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 
$k \to 0^{+}$
 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.

Double vortex condensates in the Chern-Simons-Higgs theory

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser-Trudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (aij)N × N the Cartan matrix for SU(N + 1), i.e., 






We show that 






has a lower bound in (H1(Σ))N if and only if 






This inequality is optimal. As a direct consequence, if Mj < for 4π for j = 1, 2, …, N, ΦM has a minimizer u that satisfies 






© 2001 John Wiley & Sons, Inc.

/pdf/analytic-aspects-of-the-toda-system-i-a-moser-trudinger-56e33w61ip.pdf

Analytic aspects of the Toda system : I. A Moser-Trudinger inequality

In the study of conformal structures on manifolds, application of the method of nonlinear partial differential equations may be said to have begun with the work of Poincaré. In the paper [43], Poincaré solved the uniformization problem for Riemann surfaces of genus greater than 1 by solving for a conformal metric with constant negative curvature −1. The analogous question for surfaces of positive curvature was first studied successfully by Moser in two fundamental papers [37, 38] in which he obtained, with precise constants, a sharp version of a limiting case of the Sobolev inequality that is now commonly referred to as the MoserTrudinger inequality. The inequality asserts (for a precise statement, see Section 1) that for two-dimensional domains, compactly supported functions with Dirichlet integrals bounded by a constant automatically belong to an exponentially integrable class. This inequality makes it possible to obtain bounds for the Dirichlet integral of a minimizing family of functions in a variational problem associated with the problem for prescribing the Gauss curvature of a conformal metric on the real projective plane. Subsequent development depends in part on understanding the role of the Möbius group in the critical-exponent inequalities. Since the variational functional in Moser’s approach has a natural meaning in spectral geometry, there is a beautiful development concerning the compactness of the isospectral family of metrics in low dimension. The theory is enriched by the introduction of higher-order invariants by the work of Paneitz [42] and Fefferman-Graham [26], and more recently our own work on the σ2 equation [14]. In this brief article, we will survey some work in conformal geometry in which the Moser-Trudinger inequality plays a role, as well as extensions and generalizations of this inequality to higher dimensions. In the last section we will present a new inequality on the 4-sphere that is the natural generalization of the original Moser inequality for the 2-sphere.

The inequality of Moser and Trudinger and applications to conformal geometry

We analyze solutions of the Toda system and establish an optimal Moser-Trudinger inequality

Analytic aspects of the Toda system: I. A Moser-Trudinger inequality

In the study of conformal geometry, the method of elliptic partial differential equations is playing an increasingly significant role Since the solution of the Yamabe problem, a family of conformally covariant operators (for definition, see section 2) generalizing the conformal Laplacian, and their associated conformal invariants have been introduced The conformally covariant powers of the Laplacian form a family P2k with k ∈ N and k ≤ n if the dimension n is even Each P2k has leading order term (−�) k and is equal to (−�) k if the metric is flat The curvature equations associated with these P2k operators are of interest in themselves since they exhibit a large group of symmetries The analysis of these equations is of necessity more complicated, it typically requires the derivation of an optimal Sobolev or Moser-Trudinger inequality that always occur at a critical exponent A common feature is the presence of blowup or bubbling associated to the noncompactness of the conformal group A number of techniques have been introduced to study the nature of blowup, resulting in a well developed technique to count the topological degree of such equations The curvature invariants (called the Q-curvature) associated to such operators are also of higher order However, some of the invariants are closely related with the Gauss-Bonnet-Chern integrand in even dimensions, hence of intrinsic interest to geometry For example, in dimension four, the finiteness of the Q-curvature integral can be used to conclude finiteness of topology In addition, the symmetric functions of the Ricci tensor appear in natural fashion as the lowest order terms of these curvature invariants, these equations offer the possibility to analyze the Ricci tensor itself In particular, in dimension four the sign of the Q-curvature integral can be used to conclude the sign of the Ricci tensor Therefore there is ample motivation for the study of such equations

Non-linear partial differential equations in conformal geometry

We consider a Ginzburg-Landau type functional on S2 with a 6th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg problem for the prescribed Gauss curvature equation on S2.

/pdf/multiplicity-results-for-the-two-vortex-chern-simons-higgs-12jde6dhep.pdf

Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere

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Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere