Pith. sign in

REVIEW 2 major objections 4 minor 60 references

Harmonic maps to the circle with higher dimensional singular set

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves a complete classification of circle-valued harmonic maps with prescribed codimension-two singular set and shows that three variational relaxations converge to a common renormalised interaction energy after subtracting a…

desk verdict A genuinely new classification and a mostly convincing renormalised-energy unification, but the fractional expansion in Theorem B has a repairable gap in the lower bound. read the letter →

arxiv 2411.14186 v1 pith:V3EUYTTL submitted 2024-11-21 math.DG math.AP

Marco Badran This is my paper · ORCID
classification math.DGmath.AP MSC 58E2049Q2049J45
keywords singularharmonicmapsrenormalisedenergydistributionalJacobianfractionalSobolevp-harmonicHodgeLaplaciancodimension-twosingularitiesvortexfilaments
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that a codimension-two singular set for maps from a closed Riemannian manifold to the circle carries a finite, metric-dependent interaction energy that appears in all three natural ways of making the singular problem variational. The first theorem classifies the maps that are harmonic away from the prescribed singular set and distributionally harmonic across it: they form the set $\mathcal{H}(M,\llbracket\Gamma\rrbracket)$, parametrised by the integral cohomology $H^1(M,2\pi\mathbb{Z})$, so the topology of the ambient manifold controls the multiplicity. The second theorem gives asymptotic expansions for three minimisation problems --- removing a $\delta$-tube around $\Gamma$ and prescribing the degree, minimising the $p$-energy as $p\to 2$, and minimising the $H^s$-energy as $s\to 1$ --- whose common finite part is the same renormalised energy $W_M(\Gamma)$, plus a sector-dependent desingularised energy. This matters because it converts a topologically forced singularity into a computable cost: a divergent area term of order $2\pi\mathcal{H}^{n-2}(\Gamma)/(2-p)$ and then a finite interaction between different parts of the singular set, with an explicit integral formula in Euclidean space.

What carries the argument

The load-bearing identity is the Hodge decomposition of the real one-form $ju=u^*(d\theta)=d\phi+d^*\psi+\omega$: the $d^*\psi$ part encodes the singular set through $dd^*\psi=2\pi\star\llbracket\Gamma\rrbracket$, the $d\phi$ part records harmonicity (it must vanish modulo constants for globally distributionally harmonic maps), and the harmonic part $\omega$ carries the topological sector. The renormalised energy is built from the regular part of the Green's potential: with $A=2\pi\int_\Gamma G_M^{n-2}(\cdot,y)\vec\Gamma(y)$ and $S=-\xi(|z|)\log|z|\vec\Gamma$ the subtracted logarithmic singularity in Fermi coordinates, one defines $W_M(\Gamma)=-2\pi\int_\Gamma(A+S)$. The proof machinery also includes a De Giorgi-type regularity estimate for the boundary-value problem solved by the phase in the tube-removed relaxation, which controls the difference between the true minimiser and the model harmonic map.

What would settle it

Take $M=S^1\times B^2$ with $\Gamma=S^1\times\{0\}$ and consider the admissible map $u_s(y,z)=e^{iy}z/|z|$, which has degree $+1$ around $\Gamma$; its average over $\Gamma$ is $(1/2\pi)\int_0^{2\pi}e^{iy}dy\, z/|z|=0$, so the averaged map has degree $0$. If such a map can be realised as, or perturbed into, a competitor for the fractional problem with the same energy asymptotics, the asserted step behind the lower bound (4.31) fails; if not, the missing proof must use a structural property that rules out this averaging cancellation.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes two structural facts. First, whenever the singular set $\Gamma$ is an admissible integral current that bounds an oriented hypersurface, the singular harmonic maps are exactly the maps whose angle form satisfies $ju_\alpha=d^*\psi+\omega_\alpha$, where $\psi$ is the unique two-form with $dd^*\psi=2\pi\star\llbracket\Gamma\rrbracket$ and $\omega_\alpha$ ranges over the affine lattice $[d^*\psi]+H^1(M,2\pi\mathbb{Z})$; this proves Theorem A and its integral-current version. Second, for embedded $C^{1,1}$ singular sets of codimension two, the three relaxed energies have the common expansion with divergent leading term $2\pi\mathcal{H}^{n-2}(\Gamma)\log(1/\delta)$, $2\pi\mathcal{H}^{n-2}(\Gamma)/(2-p)$, or $2\pi\mathcal{H}^{n-2}(\Gamma)/(2-2s)$, followed by the same renormalised energy $W_M(\Gamma)$ and the sector energy $e(u_\alpha)$. The renormalised energy is defined intrinsically from the Green's function of the Hodge Laplacian, and in $\mathbb{R}^3$ the interaction between two curves $\gamma_1,\gamma_2$ reduces to the double integral of $\dot\gamma_1(x)\cdot\dot\gamma_2(y)|x-y|^{-1}$.

Load-bearing premise

The fractional lower bound assumes that averaging the minimiser over the singular set $\Gamma$ preserves its degree-one winding, and this is asserted rather than proved.

Editorial extensions

If this is right

  • For two disjoint embedded $C^{1,1}$ curves $\gamma_1,\gamma_2\subset\mathbb{R}^3$, the interaction is $\int_{\gamma_1}\int_{\gamma_2}\dot\gamma_1(x)\cdot\dot\gamma_2(y)\,|x-y|^{-1}\,d\mathcal{H}^1(x)d\mathcal{H}^1(y)$, so perpendicular filaments do not interact and parallel filaments interact with orientation-dependent sign.
  • The same finite $W_M(\Gamma)$ controls the tube-removed, $p$-harmonic, and fractional minimisers, so the choice of relaxation changes only the divergent prefactor and the sector term $e(u_\alpha)$, not the geometric interaction.
  • If $\mathcal{H}(M,\llbracket\Gamma\rrbracket)$ contains a strict minimiser of the desingularised energy $e$, then for small $\delta$ the tube-removed minimiser lies in that sector, so the asymptotic expansion selects the least-energetic harmonic representative.
  • The second and third expansions give a large-cost formula for prescribing a distributional Jacobian: the least $p$-energy and $H^s$-energy needed to force $Ju=Jv$ diverge like $2\pi\mathcal{H}^{n-2}(\Gamma)/(2-p)$ and $2\pi\mathcal{H}^{n-2}(\Gamma)/(2-2s)$.
  • The paper conjectures that mixed $W^{s,p}$ relaxations with $sp\to 2$ obey the analogous expansion with $2-sp$ in the denominator, which would place the fractional and $p$-harmonic results in one family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extrapolation is that $W_M(\Gamma)$ defines an intrinsic self-interaction for arbitrary codimension-two cycles; in flat space it should equal a desingularised Coulomb-type integral, and computing it numerically for knotted curves would test the diagonal subtraction procedure explicitly.
  • If the commonality of $W_M$ survives the conjectured mixed $W^{s,p}$ relaxations, then multiplying the fractional expansion by $(1-s)$ would write nonlocal codimension-two area as local area plus $W_M$, suggesting that the nonlocal-to-local limit inherits a universal finite interaction term.
  • A topological subtlety left implicit is that averaging the fractional minimiser over $\Gamma$ could in principle reduce its degree; checking maps of the form $e^{i\varphi(y)}z/|z|$ on a tube over a circle would clarify whether the lower bound needs an additional hypothesis or a stronger structural property of minimisers.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies S1-valued harmonic maps with prescribed codimension-two singular set on a closed oriented Riemannian manifold. Theorem A/Theorem 1 classifies all such distributionally harmonic maps for any admissible integral current Γ=∂Σ, parametrising them by H1(M,2πZ) via the Hodge decomposition of the pulled-back form ju. For embedded C1,1 Γ, the paper defines a renormalised energy WM(Γ) from the regular part of the Green's potential of the Hodge Laplacian and proves in Proposition 3.1 the Dirichlet-energy expansion on Mδ. Theorem B/Theorem 3 claims matching asymptotic expansions for the tube-removed, p-energy (p↑2), and fractional Hs (s↑1) relaxations, with common leading term 2πH^{n-2}(Γ) times the relevant divergence and common renormalised energy WM(Γ)+e(uα). The proofs compare the relaxed minimisers to the fixed harmonic map uα; the tube-removed case uses a De Giorgi iteration, and the p/fractional cases compare with uδ.

