The Yang–Mills–Higgs functional on complex line bundles: Asymptotics for critical points

Main results.

In this paper, we will address two more questions about the asymptotic behaviour, as ε → 0 , of critical points satisfying (9). Our first main result describes the concentration of the rescaled energy densities. We denote by spt ⁡ μ the support of a Radon measure μ and by ℋ n - 2 the ( n - 2 ) -dimensional Hausdorff measure.

Recall that varifolds are measure-theoretic generalisations of smooth submanifolds (see, e.g., [35] and Section 5 below for details). In particular, stationary rectifiable varifolds are generalisations of minimal surfaces. Constructing non-trivial stationary, rectifiable varifolds in Riemannian manifolds is a fundamental problem in Geometric Measure Theory [3, 31].

The proof of Theorem 1 relies on a suitable monotonicity formula (see Theorem 3 below).

In our second main result, we prove compactness for the sequence { ( u ε , A ε ) } in W 1 , p ⁢ ( M , E ) × W 2 , p ⁢ ( M , T * ⁢ M ) , for any p with 1 ≤ p < n n - 1 , so long as A ε is in Coulomb gauge, that is,

where ψ ε is an exact 2-form, ζ ε is a harmonic 1-form and ∥ ζ ε ∥ L ∞ ⁢ ( M ) ≤ C M for some constant C M that depends on M only (see, e.g., [13, Lemma 2.10]).

Before illustrating the ideas of the proof, let us motivate the interest towards our results.

Background and motivation.

Ginzburg–Landau functionals of the form (1) were originally proposed as a model of superconductivity. This theory gained much popularity, as it accounts for most commonly observed effects in superconductors, and it gradually became relevant to other areas of physics, such as particle physics and gauge theories. The asymptotic regime as ε → 0 , which is known as the London limit in superconductivity theory or the strongly repulsive limit in particle physics, is characterised by the emergence of topological singularities: due to topological obstructions set by the structure of the bundle E → M , critical points develop singularities, in the asymptotic limit, which are supported on a set of dimension n - 2 . The topologically-driven energy concentration phenomenon can be detected by the distributional Jacobian, which both enjoys remarkable compactness properties and identifies the topological singularities that emerge in the limit as ε → 0 (see, e.g., [23, 1, 2]).

The asymptotic analysis of (1) in the limit as ε → 0 relies on analogous results for the “non-magnetic” version of the functional, in which the variable A is set to zero and the energy is considered as a functional of u only. The analysis of critical points of Ginzburg–Landau-type functionals without magnetic field was initiated by Bethuel, Brezis, and Hélein [7] in the 2-dimensional, Euclidean setting. The analysis was then extended to critical points on 2-dimensional Riemann surfaces [5], higher-dimensional Euclidean domains [25, 8] and, more recently, higher-dimensional manifolds [14, 38, 15, 16]. In particular, our Theorems 2 and 1 generalise results that were first obtained in [8] in the Euclidean, non-magnetic case. In all these works, a main difficulty lies in the lack of uniform energy bounds, as the natural energy scaling considered is the logarithmic one (9).

As for the gauge-invariant, “magnetic” version of the functional, the first results addressed the asymptotic behaviour of minimisers in two-dimensional Euclidean domains [10] and on Riemann surfaces [28, 32]. A detailed analysis of the asymptotics of critical points for the gauge-invariant functional in two dimensions, in the Euclidean setting and in the logarithmic energy regime, has been carried out by Sandier and Serfaty – see, e.g., [33, Chapter 13] and the references therein. Apparently, less effort has been put in studying the asymptotic behaviour of critical points in higher dimensions. Recently, this problem has been addressed in [29], in the context of Hermitian line bundles, for the self-dual version G ε self of G ε , in which the curvature term ∫ M | F ε | 2 ⁢ vol g is replaced by ε 2 ⁢ ∫ M | F ε | 2 ⁢ vol g . The main outcome of [29] is that, for a sequence { ( u ε , A ε ) } of critical points with bounded G ε self -energy, the energy densities e ε self ⁢ ( u ε , A ε ) converge, up to subsequences, to the weight measure of a limiting stationary, rectifiable, integral varifold of codimension 2 in M. Conversely, in [16], it is shown that any non-degenerate minimal submanifold of codimension 2 in M can be obtained as energy concentration set of a sequences of bounded energy critical points of G ε self .

Proofs of the main results: A sketch.

In the paper, we first address the proof of Theorem 2, which ultimately relies on the L p -compactness for the prejacobians j ⁢ ( u ε , A ε ) given by Theorem 0. Instead, the proof of Theorem 1 combines ideas from [8] in order to deal with the logarithmic energy regime and from [28, 29] to deal with the Hermitian line bundle setting. The most difficult part is to identify μ * as the weight measure of a stationary, rectifiable ( n - 2 ) -varifold. The key point is showing that, if μ * ⁢ ( ℬ R ⁢ ( x 0 ) ) is smaller than η 0 ⁢ R n - 2 , where η 0 is suitable uniform “small” constant, then μ * ⁢ ( ℬ R / 2 ⁢ ( x 0 ) ) = 0 . This is shown in Lemma 5.6. The proof goes along the lines of that of [8, Proposition VIII.1] and it is based on a clearing-out result (see the analysis in Section 4, in particular Proposition 4.1 and Proposition 4.6).

