Existence of critical points for the Ginzburg-Landau functional on Riemannian manifolds

In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if N is a compact, orientable 3-dimensional Riemannian manifold without boundary and gamma is a simple, smooth, connected, closed geodesic in N satisfying a natural nondegeneracy condition, then for every epsilon > 0 sufficiently small, ∃ a critical point uepsilon ∈ H1( N; C ) of the Ginzburg-Landau functional Eeu: =12plne N1u 2+ u2-12 2e2 and these critical points have the property that Eepsilon(uepsilon) → length(gamma) as epsilon → 0.;Using known results on R3 , we show the Ginzburg-Landau functional E epsilon defined above Gamma-converges to a functional E which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almost-minimal currents that gamma is a saddle point of E in an appropriate sense.;To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if Eepsilon Gamma-converges to E (not necessarily defined on the same Banach space as Eepsilon), v is a saddle point of E and some additional mild hypotheses are met, then there exists epsilon 0 > 0 such that for every epsilon ∈ (0, epsilon0), Eepsilon possesses a critical point u epsilon and limepsilon→0 Eepsilon (uepsilon) = E( v). Typically, E is only lower semicontinuous, therefore a suitable notion of saddle point is needed.