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A $G_2$-Hilbert functional in $G_2$-geometry

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Pith's one-line read This paper introduces the G2-Hilbert functional and proves it uniquely determines two Ricci-flow analogues on 7-manifolds.

desk verdict A genuinely new G2-Hilbert functional and two natural flows, with a coherent uniqueness argument, but the advertised saddle-point theorem is not established as printed: the K2 formula in Proposition 6.9 has a factor-three error and the spectral conclusions contradict each other. read the letter →

arxiv 2505.06872 v1 pith:A6H2IQV6 submitted 2025-05-11 math.DG

Panagiotis Gianniotis , George Zacharopoulos This is my paper · ORCID
classification math.DG MSC 53C1053C4453C2553C29
keywords G2-structuresG2-HilbertfunctionalspecialRicci-likeoperatorsgeometricflowstorsionnearlyG2structuresRicciflowanaloguesecondvariation
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

On a 7-manifold, a $G_2$-structure is a special 3-form that determines both a Riemannian metric and an orientation; its torsion $T$ measures how far it is from being integrable. This paper introduces the $G_2$-Hilbert functional $F(\varphi) = \int_M \left(\tfrac{1}{6}\,\mathrm{Scal} - \tfrac{1}{3}\,|T|^2 - \tfrac{1}{6}\,(\mathrm{tr}\,T)^2\right)d\mu$, and argues it is the unique functional that is linear in scalar curvature and quadratic in torsion and whose first variation has the shape of a special Ricci-like operator, the $G_2$ counterpart of the Ricci tensor. It then shows that torsion-free structures ($T = 0$) and nearly $G_2$ structures ($T = c\,g$) are saddle critical points of the volume-normalized functional, exactly as Einstein metrics saddle the Einstein–Hilbert functional. From the first variation the paper extracts two operators, $\hat P$ and $\tilde P$, and proposes the flows $\partial\varphi/\partial t = \hat P(\varphi)$ and $\partial\varphi/\partial t = \tilde P(\varphi)$ as $G_2$ analogues of the Ricci flow. If the argument is right, these flows fill the gap left by the Laplacian flow, which is only parabolic for closed $G_2$-structures, and provide the first variational justification for Ricci-flow-like evolutions of general $G_2$-structures.

What carries the argument

The mechanism is the class of special Ricci-like operators: second-order quasilinear operators on $G_2$-structures of the form $P(\varphi) = (-\mathrm{Ric} + a\,L_{VT}g + \mathrm{lots})\diamond\varphi + ((1+a)\,\mathrm{div}\,T - a\,\nabla\mathrm{tr}\,T + \mathrm{lots})\lrcorner\psi$, whose principal symbol, by Proposition 2.3 (from the companion classification [11]), acts as $|\xi|^2$ on the kernel of a Bianchi-type operator $\tilde B$ and vanishes only on diffeomorphism directions — a symbolic profile identical to that of the Ricci tensor on metrics. The argument runs on three interlocking pieces: the six-operator classification of [11], which fixes the admissible second-order building blocks ($\mathrm{Ric}$, $\mathrm{Scal}\,g$, $L_{VT}g$, $F$, $\mathrm{div}\,T$, $\nabla\mathrm{tr}\,T$); the diffeomorphism-invariance system (3.3) and the algebraic system (4.6), which force the coefficients $a = -1/3$, $\beta = 1/6$ and pin down the density $F$ uniquely; and the elliptic decomposition $T_\varphi\Omega^3_+ \cong \{(fg,0)\} \oplus \mathrm{Im}\,K \oplus (\ker L \cap \ker \mathrm{tr})$ built from the $G_2$-conformal Killing operator $K$, which splits the tangent space and organizes the second-variation computation at torsion-free and nearly $G_2$ structures.

What would settle it

Test the completeness of the classification: enumerate second-order quasilinear operators on $G_2$-structures whose linearizations have linearly independent principal symbols; finding a seventh such operator beyond $\mathrm{Ric}$, $\mathrm{Scal}\,g$, $L_{VT}g$, $F$, $\mathrm{div}\,T$, $\nabla\mathrm{tr}\,T$ falsifies the uniqueness of the $G_2$-Hilbert functional. Alternatively, compute the first variation of $F$ against a concrete one-parameter family of $G_2$-structures with nonzero torsion and check that the second-order terms of the gradient match the system (4.6); any mismatch in the coefficients would also settle the claim.

