- The Stability of α− Harmonic Maps with Physical Applications(arXiv)
Abstract : The first result in this study is a non-existence theorem for α−harmonic mappings. Additionally, a direct connection between the α− harmonic and harmonic maps is made possible via conformal deformation. Second, the instability of non-constant α-harmonic maps is investigated with regard to the target manifold’s Ricci curvature requirements. Next, the concept of α−stable manifolds and their physical applications are explored. Finally, it is investigated the α−stability of compact Riemannian manifolds that admit a non-isometric conformal vector field as well as the Einstein Riemannian manifolds under certain assumption on the smallest positive eigenvalue of its Laplacian operator on functions.
2. Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints(arXiv)
Abstract : An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.
3.Existence and Stability of α− harmonic Maps (arXiv)
Abstract : In this paper, we first study the α−energy functional, Euler-Lagrange operator and α-stress energy tensor. Second, it is shown that the critical points of α− energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an α−harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal α− harmonic mapsare minimal submanifolds. Then, the stability of any α−harmonic map from a Riemannian manifold to a Riemannian manifold with non-positive Riemannian curvature is demonstrated. Finally, the instability of α−harmonic maps from a compact
4. Two-dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections(arXiv)
Abstract : We study a variational model for ferronematics in two-dimensional domains, in the “super-dilute” regime. The free energy functional consists of a reduced Landau-de Gennes energy for the nematic order parameter, a Ginzburg-Landau type energy for the spontaneous magnetisation, and a coupling term that favours the co-alignment of the nematic director and the magnetisation. In a suitable asymptotic regime, we prove that the nematic order parameter converges to a canonical harmonicmap with non-orientable point defects, while the magnetisation converges to a singular vector field, with line defects that connect the non-orientable point defects in pairs, along a minimal connection.