Geometric and Analytic Aspects of Simon-Lojasiewicz Inequalities on Vector Bundles

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.