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Extended reading notes

Core claim

The paper's central claim is that a single functional, the $G_2$-Hilbert functional $F(\varphi) = \int_M \left(\tfrac{1}{6}\,\mathrm{Scal} - \tfrac{1}{3}\,|T|^2 - \tfrac{1}{6}\,(\mathrm{tr}\,T)^2\right)d\mu$, is the unique diffeomorphism-invariant functional on the space of $G_2$-structures that is linear in the scalar curvature and quadratic in the torsion and whose gradient, up to divergence terms, is a special Ricci-like operator. The uniqueness computation forces the coefficients $a = -1/3$ and $\beta = 1/6$ in the candidate operator $(-\mathrm{Ric} + a\,L_{VT}g)\diamond\varphi + ((1+a)\,\mathrm{div}\,T - a\,\nabla\mathrm{tr}\,T)\lrcorner\psi$, and the same computation singles out the density $\tfrac{1}{6}\,\mathrm{Scal} - \tfrac{1}{3}\,|T|^2 - \tfrac{1}{6}\,(\mathrm{tr}\,T)^2$. The volume-normalized functional $\tilde F(\varphi) = \mathrm{Vol}(g)^{-5/7}\,F(\varphi)$ has the torsion-free structures and the nearly $G_2$ structures ($T = c\,g$) as critical points, and the second variation (Theorem 6.1) shows these are saddle points: positive definite along the conformal class, with infinitely many negative directions among $G_2$-transverse-traceless deformations. This variational rigidity is used to distinguish two operators, $\hat P$ and $\tilde P$, whose flows are presented as the $G_2$ analogues of the Ricci flow; torsion-free structures are fixed points, nearly $G_2$ structures expand homothetically along both flows, and the induced metric evolution is the Ricci flow modified by diffeomorphisms and a term proportional to $L_{VT}g$.

Load-bearing premise

The entire uniqueness result and the well-posedness of the two flows rest on the completeness and correctness of the companion classification of second-order quasilinear operators on $G_2$-structures (reference [11]): if that classification missed an operator, or if its principal-symbol or variation formulas contain a sign error, the forced coefficient $a = -1/3$ and the parabolicity of $\hat P$ and $\tilde P$ would both collapse.

Editorial extensions

If this is right

  • Torsion-free $G_2$-structures are fixed points of both flows, while nearly $G_2$ structures expand homothetically: $\varphi(t) = (c_0^2 t + 1)^3 \varphi_0$ for (4.13) and $\varphi(t) = ((10/3)c_0^2 t + 1)^3 \varphi_0$ for (4.14).
  • Along any flow generated by a special Ricci-like operator, $\left(\tfrac{\partial}{\partial t} - \Delta\right)\mathrm{Scal} = \mathrm{lots}$ and $\left(\tfrac{\partial}{\partial t} - \Delta\right)T_{ij} = \mathrm{lots}$, so the scalar curvature and torsion evolve by heat-type equations modulo lower-order terms.
  • Modulo diffeomorphisms, both flows couple the Ricci flow with the isometric flow $\partial\varphi/\partial t = \mathrm{div}\,T \lrcorner \psi$, giving the first variational origin for Ricci-flow-like $G_2$ evolutions.
  • The saddle-point behavior mirrors the Einstein–Hilbert case: the conformal-direction second variation is the strictly positive operator $-6\Delta f + 35c^2 f$, while infinitely many $G_2$-transverse-traceless directions are negative, so minimizing $\tilde F$ cannot directly produce torsion-free or nearly $G_2$ structures.

Reading between the lines

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

  • If a later computation shows $\hat P$ or $\tilde P$ is the Ricci tensor of a canonical connection induced by the $G_2$-structure, the analogy with the Hermitian curvature flow would become exact; the paper explicitly leaves this open.
  • The sign flip — nearly $G_2$ structures expand homothetically, unlike positive-scalar-curvature Einstein metrics under Ricci flow — suggests genuinely $G_2$-specific soliton behavior that could be probed first on symmetric examples such as 3-Sasakian or homogeneous 7-manifolds.
  • The uniqueness mechanism is modular: if a similar operator classification were available for $\mathrm{Spin}(7)$-structures, the same derivation would likely produce candidate Hilbert-type functionals and flows for that geometry without a separate variational search.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper introduces a new functional on the space of G2-structures, F(φ)=∫(1/6 Scal − 1/3 |T|^2 − 1/6 (trT)^2) dμ, called the G2-Hilbert functional. It proves that this functional is uniquely determined by principles inspired by the Einstein–Hilbert functional: linearity in scalar curvature, quadratic dependence on torsion, and a special Ricci-like gradient. It then defines two flows, ∂φ/∂t=P̂(φ) and ∂φ/∂t=P̃(φ), presented as G2 analogues of the Ricci flow, and claims that torsion-free and nearly G2 structures are saddle points of the volume-normalized functional. The central variational identity (Proposition 4.1) and the uniqueness system (Section 4.1) are developed in detail, but the saddle-point analysis in Section 6 contains internal inconsistencies that affect the main advertised theorem.

Significance. If correct, the G2-Hilbert functional would provide a natural variational principle in G2-geometry, with a uniqueness statement that sharply distinguishes two candidate flows and an explicit nearly-G2 homothetic solution along both flows. The exposition of the first variation and the uniqueness argument are genuine strengths, and the paper gives concrete, falsifiable predictions about the behaviour of nearly G2 structures. However, the saddle-point theorem, which is one of the paper's central claims, is not established as printed because of arithmetic and spectral inconsistencies in Proposition 6.9. The paper also relies heavily on the unpublished preprint [11] for the classification of second-order operators, principal-symbol statements, and variation formulas that underpin the uniqueness and parabolicity claims.

major comments (3)
  1. [Section 6.4, Proposition 6.9] The displayed derivation of K2 is arithmetically inconsistent. For a transverse-traceless variation with L(h,X)=0 and trh=0, equation (6.6) gives divh = −1/2 curl(X) + cX, hence −3c divh = (3c/2)curl(X) − 3c^2 X. Substituting into K2 yields K2 = ∆X + (3c/2 + 11c/3) curl(X) + 9c^2 X = ∆X + (31c/6) curl(X) + 9c^2 X. The proposition instead prints 31c/2, which is a factor-three error. This error is load-bearing because the subsequent projection argument uses the printed coefficient to derive the projected operator B: the manipulation with K(−5c/3 X) produces the stated coefficients 49c/3 and 22c^2/3 only from the incorrect 31c/2 term. With the correct coefficient 31c/6, the remainder 67c/6 curl(X) − 5c^2/3 X is not in the image of K, so the displayed formula for B does not follow.
  2. [Section 6.4, Proposition 6.9] The spectral conclusions are self-contradictory and the saddle argument is unsupported as written. The proposition states that B has real spectrum consisting of a sequence λ_i → −∞ and then immediately states that 'there are only finitely many negative eigenvalues.' If λ_i → −∞, there are infinitely many negative eigenvalues, so the two assertions are incompatible. The final step of the saddle argument requires the finite-dimensional subspace to be the positive eigenspace, so the clause should presumably read 'only finitely many positive eigenvalues.' This is not a purely cosmetic issue: the sign of the eigenvalues in the transverse-traceless directions is exactly what determines whether φ is a saddle point rather than a local minimum or maximum. Until the arithmetic behind B and the spectral statement are corrected, Theorem 6.1's saddle-point assertion is not established.
  3. [Section 2.5 and Proposition 3.3] Several load-bearing results are quoted from the unpublished preprint [11] without proof: the classification of the six second-order quasilinear operators, Proposition 2.3 on principal symbols, Corollary 5.35 giving the variation formulas used in Proposition 3.1, and Theorem 1.1 on short-time existence. The uniqueness system (4.6) and the parabolicity interpretation of the flows (4.13)–(4.14) both depend on this classification and on the principal-symbol formula. As submitted, the manuscript cannot be fully verified without access to [11]. Please either include the precise statements and proofs of these results, or make the dependence on [11] explicit and ensure that reference is publicly available and refereed.
minor comments (3)
  1. [Section 6.4, proof of Proposition 6.9] In the computation of ∂/∂t (F(φ_t)g_t), the letter f appears in the term −7c^2 f g although no function f has been introduced in the transverse-traceless setting and h is trace-free. The intended term should involve h (or the computation should be carried out directly from the variation of the normalized functional); this appears to be a local typographical error.
  2. [Section 4.1, uniqueness assumptions] The displayed form of P2 in Assumption 2 is written as (1+a)divT + a∇trT, whereas Definition 2.2 and Proposition 3.3 use (1+a)divT − a∇trT. The final value a = −1/3 makes the two expressions coincide, but the sign inconsistency should be fixed for clarity.
  3. [Section 6.4, Proposition 6.9] The phrase 'which has infinitely many negative eigenvalues' appears before the contradictory 'only finitely many negative eigenvalues' sentence. Whichever statement is intended, the two should be reconciled so that the eigenvalue count for B is unambiguous.

Circularity Check

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No circularity: the G2-Hilbert functional and its flows are derived from explicitly stated axiomatic principles; reliance on the authors' classification in [11] is load-bearing but independent, so it does not make the derivation circular.

full rationale

The central construction is axiomatic rather than fitted. Section 4.1 states the assumptions (F linear in scalar curvature and quadratic in torsion; the gradient is a special Ricci-like operator), expands the most general such quantity as a linear combination of Scal, (trT)^2, |T|^2, <T,T^t>+<T,P(T)>, |VT|^2, and solves the resulting system (4.6). The unique solution is exhibited in the text, and the apparent delta-parameter is killed by identity (2.11), an independent contraction identity. No parameter is fitted to the saddle-point conclusion. The classification of second-order quasilinear operators, the principal-symbol formula, and the first-variation formulas are imported from [11], which shares an author with this paper. That self-citation is load-bearing, but [11] states parameter-free classification and variation theorems whose assumptions do not include the G2-Hilbert functional, the two flows, or the saddle-point result, so it counts as independent mathematical support rather than a circular step. The saddle-point theorem is then a genuine second-variation computation, not an unpacking of a definition. The apparent arithmetic inconsistencies in Proposition 6.9 (31c/2 versus 31c/6, and the simultaneous statement that B has infinitely many negative eigenvalues and only finitely many) are correctness defects that would invalidate the proof as printed, but they are not circularity: they do not reduce the claimed theorem to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No empirical free parameters appear; the coefficients (1/6, -1/3, -1/6) in the functional are derived from the axioms, not fitted. The paper's external load is the classification and existence theorems from the self-cited reference [11], and the axioms that define what counts as a 'Ricci-like' gradient.

assumptions (4)
  • domain assumption The classification and symbol analysis of second-order quasilinear operators on G2 structures in [11] (Theorem 6.76, Proposition 6.42, Corollary 5.35) are correct and complete.
    The uniqueness system (4.6) and the principal-symbol property in Proposition 2.3 are taken from [11], which is a preprint by the first author and collaborators. If this classification misses operators or has an error, the uniqueness and the parabolicity of the flows would fail.
  • domain assumption The tangent-space decompositions T_φ Ω³_+ = Im L* ⊕ ker L and the refinement using the G2 conformal Killing operator K (Propositions 3.2 and 5.2) hold, and the operators L∘L* and L∘K are elliptic.
    These decompositions are used to isolate conformal and transverse-traceless directions in the second variation computation and to prove the saddle-point behavior. They rely on standard elliptic operator theory and the structure of the differential operators involved.
  • ad hoc to paper The 'special Ricci-like operator' condition (Definition 2.2) is the correct G2 analogue of the Ricci tensor, and the principles in Section 4 (linearity in scalar curvature, quadratic torsion, special-Ricci-like gradient) are well-posed.
    The axioms are chosen to mirror the Einstein-Hilbert functional and the Bianchi identity in Riemannian geometry, but they are not forced by external data. They are natural yet tailored to the authors' goal of reproducing known G2 phenomena.
  • standard math Standard G2 identities, including the contraction identities (2.2)-(2.4), the G2-Bianchi identity (2.9), and the torsion relation T = cg for nearly G2 structures, are assumed.
    These are background facts from G2 geometry, mostly from Karigiannis [22] and Dwivedi-Gianniotis-Karigiannis [11], used throughout the derivations.

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Pith. "Pith review of A $G_2$-Hilbert functional in $G_2$-geometry." pith.science (2026). https://pith.science/paper/A6H2IQV6

@misc{pith2026250506872,
  author       = {Pith},
  title        = {Pith review of: A $G_2$-Hilbert functional in $G_2$-geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A6H2IQV6}},
  note         = {Machine review of arXiv:2505.06872}
}
abstract

In this paper we introduce a new functional on the space of $G_2$-structures which we call the $G_2$-Hilbert functional. It is uniquely determined by a few basic principles inspired by the Einstein-Hilbert functional in Riemannian Geometry, and it has similar variational behaviour with it. For instance, torsion-free and nearly $G_2$-structures are saddle critical points of the volume-normalized $G_2$-Hilbert functional. This allows us to uniquely distinguish two new flows of $G_2$-structures, which can be considered as analogues of the Ricci flow in $G_2$-geometry.

