Title: Geometric and Analytic Aspects of Simon-Lojasiewicz Inequalities on Vector Bundles Show Chinese title

Title: 向量丛上的西蒙-洛瓦斯iewicz不等式的几何与分析方面

Abstract: In real analysis, the Lojasiewicz inequalities, revitalized by Leon Simon in his pioneering work on singularities of energy minimizing maps, have proven to be monumental in differential geometry, geometric measure theory, and variational problems. These inequalities provide specific growth and stability conditions for prescribed real-analytic functions, and have found applications to gradient flows, gradient systems, and as explicated in this paper, vector bundles over compact Riemannian manifolds. In this work, we outline the theory of functionals and variational problems over vector bundles, explore applications to arbitrary real-analytic functionals, and describe the energy functional on $S^{n-1}$ as a functional over a vector bundle. Show Chinese abstract

Abstract: 在实分析中，Lojasiewicz不等式，在Leon Simon关于能量最小映射奇点的开创性工作中重新受到重视，已被证明在微分几何、几何测度论和变分问题中具有重大意义。 这些不等式为指定的实解析函数提供了特定的增长和稳定性条件，并已应用于梯度流、梯度系统，正如本文所阐明的，还应用于紧致黎曼流形上的向量丛。 在本工作中，我们概述了向量丛上的泛函和变分问题的理论，探讨了对任意实解析泛函的应用，并描述了$S^{n-1}$上的能量泛函作为向量丛上的泛函。