On a Sharp Sobolev‐Type Inequality on Two-Dimensional Compact Manifolds

Motivated by the asymptotic analysis of double vortex condensates in the Chern‐Simons‐Higgs theory, we construct a suitable minimizing sequence for a sharp Sobolev inequality “à la Moser” for two‐dimensional compact manifolds. As a consequence, we first obtain a direct proof of the sharp character of such an inequality. Secondly, and more interestingly, we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum. (Accepted April 6, 1998)     

Vortex Condensates¶for the SU(3) Chern–Simons Theory

We investigate SU(3)-periodic vortices in the self-dual Chern–Simons theory proposed by Dunne in [13, 15]. At the first admissible non-zero energy level E= 2 π, and for each (broken and unbroken) vacuum state φ₍₀₎ of the system, we find a family of periodic vortices asymptotically gauge equivalent to φ₍₀₎, as the Chern–Simons coupling parameter k→ 0. At higher energy levels, we show the existence of multiple gauge distinct periodic vortices with at least one of them asymptotically gauge equivalent to the (broken) principal embedding vacuum, when k→ 0. Received: 23 October 1999 / Accepted: 14 March 2000     

Vortices in the Maxwell‐Chern‐Simons theory

Our aim is to prove rigorously that the Chern‐Simons model of Hong, Kim, and Pac [13] and Jackiw and Weinberg [14] (the CS model) and the Abelian Higgs model of Ginzburg and Landau (the AH model, see [15]) are unified by the Maxwell‐Chern‐Simons theory introduced by Lee, Lee, and Min in [16] (MCS model). In [16] the authors give a formal argument that shows how to recover both the CS and AH models out of their theory by taking special limits for the values of the physical parameters involved. To make this argument rigorous, we consider the existence and multiplicity of periodic vortex solutions for the MCS model and analyze their asymptotic behavior as the physical parameters approach these limiting values. We show that, indeed, the given vortices approach (in a strong sense) vortices for the CS and AH models, respectively. For this purpose, we are led to analyze a system of two elliptic PDEs with exponential nonlinearities on a flat torus. © 2000 John Wiley & Sons, Inc.     

Blow-up analysis for a cosmic strings equation

In this paper we develop a blow-up analysis for solutions of an elliptic PDE of Liouville type over the plane. Such solutions describe the behavior of cosmic strings (parallel in a given direction) for a W-boson model coupled with Einstein's equation. We show how the blow-up behavior of the solutions is characterized, according to the physical parameters involved, by new and surprising phenomena. For example in some cases, after a suitable scaling, the blow-up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the “uniformization” of the Riemann sphere with conical singularities.     

Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when existence holds, and obtain unique stable solutions for area minimizing immersions. Furthermore, we find exactly when other (unstable) solutions exist and study how they blow-up. We prove our class of unstable solutions exhibit different blow-up behaviors when the surface is of genus two or greater. We establish similar results for the blow-up behavior of any general family of unstable solutions. This information allows us to consider similar minimal immersion problems when the total extrinsic curvature is also prescribed.     

Double vortex condensates in the Chern-Simons-Higgs theory

For a selfdual model introduced by Hong-Kim-Pac [18] and Jackiw-Weinberg [19] we study the existence of double vortex-condensates“bifurcating” from the symmetric vacuum state as the Chern-Simons coupling parameter k tends to zero. Surprisingly, we show a connection between the asymptotic behavior of the given double vortex as with the existence of extremal functions for a Sobolev inequality of the Moser-Trudinger's type on the flat 2-torus ([22], [1] and [15]). In fact, our construction yields to a “best” minimizing sequence for the (non-coercive) associated extremal problem, in the sense that, the infimum is attained if and only if the given minimizing sequence admits a convergent subsequence. Received: March 3, 1998 / Accepted October 23, 1998     

Multiple forced oscillations for theN-pendulum equation

We consider the periodically forcedN-pendulum equation. Forced oscillations are obtained, and their multiplicity is studied in terms of the mean value of the forcing term.     

Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model

In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions in a $2\times 2$ nonlinear elliptic system arising in the study of self-dual non-Abelian Chern–Simons vortices. We show that the system admits at least two solutions when the Chern–Simons coupling parameter $\kappa >0$ is sufficiently small; while no solution exists for $\kappa >0$ sufficiently large. As in Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), we use a variational formulation of the problem. Thus, we obtain a first solution via a constrained minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as $\kappa \rightarrow 0$ . Then we obtain a second solution by a min-max procedure of “mountain pass” type.