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A double vortex and a separate third vortex

Example

Let $e, ξ > 0$ , $κ = 2 e^{2} ξ$ , and let $z_{0} \neq z_{1}$ be any two points of $C$ . For the positive-winding divisor $D = 2 [z_{0}] + [z_{1}]$ , whose degree is $3$ , the unique scalar solution satisfies $Δ h + κ (1 - e^{h}) = 8 π δ_{z_{0}} + 4 π δ_{z_{1}}, h - 4 \log ∣ z - z_{0} ∣ extends smoothly at z_{0}, h - 2 \log ∣ z - z_{1} ∣ extends smoothly at z_{1}, h \to 0 at infinity,$ with $h < 0$ off ${z_{0}, z_{1}}$ . For $P (z) = (z - z_{0})^{2} (z - z_{1})$ and $s = 2 \log ∣ P ∣$ , the reconstructed fields $q = \sqrt{ξ} e^{(h - s) / 2} P$ and $A = - \frac{1}{2} * d (h - s)$ are smooth, have flux $6 π$ , and attain transverse energy $6 π ξ$ . The solution is specified by this unique nonlinear elliptic problem; no elementary closed-form profile is asserted.

Separately, for each $n \in Z$ and each effective divisor of degree $∣ n ∣$ , the corresponding planar minimizer has signed flux $2 π n$ and transverse energy $T_{n} = 2 π ξ ∣ n ∣$ . Use the transverse energy $E_{⊥}$ , cylinder energy $E_{L}$ , and admissible cylinder class defined in Fixed-flux cylinders and uniform end-collar energy. Fix smooth end traces $p_{j} = (A^{j}, q^{j})$ on $R^{2}$ , $j \in {0, 1}$ , with $E_{⊥} (p_{j}) < \infty$ and flux $2 π n$ . Prescribe their representatives after the globally smooth degree-zero gauge cleanup: outside a disk, $q^{j} = ρ_{j} e^{i n θ}$ with $ρ_{j} > 0$ , $ρ_{j} \to \sqrt{ξ}$ uniformly in $θ$ , $A^{j} - n d θ \in L^{2}$ , and $\lim_{R \to \infty} \int_{\partial D_{R}} (A^{j} - n d θ) = 0$ . The infimum over that admissible class with these fixed ends satisfies $T_{n} L \leq \inf E_{L} \leq T_{n} (L - 2 ℓ) + C_{0} + C_{1} (L \geq 2 ℓ),$ for a fixed collar width $ℓ > 0$ and finite constants $C_{0}, C_{1}$ independent of $L$ ; matching minimizing end traces give the exact value $T_{n} L$ . This is the fixed-flux variational estimate for the stated cylinder class.

Example references

Transverse and cylinder energies, admissible fields, and the fixed-end estimate

Critical vortex winding, square completion and tension (Tong, §§3.1–3.3)

Independent multiple-vortex existence and flux conventions (Chen–Yang, §§2–3 and 5–7)

Facts & Assumptions

Given: $e, ξ > 0$ , distinct $z_{0}, z_{1} \in C$ , and the critical Abelian-Higgs conventions of Critical Abelian Higgs fields, winding and divisor data.

[F1]

Critical Abelian Higgs fields, winding and divisor data: the scalar divisor equation has source $4 π \sum_{j} m_{j} δ_{z_{j}}$ , the flux/tension conventions are $Φ = \int B$ and $E = \int (B^{2} / (2 e^{2}) + ∣ D_{A} q ∣^{2} + e^{2} (∣ q ∣^{2} - ξ)^{2} / 2)$ .

[F2]

Planar critical vortices for arbitrary finite divisors: for each effective divisor there is a unique scalar $h$ with the stated logarithmic subtractions and decay; the reconstruction is smooth, solves the positive first-order equations, has finite energy, and has flux $2 π$ times the degree. Negative winding follows by conjugation.

[F3]

Global energy minima for critical Abelian Higgs vortices: a reconstructed fixed-divisor solution attains the absolute transverse minimum $2 π ξ ∣ n ∣$ in its winding sector.

[F4]

Fixed-flux cylinders and uniform end-collar energy: for fixed smooth finite-energy end traces of flux $2 π n$ , after the stated global degree-zero gauge cleanup into the canonical exterior gauge with the angular one-form limit, the infimum over its admissible cylinder class obeys $T_{n} L \leq \inf E_{L} \leq T_{n} (L - 2 ℓ) + C_{0} + C_{1}$ for $L \geq 2 ℓ$ , and identical minimizing ends attain $T_{n} L$ .

Proof

Given: The divisor $D = 2 [z_{0}] + [z_{1}]$ and the assumptions above.

1.1F1F2givenalgebra

Its multiplicities give degree $N = 2 + 1 = 3$ and distributional source $4 π (2 δ_{z_{0}} + δ_{z_{1}}) = 8 π δ_{z_{0}} + 4 π δ_{z_{1}}$ . The center subtraction coefficients are $2 m_{0} = 4$ and $2 m_{1} = 2$ . Applying [F2] gives the unique scalar $h$ , its smooth subtractions, $h \to 0$ , and strict negativity away from the divisor.

1.2F2F3givenalgebra

Apply [F2] to the explicit effective divisor $∣ n ∣ [0]$ of degree $∣ n ∣$ (the empty divisor if $n = 0$ ) and conjugate when $n < 0$ . Its signed flux is $2 π n$ , and [F3] gives $T_{n} = 2 π ξ ∣ n ∣$ .

2.1F1F2F3step 1.1algebra

Here $P = (z - z_{0})^{2} (z - z_{1})$ and $s = 2 \log ∣ P ∣$ . The reconstruction in [F2] extends smoothly over both centers, solves the positive critical equations, and has flux $2 π N = 6 π$ . Since it is a fixed-divisor reconstructed solution, [F3] gives its attained absolute transverse energy $2 π ξ N = 6 π ξ$ . The argument determines the profile by the unique scalar boundary-value problem and supplies no elementary closed form.

3.1F4step 1.2givenalgebra∎

Fix the smooth end traces specified in the Example. Their finite transverse energy, winding $n$ , flux $2 π n$ , canonical exterior gauge and angular one-form limit are exactly the end hypotheses of [F4]; the infimum uses its admissible cylinder class. Applying [F4] with the value of $T_{n}$ from step 1.2 gives the stated cylinder bounds for every $L \geq 2 ℓ$ , for any selected divisor of degree $∣ n ∣$ ; if both traces are the same selected planar minimizer, [F4] gives the exact translation-invariant minimum. No step assumes radial symmetry or a unit-winding profile.

Proof sources

Arbitrary-divisor scalar equation and reconstruction (Taubes attribution; local proof in supplier)

Critical tension normalization (Tong, §§3.1–3.3)

A double vortex and a separate third vortex

Example

Example references

Facts & Assumptions

Proof

Proof sources

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Dependency tree · 2 levels

Sources