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Works this paper leans on

35 extracted references · 31 canonical work pages

  1. [11]

    Shubham Dwivedi, Panagiotis Gianniotis, and Spiro Karigiannis,Flows ofg 2-structures, ii: Curvature, torsion, symbols, and functionals, 2023, arXiv:2311.05516

    arXiv 2023

  2. [1]

    Ann.365(2016), no

    Bernd Ammann, Hartmut Weiss, and Frederik Witt,A spinorial energy functional: critical points and gradient flow, Math. Ann.365(2016), no. 3-4, 1559–1602. MR 3521099

    work page 2016

  3. [2]

    Leonardo Bagaglini,The energy functional ofG 2-structures compatible with a background metric, J. Geom. Anal. 31(2021), no. 1, 346–365. MR 4203649

    work page 2021

  4. [3]

    Math.364(2020), 107030,

    Lucio Bedulli and Luigi Vezzoni,Stability of geometric flows of closed forms, Adv. Math.364(2020), 107030,

    work page 2020

  5. [4]

    Besse,Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008, Reprint of the 1987 edition

    Arthur L. Besse,Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008, Reprint of the 1987 edition. MR 2371700

    2008

  6. [5]

    Robert Bryant and Feng Xu,Laplacian flow for closedg 2-structures: Short time behavior, 2011, arXiv:1101.2004

    arXiv 2011

  7. [6]

    Bryant,Some remarks onG 2-structures, Proceedings of G¨ okova Geometry-Topology Conference 2005, G¨ okova Geometry/Topology Conference (GGT), G¨ okova, 2006, pp

    Robert L. Bryant,Some remarks onG 2-structures, Proceedings of G¨ okova Geometry-Topology Conference 2005, G¨ okova Geometry/Topology Conference (GGT), G¨ okova, 2006, pp. 75–109. MR 2282011

    work page 2005

  8. [7]

    Gao Chen,Shi-type estimates and finite-time singularities of flows ofG 2 structures, Q. J. Math.69(2018), no. 3, 779–797. MR 3859207

    work page 2018

Show all 35 references
  1. [8]

    110, American Mathematical Society, Providence, RI, 2004

    Bennett Chow and Dan Knopf,The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. MR 2061425

    2004

  2. [9]

    Topol.19(2015), no

    Diarmuid Crowley and Johannes Nordstr¨ om,New invariants ofG 2-structures, Geom. Topol.19(2015), no. 5, 2949–2992. MR 3416118

    2015

  3. [10]

    Shubham Dwivedi, Panagiotis Gianniotis, and Spiro Karigiannis,A gradient flow of isometricG 2-structures, J. Geom. Anal.31(2021), no. 2, 1855–1933. MR 4215279

    2021

  4. [12]

    Shubham Dwivedi, Eric Loubeau, and Henrique S´ a Earp,Harmonic flow ofSpin(7)-structures, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)25(2024), no. 1, 151–215. MR 4732637

    2024

  5. [13]

    S´ a Earp, Julieth Saavedra, and Caleb Suan,Laplacian coflows ofg2-structures on contact calabi–yau 7-manifolds, 2024, arXiv:2406.15254

    Henrique N. S´ a Earp, Julieth Saavedra, and Caleb Suan,Laplacian coflows ofg2-structures on contact calabi–yau 7-manifolds, 2024, arXiv:2406.15254

    2024 arXiv

  6. [14]

    Evans,Partial differential equations, Graduate Studies in Mathematics, vol

    Lawrence C. Evans,Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathe- matical Society, Providence, RI, 1998. MR 1625845

    1998

  7. [15]

    Anna Fino and Alberto Raffero,A class of eternal solutions to theG 2-Laplacian flow, J. Geom. Anal.31(2021), no. 5, 4641–4660. MR 4244880

    2021

  8. [16]

    Math.248(2013), 378–415

    Sergey Grigorian,Short-time behaviour of a modified Laplacian coflow ofG 2-structures, Adv. Math.248(2013), 378–415. MR 3107516

    2013

  9. [17]

    Appl.46 (2016), 39–78

    ,Modified Laplacian coflow ofG 2-structures on manifolds with symmetry, Differential Geom. Appl.46 (2016), 39–78. MR 3475531

    2016

  10. [18]

