Neckpinch — Generalized & G₂ flows

Generalized Ricci Solitons on Heisenberg and Oscillator Algebras

Sun, 01 Nov 2026 00:00:00 GMT

Vicente Cortés, Shubham Dwivedi, Lorenz Jetter — We prove that on a connected, simply-connected nilpotent Lie group, the notions of left-invariant generalized Ricci soliton, semi-algebraic and algebraic generalized Ricci soliton coincide. Moreover, every such soliton has harmonic torsion. We classify all algebraic solitons with positive definite metric on the five-dimensional Heisenberg algebra, the four-dimensional oscillator algebra and the four-dimensional split-oscillator algebra up to automorphisms and rescalings. We find that there is only a finite number of algebraic solitons, all of which have harmonic torsion.

Backward Pseudolocality of G₂-Laplacian Flow

Wed, 28 Oct 2026 00:00:00 GMT

Weiyong He, Longhuan Yan — In this paper, we establish a new estimate on the time-derivative of pointed Nash entropy and use it to give a new proof of backward pseudolocality for Ricci flow. The proof does not rely on Bamler's compactness theorem and structure theory concerning limits of Ricci flows. Our methods also work for other reasonable geometric flows, like the G₂-Laplacian, at a scale depending on its deviation from Ricci flow and its scalar curvature.

Pseudolocality of G₂-Laplacian Flow

Wed, 28 Oct 2026 00:00:00 GMT

Weiyong He, Longhuan Yan — In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local μ-entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the G₂-Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

Chern holonomy under the first and second Chern–Ricci flows

Mon, 26 Oct 2026 00:00:00 GMT

Yuan Liao — We compare the Chern holonomy of the first and second Chern-Ricci flows on compact complex manifolds. For the first flow, we construct an explicit immortal pluriclosed solution on a complex two-torus with full and restricted holonomy U(2), except at one finite time when both groups are U(1)²; its smooth limit has holonomy SU(2). The solution is uniformly nondegenerate and has bounds of every fixed differential order for all time. In contrast, the second Chern-Ricci flow preserves full and restricted Chern holonomy under the natural unitary identification, including the initial time and nonclosed groups. The proof combines preservation of a connection reduction, positive-time analyticity, and a path-transport comparison. As an application, the optimal minimum mean curvature on a holomorphic bundle is unchanged when metrics are restricted to any realizable full and restricted holonomy class. An explicit projected-curvature source explains the failure of the corresponding Ricci-flow holonomy argument.

Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

Tue, 06 Oct 2026 00:00:00 GMT

Shaochuang Huang, Zhuo Peng — In this paper, we study strong uniqueness for smooth instantaneously complete Chern-Ricci flows on noncompact complex manifolds. Under suitable reference metric and positivity assumptions, we establish strong uniqueness for initial Hermitian metrics which are Kähler outside a compact set. The initial metric may be incomplete, and no curvature, metric comparison, or growth bounds are imposed on the evolving solutions. In particular, we obtain strong uniqueness from complete initial metrics in this class with bounded Chern-Ricci curvature. We also prove stationarity of flows from the Euclidean metric on ℂⁿ and uniqueness of the instantaneously complete flow from the Euclidean metric on the complex unit ball for all positive time. Finally, on complex surfaces, we show that a Ricci flow remaining Hermitian for a fixed complex structure is Kähler at all times if it is Kähler at one time, and derive a corresponding strong uniqueness result for Ricci flows in this class.

Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

Mon, 05 Oct 2026 00:00:00 GMT

Huibin Chen, Zhiqi Chen, Fuhai Zhu — We prove harmonicity for nonshrinking algebraic generalized Ricci solitons and for all algebraic generalized Ricci solitons on solvable Lie algebras. In the solvable setting, we characterize expanding solitons by normal extensions of generalized nilsolitons, and derive a sharp rank bound.

Generalized scalar curvature and Ricci flow on compact Lie groups

Thu, 01 Oct 2026 00:00:00 GMT

Zhiqi Chen, Jianhua Yu, Huibin Chen, Junna Ni — Let G be a compact connected semisimple Lie group, let Q be a bi-invariant metric and let ρ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with [H₀]=[ρ] exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form [H₀]=[ρ] and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

Almost Hermitian geometry and generalized Ricci flow

Thu, 01 Oct 2026 00:00:00 GMT

Anna Fino, Julieth Saavedra — We study almost Hermitian manifolds whose lowered Nijenhuis tensor is totally skew-symmetric and whose characteristic (Bismut) torsion is closed, the almost-SKT condition. This extends the SKT (pluriclosed) condition beyond the integrable setting. For a fixed almost complex structure, these constraints are linear in the metric. We introduce an overdetermined ellipticity condition which, on compact manifolds, makes the admissible infinitesimal deformation space finite-dimensional and yields short-time existence for a constrained Bismut–Ricci flow. In real dimension six, almost-SKT structures satisfy a dichotomy: either the almost complex structure is integrable, or the skew Nijenhuis tensor is nowhere vanishing with constant norm and the Bismut Ricci form vanishes. Hence, on compact non-integrable almost-SKT six-manifolds, the constrained Bismut–Ricci flow is stationary. In dimension eight, ellipticity always fails, while in higher dimensions Nijenhuis rigidity holds on an open dense locus. In arbitrary dimension, we characterize when total skew-symmetry of the lowered Nijenhuis tensor determines the compatible metric pointwise up to scale. On compact manifolds, with closed torsion, this implies uniqueness of the almost-SKT metric up to scaling; the constrained Bismut–Ricci flow is then an explicit homothetic solution. We also relate this flow to generalized Ricci flow and characterize the infinitesimal obstruction to lifting the evolution when the almost complex structure varies. Finally, we construct non-integrable almost-SKT structures with nowhere vanishing Nijenhuis tensor and non-flat characteristic connection on compact six-manifolds, including S¹× S²× S³ and S³× S³. In these families, the metric is fixed and the compatible almost complex structures form an S²-family. These structures have vanishing Bismut Ricci form and are therefore almost-CYT.

Solvable Algebraic Generalized Ricci Solitons in low dimensions

Thu, 01 Oct 2026 00:00:00 GMT

Lorenzo Mucciante — We prove new structural results for algebraic generalized Ricci solitons on solvable Lie algebras and we discuss the existence of examples in low dimensions. We establish that all non-trivial solvable algebraic generalized Ricci solitons must be expanding and give necessary conditions for the existence of such solitons on one-dimensional extensions of abelian Lie algebras. Finally, we classify algebraic generalized Ricci solitons on three-dimensional Lie algebras, and algebraic generalized Ricci solitons on four-dimensional solvable unimodular Lie algebras, up to isometry and scaling.

Cohomogeneity one solutions of the IIB system

Wed, 23 Sep 2026 00:00:00 GMT

Lorenzo Foscolo, Mario Garcia-Fernandez — The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.

Chern-Ricci flow on Kato surfaces

Wed, 16 Sep 2026 00:00:00 GMT

Daniele Angella, Mauricio Corrêa — Let S be a Kato surface and D its maximal reduced divisor of rational curves. On S∖ D we construct Hermitian metrics which are flat along the leaves of the canonical foliation and study their evolution under the Chern-Ricci flow. For Enoki surfaces, we construct an immortal normalised solution which, on every compact sublevel of the natural exhaustion, converges in the Gromov-Hausdorff sense to the elliptic base endowed with an explicit flat metric. For Kato surfaces of intermediate type, the affine Green model and its finite-index extension give, under the assumption 0<μ<2, an explicit normalised solution which, on every compact Green block, collapses in the Gromov-Hausdorff sense to a circle. In the Enoki case the limiting area is 2πb₂(S); in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.

Harmonicity and Existence of Algebraic Generalized Ricci Solitons

Thu, 10 Sep 2026 00:00:00 GMT

Huibin Chen, Zhiqi Chen, Fuhai Zhu — In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions 6m+d for m>d≥1. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra mathfrak m₂(n), n≥5, admits a generalized nilsoliton if and only if 5≤ n≤8.

Topology of low-dimensional generalized Ricci solitons and string backgrounds

Wed, 26 Aug 2026 00:00:00 GMT

Jeffrey Streets — Adapting ideas of, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in. Using this we show that low dimensional GRS are diffeomorphic to S³ / Γ or S³ × S¹ / Γ. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion G₂, strong torsion Spin(7)-manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered 5-manifolds over complex orbifolds.