Significance. If the central claims are made rigorous, this is an important contribution: it gives a Hodge-theoretic classification of singular harmonic maps, and it shows that three different relaxations share the same renormalised interaction energy, providing a higher-dimensional analogue of the Bethuel–Brezis–Hélein theory. The explicit expression in R3 (Eq. (1.2)) answers a question of Rivière and connects the renormalised energy to magnetic inductance; the fractional case also connects to a recently introduced nonlocal area in codimension two. The paper's structural clarity and its intrinsic Green's-function formulation are genuine strengths. However, the proof of the fractional expansion currently contains a load-bearing gap, so the significance is conditional on the repair described below.

major comments (2)
  1. [§4.5, Eq. (4.31)] The lower bound (4.31) is not established by the displayed argument. After deriving [us]^2_{Hs(Bδ×Γ)} ≥ H^{n-2}(Γ)[ū_s]^2_{Hs(Bδ)}, the proof states that the degree condition on us implies that the Γ-average ū_s has degree +1, and then applies [CFP24, Lemma 3.6 and Proposition 4.1]. This implication is false as stated: ū_s(z)=average over Γ of us(z,·) is merely a unit-disk-valued map, its degree is not even defined unless it avoids 0, and averaging S1-valued degree-one maps can destroy the winding. For instance, u(z,θ)=e^{i(arg z+θ)} on B1×S1 has degree +1 on every fiber but its Γ-average is identically zero. Since (4.31) is the only step in the proof of Theorem 3 that produces the leading constant 2πH^{n-2}(Γ)/(2-2s), expansion (4.22) is not proven as written. The gap is likely repairable, for example by applying the fiberwise [CFP24] lower bound to us(·,y) and integrating over Γ using the spectral identity (4.33), but that argument must be supplied.
  2. [§4.1, Proposition 4.1] The identification [ω2]=[d*ψ] is asserted with the phrase 'one sees that' after invoking the degree condition and properties of the distributional degree. This step is load-bearing because Proposition 4.1 justifies the reduction of the tube-removed minimisation (4.5) to the sectors W^{1,2}_α, and the same sector decomposition is reused in §4.2–4.3 for the p- and fractional relaxations. Please provide the full cohomological argument, including the relation between deg(u,Γ), the class of ω2 in H1(M\Γ), and the class of d*ψ.
minor comments (4)
  1. [§4.3] In the double minimisation display, 'min_{u∈Hs_α} Ep(u)' should read 'min_{u∈Hs_α} Es(u)', since the surrounding discussion concerns the fractional energy Es.
  2. [§4.5] Just before Eq. (4.31), the sentence 'By the degree condition on us, if follows' contains a typo; it should be 'it follows'.
  3. [Lemma 3.3(4)] The existence of the model map u* with u*|Tδ = z/|z| is imported from [BD24, Appendix A] without stating the precise hypotheses; adding a short lemma stating the model-map result would improve self-containedness.
  4. [Lemma 3.4] The cut-off ξ and the radius δ0 in the definition of S should be tied explicitly to the Fermi-coordinate tube; as written, the choice of cut-off is implicit and the invariance of WM(Γ) under this choice deserves at least a sentence.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the renormalised energy and the Theorem B expansions are derived by explicit computation and comparison, not assumed; the only flagged weakness is a non-circular proof gap in the fractional lower bound.

full rationale

I walked the derivation chain from W_M(Γ) through Theorem B. W_M(Γ) is introduced in §3.1 as the desingularisation of the Green-potential integral (3.13), namely W_M(Γ) := -2π∫Γ R with R = A + S, and is then shown, not assumed, to appear in expansion (3.15) via Stokes' theorem and Lemma 3.5; the leading logarithmic term comes from S and the finite part from R. The tube-removed, p-energy and fractional expansions are obtained by comparison with (3.15)/(4.13) through minimality, interpolation and regularity estimates, so the common constant 2πH^{n-2}(Γ) and the interaction energy W_M(Γ) are not imported from the definitions of the relaxations. The only self-citation entering the proof, [BD24, Appendix A], is used in Lemma 3.3(4) solely to supply a local model map u∗ with u∗|_{Tδ} = z/|z|; that is an external construction in the same authors' published work and is not equivalent to Theorem A or Theorem B. The reviewer-noted weakness in §4.5 is real but is not circular: the assertion that the Γ-average ar u_s inherits degree +1 is an unproved mathematical claim and a potential gap in establishing (4.31), not a reduction of the desired energy constant to an input of the argument. Thus no circular step can be exhibited, and the circularity score is 1 rather than 4 or higher.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central construction uses standard Hodge theory and geometric measure theory; the only domain-specific input is that the prescribed singular set is an admissible integral current that is a boundary, and, for the energy part, that it is C^{1,1} embedded. No free parameters are fitted to data. The paper relies on the author's earlier work [BD24] for a model map and on [CFP24] for fractional estimates; neither contains the target results.

assumptions (6)
  • standard math Standard Hodge decomposition and the existence of the Hodge-Laplacian Green's function on closed Riemannian manifolds (Section 2.1).
    Used repeatedly to define ψ and the renormalised energy; this is a standard background tool.
  • domain assumption The prescribed singular set Γ is an admissible current, i.e. Γ = ∂S for an integral (n-1)-current S (Definition 3).
    This boundary condition is necessary for the existence of an S1-valued map with that singular set, and it is needed to solve ΔA = 2π⟦Γ⟧ in Lemma 3.1.
  • domain assumption Γ is an embedded C^{1,1} submanifold for the renormalised energy results.
    Fermi coordinates, the tubular neighbourhood estimates, and the regularity of the regular part R in Lemma 3.4 all require a C^{1,1} embedded submanifold.
  • domain assumption The distributional Jacobian J u tracks exactly the prescribed singular set, in the sense of Alberti-Baldo-Orlandi [ABO03] and Brezis-Mironescu [BM21].
    The paper's notion of 'singular set' for W^{1,p} and H^s maps is defined through the distributional Jacobian, and the equivalence with the degree around Γ is imported from the cited literature.
  • ad hoc to paper Existence of a smooth model map u* with u*|Tδ = z/|z| in Fermi coordinates, taken from Badran-Del Pino [BD24, Appendix A].
    This is a self-cited auxiliary construction used in Lemma 3.3(4) and in the comparison arguments; it is a tool, not the target result.
  • domain assumption Fractional H^s lower bounds for degree-one maps on the disk are taken from Caselli-Freguglia-Picenni [CFP24, Lemma 3.6, Proposition 4.1, Lemma A.1].
    Used directly in the fractional relaxation expansion (4.31) and (4.32); the paper relies on external estimates for the disk-to-circle fractional energy.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Harmonic maps to the circle with higher dimensional singular set." pith.science (2026). https://pith.science/paper/V3EUYTTL

@misc{pith2026241114186,
  author       = {Pith},
  title        = {Pith review of: Harmonic maps to the circle with higher dimensional singular set},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3EUYTTL}},
  note         = {Machine review of arXiv:2411.14186}
}
abstract

In a closed, oriented ambient manifold $(M^n,g)$ we consider the problem of finding $\mathbb{S}^1$-valued harmonic maps with prescribed singular set. We show that the boundary of any oriented $(n-1)$-submanifold can be realised as the singular set of an $\mathbb{S}^1$-valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set $\Gamma$ is also embedded and $C^{1,1}$, we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume $\mathcal{H}^{n-2}(\Gamma)$ plus a lower-order "renormalised energy" -- common to all relaxations -- describing an energetic interaction between different components of the singular set.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 35 canonical work pages

  1. [1]

    R. A. Adams and J. Fournier , Cone conditions and properties of S obolev spaces, J. Math. Anal. Appl. 61 no. 3 (1977), 713--734. 463902 . doi:10.1016/0022-247X(77)90173-1

    doi:10.1016/0022-247x(77)90173-1 1977

  2. [2]

    Alberti , S

    G. Alberti , S. Baldo , and G. Orlandi , Functions with prescribed singularities, J. Eur. Math. Soc. (JEMS) 5 no. 3 (2003), 275--311. 2002215 . doi:10.1007/s10097-003-0053-5

    doi:10.1007/s10097-003-0053-5 2003

  3. [3]

    Alberti , S

    G. Alberti , S. Baldo , and G. Orlandi , Variational convergence for functionals of G inzburg- L andau type, Indiana Univ. Math. J. 54 no. 5 (2005), 1411--1472. 2177107 . doi:10.1512/iumj.2005.54.2601

    work page doi:10.1512/iumj.2005.54.2601 2005

  4. [4]

