n-harmonic maps with applications to geometry of conformal structures on manifolds

n 调和映射及其在流形上共形结构几何中的应用

• 批准号: 12640212
• 负责人: NAKAUCHI Nobumitsu
• 金额: $ 2.3万
• 依托单位: Yamaguchi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640212/
• 关键词: n-harmonic map; p-harmonic map; harmonic map; global analysis; variational problem; elliptic equation; 共形幾何構造; Concentration phenomenon

We studied n-harmonic maps and more generally p-harmonic maps for applications to the geometry of conformal structures. Nakauchi and Kawai gave some estimates for the first eigenvalue of the p-Lapalacian. We obtained an estimate of Lichnerowicz type if the Ricci curvature is bounded from below by a positive constant, and an estimate of Li-Yau type if the Ricci curvature is non-negative. In our proof of these estimates, we used Bochner-Weitzonbeck formulas of different types. The estimate of Lichnerowicz type was proved for more general class. Kawai investigated Kato type inequalities for p-harmonic maps, and got some estimates. Takakuwa studied behaviors of solution of certain nonlinear elliptic systems, and gave an estimate on the first order derivatives. He also proved uniqueness of solutions of such elliptic systems on bounded domains in the Euclidean space, using a kind of Pohozaev identity. With such supports from various spects, Nakauchi studied n-harmonic maps, and gave a relationship between n-harmonic maps and conformal Killing vector fields.

我们研究了n-调和映射和更一般的p-调和映射在共形结构几何中的应用。Nakauchi和Kawai给出了p-Lapalacian的第一特征值的一些估计。当Ricci曲率下有界时，得到了Lichnerowicz型的估计;当Ricci曲率非负时，得到了Li-Yau型的估计.在我们证明这些估计时，我们使用了不同类型的Bochner-Weitzonbeck公式。对更一般的类证明了Lichnerowicz型估计。Kawai研究了p-调和映射的Kato型不等式，得到了一些估计. Takakuwa研究了一类非线性椭圆型方程组解的性态，给出了一阶导数的估计。他还证明了唯一性的解决方案，这种椭圆系统的有界域在欧几里德空间，使用一种Pohozaev身份。Nakauchi在各种理论的支持下，研究了n-调和映射，并给出了n-调和映射与共形Killing向量场之间的关系。

项目成果

Nobumitsu Nakauchi: "Regularity of minimizing p-harmonic maps hit the sphere"Nonlinear Analysis. 47. 1051-1057 (2001)

Nobumitsu Nakauchi：“最小化 p 谐波映射击中球体的规律性”非线性分析。

Nobumitsu Nakauchi: "Regularity of minimizing p-harmonic maps into the sphere"Nonlinear Analysis. 47. 1051-1057 (2001)

Nobumitsu Nakauchi：“最小化 p 谐波映射到球体的规律性”非线性分析。

Makoto Masumoto: "Extremal lengths of homology classes on compact Riemann surfaces"Nonlinear Anal. 47. 5491-5500 (2001)

Makoto Masumoto：“紧凑黎曼曲面上同调类的极值长度”非线性肛门。

Katsuhiro Komiya: "Equivariant K - theory Euler Classes and maps of representation spheres"Osaka J. Math. 38. 321-328 (2001)

Katsuhiro Komiya：“等变 K - 理论欧拉类和表示球面图”Osaka J. Math。

Hiroo Naitoh: "Grassmann geometries on compact symmetric spaces of exceptional type"Japanese J.Math.. 26. (2000)

Hiroo Naitoh：“特殊类型的紧对称空间上的格拉斯曼几何”日本 J.Math.. 26. (2000)