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

非线性边值问题解的分叉曲线确定非线性项

• 批准号: 05640157
• 负责人: TSUBOI Kenji
• 金额: $ 0.96万
• 依托单位: Tokyo University of Fisheries
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1993
• 资助国家: 日本
• 起止时间: 1993 至 1994
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-05640157/
• 关键词: 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

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.

利用 Atiyah-Singer 指数和 Atiyah-Bott-Lefschetz-Singer 不动点公式，我们得到了提升 Futaki 不变量的显式计算公式。 （Futaki 不变量是李代数同态，提升的 Futaki 不变量是将 Futaki 不变量提升到李群同态。）使用上面的计算公式，我们得到以下结果。 (1) 具有正第一 Chern 类且具有一般复结构的复二维凯勒曲面的提升 Futaki 不变量消失当且仅当该表面允许 爱因斯坦-凯勒度量。(2) 完全交集的某个一般自同构的提升二木不变量消失。 （注意，每个完全交集都期望承认爱因斯坦-凯勒度量。）此外，我们定义了椭圆算子的行列式，获得了行列式的显式计算公式，并证明了维滕完整公式作为计算公式的应用的特殊情况。此外，利用椭圆算子的行列式，我们定义了一个李群同态，它是一个提升 对Futaki-Morita不变量多项式（Futaki不变量的推广）进行了求解，得到了李群同态的显式计算公式。

项目成果

Kenji Tsuboi: "The Atiyar-Singer index theorem for G-equivalent Real elliptic families" Mathematical Journal of Okayama University.

Kenji Tsuboi：“G 等价实椭圆族的 Atiyar-Singer 指数定理”冈山大学数学杂志。

Yutaka Kamimura: "An inverse problem in bifurcation theory,II" Journal of the Mathematical Society of Japan. 46. 89-110 (1994)

Yutaka Kamimura：“分岔理论中的反问题，II”日本数学会杂志。

Kenji Tsuboi: "On the determinant and the holonomy of equivariant elliptic operators" Proceeding of the American Mathematical Society.

Kenji Tsuboi：“论等变椭圆算子的行列式和完整性”美国数学会学报。

Kenji Tsuboi: "The lifted Futaki invoriants and the Spin^C-Dirac operators" Osaka J.of Math.(to oppear).

Kenji Tsuboi：“提升的 Futaki 不变量和 Spin^C-Dirac 算子”Osaka J.of Math.（反对）。

Kenji Tsuboi: "On the determinant and the holonomt of eguivariaht elliptic operators" Proc.of Amer.Math.Soc.(to appear).

Kenji Tsuboi：“关于 eguivariaht 椭圆算子的行列式和完整函数”Proc.of Amer.Math.Soc.（即将出现）。