    Badran and M

    M. Badran and M. Del Pino , Entire solutions to 4 dimensional G inzburg- L andau equations and codimension 2 minimal submanifolds, Adv. Math. 435 no. part A (2023), Paper No. 109365, 73. 4658832 . doi:10.1016/j.aim.2023.109365

    work page arXiv 2023

  5. [5]

    Badran and M

    M. Badran and M. Del Pino , Solutions of the G inzburg- L andau equations concentrating on codimension-2 minimal submanifolds, J. Lond. Math. Soc. (2) 109 no. 1 (2024), Paper No. e12851, 31. 4754434 . doi:10.1112/jlms.12851

    work page doi:10.1112/jlms.12851 2024

  6. [6]

    Bethuel , H

    F. Bethuel , H. Brezis , and F. H\' e lein , Asymptotics for the minimization of a G inzburg- L andau functional, Calc. Var. Partial Differential Equations 1 no. 2 (1993), 123--148. 1261720 . doi:10.1007/BF01191614

    doi:10.1007/bf01191614 1993

  7. [7]

    Bethuel , H

    F. Bethuel , H. Brezis , and F. H\' e lein , Ginzburg- L andau vortices , Progress in Nonlinear Differential Equations and their Applications 13, Birkh\" a user Boston, Inc., Boston, MA, 1994. 1269538 . doi:10.1007/978-1-4612-0287-5

    doi:10.1007/978-1-4612-0287-5 1994

  8. [8]

    Brezis and P

    H. Brezis and P. Mironescu , Sobolev maps to the circle---from the perspective of analysis, geometry, and topology, Progress in Nonlinear Differential Equations and their Applications 96, Birkh\" a user/Springer, New York, [2021] 2021. 4390036 . doi:10.1007/978-1-0716-1512-6

    doi:10.1007/978-1-0716-1512-6 2021

Show all 60 references
  1. [9]

    Caffarelli , J.-M

    L. Caffarelli , J.-M. Roquejoffre , and O. Savin , Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 no. 9 (2010), 1111--1144. 2675483 . doi:10.1002/cpa.20331

    2010 doi

  2. [10]

    Caselli , E

    M. Caselli , E. Florit-Simon , and J. Serra , Fractional S obolev spaces on R iemannian manifolds, 2024. Available at https://arxiv.org/abs/2402.04076

    2024 arXiv

  3. [11]

    Caselli , E

    M. Caselli , E. Florit-Simon , and J. Serra , Yau's conjecture for nonlocal minimal surfaces, 2024. Available at https://arxiv.org/abs/2306.07100

    2024 arXiv

  4. [12]

    Caselli , M

    M. Caselli , M. Freguglia , and N. Picenni , A nonlocal approximation of the area in codimension two, 2024. Available at https://arxiv.org/abs/2406.13696

    2024 arXiv

  5. [13]

    Chan , S

    H. Chan , S. Dipierro , J. Serra , and E. Valdinoci , Nonlocal approximation of minimal surfaces: optimal estimates from stability, 2023. Available at https://arxiv.org/abs/2308.06328

    2023

  6. [14]

    Chodosh and C

    O. Chodosh and C. Mantoulidis , Minimal surfaces and the A llen- C ahn equation on 3-manifolds: index, multiplicity, and curvature estimates, Ann. of Math. (2) 191 no. 1 (2020), 213--328. 4045964 . doi:10.4007/annals.2020.191.1.4

    2020 doi

  7. [15]

    Colinet , R

    A. Colinet , R. L. Jerrard , and P. Sternberg , Solutions of the G inzburg- L andau equations with vorticity concentrating near a nondegenerate geodesic, 2021. Available at https://arxiv.org/abs/2101.03575

    2021 arXiv

  8. [16]

    Contreras and R

    A. Contreras and R. L. Jerrard , Nearly parallel vortex filaments in the 3 D G inzburg- L andau equations, Geom. Funct. Anal. 27 no. 5 (2017), 1161--1230. 3714719 . doi:10.1007/s00039-017-0425-8

    2017 doi

  9. [17]

    Da Lio and T

    F. Da Lio and T. Rivi\`ere , Three-term commutator estimates and the regularity of 12 -harmonic maps into spheres, Anal. PDE 4 no. 1 (2011), 149--190. 2783309 . doi:10.2140/apde.2011.4.149

