Quantitative Regularity for p-Minimizing Maps Through a Reifenberg Theorem

In this article we extend to arbitrary p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $$p=2$$ p = 2 . We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done in Cheeger and Naber (Commun Pure Appl Math 66(6): 965–990, 2013). Then, adapting the work of Naber and Valtorta (Ann Math (2) 185(1): 131–227, 2017), we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is k-rectifiable for a k which only depends on p and the dimension of the domain.

Energy-minimizing maps from manifolds with nonnegative Ricci curvature

The energy of any [Formula: see text] representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger–Gromoll splitting theorem.

NUMERICAL BIFURCATION OF EQUILIBRIA OF NEMATIC CRYSTALS BETWEEN NON-CO-AXIAL CYLINDERS

Calculations are presented giving the stable equilibrium states of nematic liquid crystals confined between non-co-axial circular cylinders subject to the condition that the director be normal to the bounding surfaces. The one-constant theory of nematics is employed and emphasis is laid on cases for which the minimizing director field does not lie in the plane perpendicular to the cylinders. Calculations are made of the force that the nematic exerts on the cylinders, tending to make them co-axial. The numerical problem faced is that of finding, for the Dirichlet energy, minimizing maps to S2, from a region in ℝ2 bounded by non-concentric circles. The method employed is a finite-element implementation of a projected gradient method for which the energy decreases at each iteration.