和文)」より
• [Publications] Kenji Tsuboi: "The lifted Futaki invariants and the Spin^C-Dira operators" to appear in Osaka J.of Math.
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] Kenji Tsuboi: "On the determinant and the holonomy of equivariant elliptic operators" to appear in Proc.of Amer.Math.Soc.
  • Description: 「研究成果報告書概要(欧文)」より
• [Publications] Kenji Tsuboi: "The lifted Futaki invariants and the Spin^C-Dirac operators" Osaka J.of Math.(to appear).
• [Publications] Kenji Tsuboi: "On the determinant and the holonomy of eguivariant elligticoperators" Proc.of Amer.Math.Soc.(to appear).
• [Publications] Yutaka Kamimura: "An inverse problem in bifurcation theory,II" Journal of the Mathematical Society of Japan. 46. 89-110 (1994)
• [Publications] Kenji Tsuboi: "The Atiyar-Singer index theorem for G-equivalent Real elliptic families" Mathematical Journal of Okayama University.
• [Publications] Kenji Tsuboi: "On the determinant and the holonomy of equivariant elliptic operators" Proceeding of the American Mathematical Society.