    Math.308(2017), 142–207

    ,G 2-structures and octonion bundles, Adv. Math.308(2017), 142–207. MR 3600058 56 PANAGIOTIS GIANNIOTIS AND GEORGE ZACHAROPOULOS

    2017

  11. [19]

    ,Estimates and monotonicity for a heat flow of isometricG 2-structures, Calc. Var. Partial Differential Equations58(2019), no. 5, Paper No. 175, 37. MR 4018307

    2019

  12. [20]

    Commun., vol

    ,Flows of co-closedG 2-structures, Lectures and surveys onG 2-manifolds and related topics, Fields Inst. Commun., vol. 84, Springer, New York, [2020]©2020, pp. 271–286. MR 4295863

    2020

  13. [21]

    Nigel Hitchin,The geometry of three-forms in six and seven dimensions, 2000, arXiv:math/0010054

    2000 arXiv

  14. [22]

    Spiro Karigiannis,Flows ofG 2-structures. I, Q. J. Math.60(2009), no. 4, 487–522. MR 2559631

    2009

  15. [23]

    Appl.30(2012), no

    Spiro Karigiannis, Benjamin McKay, and Mao-Pei Tsui,Soliton solutions for the Laplacian co-flow of some G2-structures with symmetry, Differential Geom. Appl.30(2012), no. 4, 318–333. MR 2926272

    2012

  16. [24]

    Lotay,Geometric flows ofG 2-structures on 3-Sasakian 7-manifolds, J

    Aaron Kennon and Jason D. Lotay,Geometric flows ofG 2-structures on 3-Sasakian 7-manifolds, J. Geom. Phys. 187(2023), Paper No. 104793, 18. MR 4562794

    2023

  17. [25]

    Lotay,Spacelike mean curvature flow, J

    Ben Lambert and Jason D. Lotay,Spacelike mean curvature flow, J. Geom. Anal.31(2021), no. 2, 1291–1359. MR 4215263

    2021

  18. [26]

    Jorge Lauret,Laplacian flow of homogeneousG 2-structures and its solitons, Proc. Lond. Math. Soc. (3)114 (2017), no. 3, 527–560. MR 3653239

    2017

  19. [27]

    Appl.54(2017), 345–360

    ,Laplacian solitons: questions and homogeneous examples, Differential Geom. Appl.54(2017), 345–360. MR 3693936

    2017

  20. [28]

    Christopher Lin,Laplacian solitons and symmetry inG 2-geometry, J. Geom. Phys.64(2013), 111–119. MR 3004019

    2013

  21. [29]

    Lotay and Yong Wei,Laplacian flow for closedG 2 structures: Shi-type estimates, uniqueness and compactness, Geom

    Jason D. Lotay and Yong Wei,Laplacian flow for closedG 2 structures: Shi-type estimates, uniqueness and compactness, Geom. Funct. Anal.27(2017), no. 1, 165–233. MR 3613456

    2017

  22. [30]

    ,Laplacian flow for closedG 2 structures: real analyticity, Comm. Anal. Geom.27(2019), no. 1, 73–109. MR 3951021

    2019

  23. [31]

    Differential Geom.111(2019), no

    ,Stability of torsion-freeG 2 structures along the Laplacian flow, J. Differential Geom.111(2019), no. 3, 495–526. MR 3934598

    2019

  24. [32]

    S´ a Earp,Harmonic flow of geometric structures, Ann

    Eric Loubeau and Henrique N. S´ a Earp,Harmonic flow of geometric structures, Ann. Global Anal. Geom.64 (2023), no. 4, Paper No. 23, 42. MR 4656808

    2023

  25. [33]

    Moreno and Julieth P

    Andr´ es J. Moreno and Julieth P. Saavedra,On the Laplacian coflow of invariantG 2-structures and its solitons, Mat. Contemp.60(2024), 185–224. MR 4729083

    2024

  26. [34]

    Jeffrey Streets and Gang Tian,Hermitian curvature flow, J. Eur. Math. Soc. (JEMS)13(2011), no. 3, 601–634. MR 2781927

    2011

  27. [35]

    Math.231(2012), no

    Hartmut Weißand Frederik Witt,A heat flow for special metrics, Adv. Math.231(2012), no. 6, 3288–3322. MR 2980500 Panagiotis Gianniotis: Department of Mathematics, National and Kapodistrian University of Athens Email address:pgianniotis@math.uoa.gr George Zacharopoulos: Departm...

    2012

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