    2011 doi

  10. [18]

    D\' a vila , M

    J. D\' a vila , M. Del Pino , M. Medina , and R. Rodiac , Interacting helical vortex filaments in the three-dimensional G inzburg- L andau equation, J. Eur. Math. Soc. (JEMS) 24 no. 12 (2022), 4143--4199. 4493621 . doi:10.4171/jems/1175

    2022 doi

  11. [19]

    D\' a vila , M

    J. D\' a vila , M. Del Pino , and J. Wei , Nonlocal s -minimal surfaces and L awson cones, J. Differential Geom. 109 no. 1 (2018), 111--175. 3798717 . doi:10.4310/jdg/1525399218

    2018

  12. [20]

    De Philippis and A

    G. De Philippis and A. Pigati , Non-degenerate minimal submanifolds as energy concentration sets: a variational approach, Comm. Pure Appl. Math. 77 no. 8 (2024), 3581--3627. 4764749

    2024

  13. [21]

    Del Pino and M

    M. Del Pino and M. Kowalczyk , Renormalized energy of interacting G inzburg- L andau vortex filaments, J. Lond. Math. Soc. (2) 77 no. 3 (2008), 647--665. 2418297 . doi:10.1112/jlms/jdm126

    2008 doi

  14. [22]

    Fern\' a ndez-Real and X

    X. Fern\' a ndez-Real and X. Ros-Oton , Regularity theory for elliptic PDE , Zurich Lectures in Advanced Mathematics 28, EMS Press, Berlin, [2022] 2022. 4560756 . doi:10.4171/zlam/28

    2022 doi

  15. [23]

    Florit-Simon , Weyl L aw and convergence in the classical limit for min-max nonlocal minimal surfaces, 2024

    E. Florit-Simon , Weyl L aw and convergence in the classical limit for min-max nonlocal minimal surfaces, 2024. Available at https://arxiv.org/abs/2406.12162

    2024

  16. [24]

    Gaspar and M

    P. Gaspar and M. A. M. Guaraco , The A llen- C ahn equation on closed manifolds, Calc. Var. Partial Differential Equations 57 no. 4 (2018), Paper No. 101, 42. 3814054 . doi:10.1007/s00526-018-1379-x

    2018 doi

  17. [25]

    Gray , Tubes, second ed., Progress in Mathematics 221, Birkh\" a user Verlag, Basel, 2004, With a preface by Vicente Miquel

    A. Gray , Tubes, second ed., Progress in Mathematics 221, Birkh\" a user Verlag, Basel, 2004, With a preface by Vicente Miquel. 2024928 . doi:10.1007/978-3-0348-7966-8

    2004 doi

  18. [26]

    Grey and G

    W. Grey and G. Sinnamon , The inclusion problem for mixed norm spaces, Trans. Amer. Math. Soc. 368 no. 12 (2016), 8715--8736. 3551586 . doi:10.1090/tran6665

    2016 doi

  19. [27]

    M. A. M. Guaraco , Min-max for phase transitions and the existence of embedded minimal hypersurfaces, J. Differential Geom. 108 no. 1 (2018), 91--133. 3743704 . doi:10.4310/jdg/1513998031

    2018

  20. [28]

    Hardt and F.-H

    R. Hardt and F.-H. Lin , Mappings minimizing the L^p norm of the gradient, Comm. Pure Appl. Math. 40 no. 5 (1987), 555--588. 896767 . doi:10.1002/cpa.3160400503

    1987 doi

  21. [29]

    Hardt and F

    R. Hardt and F. Lin , Singularities for p -energy minimizing unit vectorfields on planar domains, Calc. Var. Partial Differential Equations 3 no. 3 (1995), 311--341. 1385291 . doi:10.1007/BF01189395

    1995 doi

  22. [30]

    Hardt , F

    R. Hardt , F. Lin , and C. Wang , Singularities of p -energy minimizing maps, Comm. Pure Appl. Math. 50 no. 5 (1997), 399--447. 1443054 . doi:10.1002/(SICI)1097-0312(199705)50:5<399::AID-CPA1>3.0.CO;2-4

    1997 doi

  23. [31]

