Owen Drummond | Searcharxiv

arXiv2024

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.

math.DG↗

arXiv2024

The Top Dimensional Singular Set $\text{sing}_{*}u$

In this survey, we examine the properties of the top dimensional singular set $\text{sing}_{*}(u)$, including several theorems, geometric properties, and relation to homogeneous degree zero minimizers.

math.AP↗

arXiv2024

Geometric Analysis of Energy Minimizing Maps: Tangent Maps and Singularities

Energy minimizing maps (E.M.M.s) play a central role in the calculus of variations, partial differential equations (PDEs), and geometric analysis. These maps are often embedded into $C^\infty$ Riemannian manifolds to minimize the Dirichlet Energy functional under certain prescribed conditions. For understanding physical phenomena where systems naturally evolve to states of minimal energy, the geometric analysis of these maps has provided elucidating insights. This paper explores the geometric and analytic properties of energy minimizing maps, tangent maps, and the singular set (sing(u)). We begin by establishing key concepts from analysis, including the Sobolev Space $W^{1,2}$ harmonic functions, and Hausdorff dimension. Significant results about the density function, its upper semi-continuity, and the compactness theorem for tangent maps, and theorems for homogeneous degree zero minimizers are presented. Also analyzed in detail is the singular set (sing(u)), its Hausdorff dimension, and geometric structure. We conclude with open problems that are rich in research potential, and the far-reaching implications if these problems are to be solved.

math.AP↗