A crucial rôle is played by the following (almost) ( n - 2 ) -monotonicity inequality. Let inj ⁡ ( M ) be the injectivity radius of M. For x 0 ∈ M and r ∈ ( 0 , inj ⁡ ( M ) ) , we denote by ℬ r ⁢ ( x 0 ) the geodesic ball of radius r centred at x 0 . Given ( u ε , A ε ) ∈ ℰ , we define

and

where ν is the exterior unit normal to ∂ ⁡ B ρ ⁢ ( x 0 ) , i ν denotes the contraction against ν (see (18) below), F ε := F A ε and g ^ is the metric induced by g on ∂ ⁡ ℬ ρ ⁢ ( x 0 ) .

The proof of Theorem 3 is delicate. A classical argument (see, e.g., Proposition 3.3 below), based on the Pohozaev identity, proves that

for any x 0 ∈ M , ρ ∈ ( 0 , inj ⁡ ( M ) ) and some uniform constant C > 0 . The exponent of n - 4 at the denominator appears for scaling reasons (heuristically, the curvature F ε scales as a second derivative, so the integral of | F ε | 2 scales as a length to the power of n - 4 ). Inequality (17) is known to be sharp in the context of non-Abelian Yang–Mills–Higgs theories [36]. However, in our Abelian setting, Theorem 3 gives an (almost) monotonicity formula for the energy rescaled by ρ n - 2 . A similar result is obtained in [29, Section 4], for the Ginzburg–Landau functional in the self-dual scaling. However, the arguments of [29] fall apart in our setting, because we work with a different scaling of the energy. Instead, we exploit the properties of the Euler–Lagrange equations. In particular, our arguments rely on equation (7), which can be reformulated as a uniformly elliptic equation for the variable A ε . By elliptic regularity estimates, we can control the curvature terms and improve on the monotonicity, obtaining in the end Theorem 3. As a byproduct of our arguments, we obtain a bound for the Morrey–Campanato L q , 2 -norm of F ε , in terms of the rescaled energy (14), for any q < n (see Proposition 3.8 below).

Back on the proof of Theorem 1, the clearing-out property combined with the energy bound (9) implies a two-sided estimate for the ( n - 2 ) -density of μ * , see Lemma 5.7. Following ideas in [8, pp. 498–499] (extended first in the gauge-invariant setting in [33] and to the context of Hermitian line bundles in [29, Section 6]), the conclusion is achieved once we characterise μ * in terms of the stress-energy tensor fields T ε , defined in (5.3) below. Indeed, by criticality of ( u ε , A ε ) , each tensor field T ε is stationary (i.e., divergence-free). Since stationarity is a linear condition, their weak limit T * is still stationary. On the other hand, T * is naturally identified with an ( n - 2 ) -generalised varifold in M whose weight measure | T * | is absolutely continuous with respect to μ * . Then, by Lemma 5.7, we can apply a suitable version of the Ambrosio-Soner rectifiability criterion [4, Theorem 3.1] to deduce that T * is also rectifiable and that μ * = | T * | , whence the conclusion of Theorem 1 follows.

Further comments and open questions.

We did not address specifically the problem of the existence, in full generality, of non-minimising critical points of G ε . We believe that they could be constructed by adapting the general framework developed in [24].

Another interesting question that we have not addressed in this paper is whether the varifold 1 π ⁢ μ * is integral or not. For the magnetic Ginzburg–Landau functional in the self-dual scaling, it is known that the measure 1 π ⁢ μ * is integral [29]. By contrast, for critical points of the non-magnetic Ginzburg–Landau functional (5.4) in Euclidean domains, the situation is more involved [30] – that is, the density of the limiting varifold 1 π ⁢ μ * at a generic point is either 1 or possibly a non-integer number larger or equal than 2. Unfortunately, the arguments in [29] are really tailored to the bounded energy regime and do not carry over to our setting.

Finally, it would be interesting to study, along the lines of [16], whether any non-degenerate minimal codimension 2 submanifold of M can be obtained as energy concentration set of critical points of G ε , for an appropriate choice of the total space E of the bundle E → M . We leave all these questions to future investigation.

Organisation of the paper.

The paper is organised as follows. In Section 1 we derive several a priori estimates for gauge-invariant quantities, such as | u ε | , | D A ε ⁡ u ε | , and | F ε | , that are used throughout the whole paper. In Section 2 we prove Theorem 2. In Section 3 we prove the almost ( n - 2 ) -monotonicity inequality, i.e., Theorem 3. In Section 4, the longest and by far the most technical of the paper, we prove the clearing-out property with its consequences (Propositions 4.1 and 4.6). In the last section, Section 5, we complete the proof of Theorem 1.