    Hatcher , Algebraic topology, Cambridge University Press, Cambridge, 2002

    A. Hatcher , Algebraic topology, Cambridge University Press, Cambridge, 2002. 1867354

    2002

  24. [32]

    J. E. Hutchinson and Y. Tonegawa , Convergence of phase interfaces in the van der W aals- C ahn- H illiard theory, Calc. Var. Partial Differential Equations 10 no. 1 (2000), 49--84. 1803974 . doi:10.1007/PL00013453

    2000 doi

  25. [33]

    Ignat and R

    R. Ignat and R. L. Jerrard , Renormalized energy between vortices in some G inzburg- L andau models on 2-dimensional R iemannian manifolds, Arch. Ration. Mech. Anal. 239 no. 3 (2021), 1577--1666. 4215198 . doi:10.1007/s00205-020-01598-0

    2021 doi

  26. [34]

    J. D. Jackson , Classical electrodynamics, second ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. 436782

    1975

  27. [35]

    R. L. Jerrard and H. M. Soner , Functions of bounded higher variation, Indiana Univ. Math. J. 51 no. 3 (2002), 645--677. 1911049 . doi:10.1512/iumj.2002.51.2229

    2002 doi

  28. [36]

    R. L. Jerrard and H. M. Soner , The J acobian and the G inzburg- L andau energy, Calc. Var. Partial Differential Equations 14 no. 2 (2002), 151--191. 1890398 . doi:10.1007/s005260100093

    2002 doi

  29. [37]

    R. L. Jerrard and H. M. Soner , Limiting behavior of the G inzburg- L andau functional, J. Funct. Anal. 192 no. 2 (2002), 524--561. 1923413 . doi:10.1006/jfan.2001.3906

    2002

  30. [38]

    Y. Liu , X. Ma , J. Wei , and W. Wu , Entire solutions of the magnetic G inzburg- L andau equation in R ^4 , 2021. Available at https://arxiv.org/abs/2108.02754

    2021 arXiv

  31. [39]

    Y. Liu , J. Wei , and Z. Ye , Stable solutions of u(1) yang-mills-higgs model in R ^4 , 2024. Available at https://arxiv.org/abs/2411.05447

    2024 arXiv

  32. [40]

    Marx-Kuo , Second inner variations of energy and index of codimension 2 minimal submanifolds, 2023

    J. Marx-Kuo , Second inner variations of energy and index of codimension 2 minimal submanifolds, 2023. Available at https://arxiv.org/abs/2307.09733

    2023 arXiv

  33. [41]

    J. Mesaric , Existence of critical points for the G inzburg- L andau functional on R iemannian manifolds , ProQuest LLC, Ann Arbor, MI, 2009, Thesis (Ph.D.)--University of Toronto (Canada). 2753145 . Available at http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_...

    2009

  34. [42]

    Monteil , R

    A. Monteil , R. Rodiac , and J. Van Schaftingen , Ginzburg- L andau relaxation for harmonic maps on planar domains into a general compact vacuum manifold, Arch. Ration. Mech. Anal. 242 no. 2 (2021), 875--935. 4331018 . doi:10.1007/s00205-021-01695-8

    2021 doi

  35. [43]

    Monteil , R

    A. Monteil , R. Rodiac , and J. Van Schaftingen , Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains, Math. Ann. 383 no. 3-4 (2022), 1061--1125. 4458397 . doi:10.1007/s00208-021-02204-8

    2022 doi

  36. [44]

    Parise , A

    D. Parise , A. Pigati , and D. Stern , Convergence of the self-dual U(1) - Y ang- M ills- H iggs energies to the (n-2) -area functional, Comm. Pure Appl. Math. 77 no. 1 (2024), 670--730. 4666634 . doi:10.1002/cpa.22150

    2024 doi

  37. [45]

    Petersen , Riemannian geometry, second ed., Graduate Texts in Mathematics 171, Springer, New York, 2006

    P. Petersen , Riemannian geometry, second ed., Graduate Texts in Mathematics 171, Springer, New York, 2006. 2243772

    2006

  38. [46]

    Pigati and D

    A. Pigati and D. Stern , Minimal submanifolds from the abelian H iggs model, Invent. Math. 223 no. 3 (2021), 1027--1095. 4213771 . doi:10.1007/s00222-020-01000-6

    2021 doi

  39. [47]

