Critical points of the Ginzburg–Landau system on a Riemannian surface

Instability of Ginzburg—Landau vortices on manifolds

“…Using the direct method one can show that there exists a minimizer v * of E. Then since g is a conformal metric, v * • h −1 satisfies (2.5). The renormalized energy is defined by [1] establishes that W can be written as…”

“…Since ∇f · X and d dt | t=0 |Jac χ t | are bounded in K \ n i=1 B i , one can apply Lemma 3.2 in [1] which asserts that e 2f ε (1 − |u ε | 2 ) 2 converges to a measure supported on…”

“…To prove this, we apply Serfaty's abstract result [10] (see theorem 2.5), showing again that the renormalized energy has no stable critical points on such manifolds. For Ginzburg-Landau posed on a 2-manifold, Baraket [1] generalizes the work of [4] to identify the renormalized energy on compact 2-manifolds without boundary. We should note that, for Ginzburg-Landau in three-dimensional domains, stable vortex solutions do exist [9].…”

The generalized point-vortex problem and rotating solutions to the Gross–Pitaevskii equation on surfaces of revolution

“…To study (1.1) on M, we will appeal to a result of [1] where the author identifies W on a Riemannian 2-manifold. For a compact, simply-connected surface without boundary, one can apply the Uniformization Theorem to assert the existence of a conformal map h : M → R 2 {∞}, so that the metric g is given by e 2f (dx 2 1 + dx 2 2 ), (1.4) for some smooth function f .…”

“….., b n ) with associated degrees d = (d 1 , d 2 , ..., d n ), a result in [1] identifies the renormalized energy as…”