The paper is completed by two appendices containing technical results concerning Poincaré and trace-type inequalities for differential forms (Appendix A) and elliptic estimates for the Hodge Laplacian (Appendix B). These results are certainly known to experts but we provide full proofs because they are crucial to our arguments and we could not find the exact statements we needed in the available literature.

Notation.

In this paper, we use the symbol { X ε } to denote a family of objects indexed by the parameter ε > 0 . Although in some instances { X ε } may also be considered as a continuous family, we always look at it as a sequence. In other words, { X ε } is a shorthand for the sequence { X ε k } k , where ε k → 0 as k → + ∞ . Usually, for the sake of a lighter notation, we do not relabel subsequences.

In inequalities like A ≲ B , the symbol ≲ means that there exists a constant C, independent of A and B, such that A ≤ C ⁢ B . In particular, dealing with sequences indexed by ε, we use ≲ to denote inequality up to a constant independent of ε. Whenever it is relevant, we keep track of the dependences of the implicit constants. In most cases, they will depend on the manifold ( M , g ) and on the constant Λ in the energy bound (9). Since we only consider Riemannian manifolds ( M , g ) as base spaces of E → M and the metric g on M is fixed from the very beginning, we will sometimes write M as a shorthand for ( M , g ) .

We denote Λ k ⁢ T * ⁢ M the bundle of k-forms on M. Sobolev spaces of sections of a bundle E → M are denoted W k , p ⁢ ( M , E ) . In particular, W 1 , 2 ⁢ ( M , T * ⁢ M ) is the Sobolev space of one-forms on M of class W 1 , 2 . More details on such spaces can be found, for instance, in [13, Appendix A]. If ω ∈ Λ k ⁢ T * ⁢ M is a k-form on M and X is a vector field, we denote by i X ⁢ ω the contraction of ω with X, defined by setting

for any vector fields X 1 , … , X k - 1 on M.

Finally, as mentioned above, we will always write inj ⁡ ( M ) for the injectivity radius of M and ℬ r ⁢ ( x 0 ) for the geodesic ball in M of centre x 0 ∈ M and r > 0 .

L ∞ -bounds.

Let ( u ε , A ε ) ∈ ℰ be a critical point of G ε . Elliptic regularity theory implies that, for each ε > 0 , the pair ( u ε , A ε ) is smooth up to a suitable gauge transformation (see, e.g., [22] or [29, Appendix]). In particular, gauge-invariant quantities such as | u ε | or | D A ε ⁡ u ε | are continuous (and their squares are smooth). Below, we prove several a priori estimates on u ε , D A ε ⁡ u ε , and F A ε . We will apply these estimates to obtain compactness results. For the reader’s convenience, we provide self-contained proofs of the estimates we need. First, we prove:

In the proof Proposition 1.2, we will make use of an auxiliary result, Lemma 1.4 below. Before stating it, we recall that for any ( u , A ) ∈ ℰ , the covariant derivative D A ⁡ u belongs to L 2 ⁢ ( M , T * ⁢ M ⊗ E ) . This allows us to define a linear functional D A * ⁡ D A ⁡ u : C ∞ ⁢ ( M , E ) → ℝ by setting, for any test section v ∈ C ∞ ⁢ ( M , E ) ,

Since such linear functional is sequentially continuous with respect to convergence of test sections on E, it defines a distribution on E → M . Furthermore, the right-hand side of (1.4) still makes sense if we plug in it any measurable section v such that D A ⁡ v ∈ L 2 ⁢ ( M , T * ⁢ M ⊗ E ) .

Similarly, for ( u , A ) ∈ ℰ , we can define a linear functional 〈 D A * ⁡ D A ⁡ u , u 〉 : C ∞ ⁢ ( M , ℝ ) → ℝ by setting, for any test function φ ∈ C ∞ ⁢ ( M , ℝ ) ,

As it is immediately checked, the functional 〈 D A * ⁡ D A ⁡ u , u 〉 is sequentially continuous with respect to convergence of test functions, hence it is a well-defined distribution on M. In particular, both D A * ⁡ D A ⁡ u and 〈 D A * ⁡ D A ⁡ u , u 〉 are well-defined distributions for any ( u , A ) ∈ ℰ . We are now ready to state Lemma 1.4.

Next, we state an L ∞ -bound for D A ε ⁡ u ε . Precisely:

The proof of Lemma 1.6 is postponed to Appendix C.

Now, we provide a bound on the L ∞ ⁢ ( M ) -norm of { F ε } .

Compactness for gauge-invariant quantities.

In this subsection, we apply the bounds obtained above, as well as results from [13], to prove compactness results for gauge-invariant quantities, such as the norm | u ε | and the pre-Jacobian j ⁢ ( u ε , A ε ) of a sequence of critical points ( u ε , A ε ) that satisfies (9).