    Pigati and D

    A. Pigati and D. Stern , Quantization and non-quantization of energy for higher-dimensional G inzburg- L andau vortices, Ars Inven. Anal. (2023), Paper No. 3, 55. 4603940

    2023

  40. [48]

    Rivi\`ere , Line vortices in the U (1) - H iggs model, ESAIM Contr\^ o le Optim

    T. Rivi\`ere , Line vortices in the U (1) - H iggs model, ESAIM Contr\^ o le Optim. Calc. Var. 1 (1995/96), 77--167. 1394302 . doi:10.1051/cocv:1996103

    1995 doi

  41. [49]

    Schwarz , Hodge decomposition---a method for solving boundary value problems, Lecture Notes in Mathematics 1607, Springer-Verlag, Berlin, 1995

    G. Schwarz , Hodge decomposition---a method for solving boundary value problems, Lecture Notes in Mathematics 1607, Springer-Verlag, Berlin, 1995. 1367287 . doi:10.1007/BFb0095978

    1995 doi

  42. [50]

    Scott , L^p theory of differential forms on manifolds, Trans

    C. Scott , L^p theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 no. 6 (1995), 2075--2096. 1297538 . doi:10.2307/2154923

    1995 doi

  43. [51]

    Serra , Nonlocal minimal surfaces: recent developments, applications, and future directions, SeMA J

    J. Serra , Nonlocal minimal surfaces: recent developments, applications, and future directions, SeMA J. 81 no. 2 (2024), 165--191. 4743530 . doi:10.1007/s40324-023-00345-1

    2024 doi

  44. [52]

    L. Simon , Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. 756417

    1983

  45. [53]

    Stefanov and R

    A. Stefanov and R. H. Torres , Calder\' o n- Z ygmund operators on mixed L ebesgue spaces and applications to null forms, J. London Math. Soc. (2) 70 no. 2 (2004), 447--462. 2078904 . doi:10.1112/S0024610704005502

    2004 doi

  46. [54]

    Stern , p -harmonic maps to S^1 and stationary varifolds of codimension two, Calc

    D. Stern , p -harmonic maps to S^1 and stationary varifolds of codimension two, Calc. Var. Partial Differential Equations 59 no. 6 (2020), Paper No. 187, 47. 4160864 . doi:10.1007/s00526-020-01859-6

    2020 doi

  47. [55]

    Stern , Existence and limiting behavior of min-max solutions of the G inzburg- L andau equations on compact manifolds, J

    D. Stern , Existence and limiting behavior of min-max solutions of the G inzburg- L andau equations on compact manifolds, J. Differential Geom. 118 no. 2 (2021), 335--371. 4278697 . doi:10.4310/jdg/1622743143

    2021

  48. [56]

    Struwe , On the asymptotic behavior of minimizers of the G inzburg- L andau model in 2 dimensions, Differential Integral Equations 7 no

    M. Struwe , On the asymptotic behavior of minimizers of the G inzburg- L andau model in 2 dimensions, Differential Integral Equations 7 no. 5-6 (1994), 1613--1624. 1269674

    1994

  49. [57]

    Tonegawa , On stable critical points for a singular perturbation problem, Comm

    Y. Tonegawa , On stable critical points for a singular perturbation problem, Comm. Anal. Geom. 13 no. 2 (2005), 439--459. 2154826 . Available at http://projecteuclid.org/euclid.cag/1117656992

    2005

  50. [58]

    Tonegawa and N

    Y. Tonegawa and N. Wickramasekera , Stable phase interfaces in the van der W aals-- C ahn-- H illiard theory, J. Reine Angew. Math. 668 (2012), 191--210. 2948876 . doi:10.1515/crelle.2011.134

    2012 doi

  51. [59]

    Van Schaftingen and B

    J. Van Schaftingen and B. Van Vaerenbergh , Asymptotic behavior of minimizing p -harmonic maps when p 2 in dimension 2, Calc. Var. Partial Differential Equations 62 no. 8 (2023), Paper No. 229, 45. 4642640 . doi:10.1007/s00526-023-02568-6

    2023 doi

  52. [60]

    F. W. Warner , Foundations of differentiable manifolds and L ie groups , Graduate Texts in Mathematics 94, Springer-Verlag, New York-Berlin, 1983, Corrected reprint of the 1971 edition. 722297

    1983

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.