Homogeneous Generalized Ricci flows II

Wed, 26 Aug 2026 00:00:00 GMT

Elia Fusi, Ramiro A. Lafuente, James Stanfield — We relate the generalized Ricci curvature of left-invariant generalized metrics on a Lie group with the Ricci curvature of certain metric Lie algebras. As an application, we prove subconvergence of rescaled generalized Ricci flows to expanding generalized Ricci solitons on simply connected Lie groups with positive semi-definite Killing form. Under the same hypotheses, subconvergence of pluriclosed flows to expanding pluriclosed solitons is also shown. We further prove that left-invariant Bismut-Ricci flat metrics arise as critical points for the generalized scalar curvature functional on unimodular Lie groups. In view of this, we establish dynamical stability for the generalized Ricci flow of so-called standard bi-invariant, Bismut-flat metrics on compact, simply connected, semisimple Lie groups and construct examples of dynamically unstable Bismut-flat metrics which are non-standard.

Hermitian curvature flow and HKT geometry

Fri, 14 Aug 2026 00:00:00 GMT

Beatrice Brienza, Jeffrey Streets — We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Ampère equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.

Sasaki with torsion manifolds and string backgrounds

Sun, 09 Aug 2026 00:00:00 GMT

Beatrice Brienza, Anna Fino, Udhav Fowdar, Gueo Grantcharov — Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a ∇-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact ∇-Einstein manifold in dimension 5 and 7. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with S¹.

Long-time existence of some geometric flows with bounded scalar curvature

Tue, 09 Jun 2026 00:00:00 GMT

Chuanhuan Li, Yi Li — The analysis of singularities in scalar curvature is a classical problem. In this survey, we investigate the behavior of scalar curvature under several geometric flows, with a focus on three specific cases: the Ricci flow, the Kähler-Ricci flow coupled with (1,1)-forms, and the Laplacian flow.

On the structure of complete G₂-solitons

Thu, 04 Jun 2026 00:00:00 GMT

Haozhao Li, Yuanqing Ma, Kai Zheng — In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed G₂-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed C^(1,α) convergence. Our principal result shows that this C^(1,α) convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive 3-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to G₂-solitons. Since the available C^(1,α) control does not directly close an elliptic bootstrap for the G₂-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy ε-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed C^(1,α) convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with G₂-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted L^(7/2) -curvature bound then yields pointed C^∞ compactness.

Cylindrical generalized Ricci solitons in three dimensions

Tue, 02 Jun 2026 00:00:00 GMT

Miguel Pino Carmona — We construct an explicit two-parameter family of complete, non-compact, three-dimensional, smooth steady gradient generalized Ricci solitons with SO(2)×ℝ symmetry, providing a cylindrical counterpart to the spherically symmetric solitons recently found by Podestà and Raffero. The family is parametrized by a flux constant k>0 and a conserved quantity 𝓒≥ 0. For 𝓒=0, the asymptotic geometry exhibits power-law decay; for 𝓒>0, the metric converges exponentially fast to a flat cylinder of finite radius.

Brane flows

Fri, 29 May 2026 00:00:00 GMT

Georgios Papadopoulos, Kostas Skenderis — Based on effective D-brane actions, we present a generalisation of the Ricci flow that includes the flow of a theory with a n-form field strength for n≥ 0. This is a generalisation of both the Ricci flows and the generalised Ricci flows. Following Perelman, we show that flows that keep a suitable field-dependent volume fixed are monotonic. We also show that all steady brane flow solitons are gradient solitons and use this to demonstrate that on some occasions this implies the existence of a Killing vector field that leaves all the other fields invariant. Particular cases of gradient solitons are NS5 and D5 branes, and the volume which is kept fixed in these cases is the T-duality invariant volume (NS5 brane) or its S-dual (D5 brane). We also generalise the above analysis to gravitational actions coupled to form gauge potentials that also exhibit a Chern-Simons type term. We find an alteration is required in the adaptation of Perelman's modification to this case, which yields a new functional that also exhibits a Chern-Simons term. Under suitable assumptions, we proceed to prove the monotonicity of the flow and that all steady flow solitons are gradient solitons. We also explore the consequences of the last statement on the geometry of solitons.

Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

Thu, 21 May 2026 00:00:00 GMT

Haoyuan Sun — We prove uniform diameter estimates, volume non-collapsing estimates and Gromov-Hausdorff convergence for the normalized Chern-Ricci flow on smooth complex minimal surfaces of general type, starting from an arbitrary Hermitian metric. This removes the local Kahler assumption near the null locus used in our previous work and confirms the Tosatti-Weinkove conjecture in complex dimension two. The main analytic ingredients are a surface torsion estimate, a uniform total variation bound for Delta |G|, a Green-weighted L^2 estimate for the torsion, and a linear iteration of real Poisson equations, which together give the required Green function estimates.

On the Calabi estimate of geometric flows of Hermitian metrics

Mon, 20 Apr 2026 00:00:00 GMT

Marco Gallo, Luigi Vezzoni — We establish a general result ensuring a C¹ a priori bound for smooth curves of Hermitian metrics. As a main application, we obtain a new regularity result for Hermitian curvature flows, and in particular for the second Chern-Ricci flow.

Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

Mon, 06 Apr 2026 00:00:00 GMT

Haoyuan Sun — We establish uniform diameter estimates and volume non-collapsing estimates for the Chern-Ricci flow on smooth Hermitian minimal models of general type, assuming the initial metric is Kähler in a neighborhood of the null locus of the canonical bundle. This yields subsequential Gromov-Hausdorff convergence, partially resolving a conjecture of Tosatti and Weinkove. When the underlying manifold is Kähler, we further prove the uniqueness of the limit space. Analytically, we overcome the difficulties posed by non-Kähler torsion in the Green's formula by exploiting our local Kähler assumption, successfully adapting recent estimates of Kähler Green's function to the Hermitian setting. To prove the uniqueness of the limit, we introduce Perelman's reduced length to the Chern-Ricci flow. By establishing a uniform Chern scalar curvature bound and an almost monotonicity formula for the reduced volume, we deduce an almost-avoidance principle for the singular set, allowing us to effectively compare the flow distance with the canonical limit distance.

Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds

Thu, 26 Mar 2026 00:00:00 GMT

Quang-Tuan Dang — We study the stability and Hölder continuity of solutions to degenerate complex Monge–Ampère equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when the reference form is the pullback of a Hermitian metric. As a consequence, we establish a uniform diameter bound for the twisted Chern–Ricci flow.

Pluriclosed flow on Oeljeklaus-Toma manifolds

Fri, 12 Dec 2025 00:00:00 GMT

Jeffrey Streets, Xiaokang Wang — We establish global existence of the pluriclosed flow with arbitrary initial data on Oeljeklaus-Toma manifolds, and Gromov-Hausdorff convergence of blowdown limits to a torus under natural conjectural bounds on the flow at infinity. In the case of generalized Kähler-Ricci flow we prove refined a priori estimates in support of these conjectural bounds.

Flows of conformally coclosed G₂-structures with dilaton

Wed, 26 Nov 2025 00:00:00 GMT

Spiro Karigiannis, Sébastien Picard, Caleb Suan — We study flows of G₂-structures guided by the principle of dimensional reduction: natural geometric flows in G₂-geometry reduce to natural flows in complex geometry. Our main examples are the G₂-Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The G₂-lift of the anomaly flow deforms conformally coclosed G₂-structures. We compare the G₂-anomaly flow to the G₂-Laplacian coflow, and investigate short-time existence and fixed points.

The canonical symmetry reduction of string backgrounds

Tue, 25 Nov 2025 00:00:00 GMT

Aaron Kennon, Jeffrey Streets — String backgrounds, defined here as metric connections with skew-symmetric torsion and reduced holonomy, yield generalized Ricci solitons relative to the Lee vector field. By a variational argument using the string action, they are also gradient generalized Ricci solitons relative to a potential function. These two observations combine to yield a canonical symmetry, and in this work we derive fundamental features of the transverse geometry, and rigidity phenomena. We prove in a unified conceptual fashion that the transverse geometry satisfies the string generalized Ricci soliton equations (a simplified Hull-Strominger system) in many settings including almost Hermitian, almost contact, SU(3), G₂, and Spin(7) geometry. We also show that the transverse geometry is always conformally co-closed, with the conformal factor given by the associated soliton potential.

The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe

Wed, 29 Oct 2025 00:00:00 GMT

Severin Bunk, Miguel Pino Carmona, C. S. Shahbazi — We prove well-posedness of the analytic Cauchy problem for gradient generalized Ricci solitons on an abelian bundle gerbe and solve the initial data equations on every compact Riemann surface. Along the way, we provide a novel characterization of the self-similar solutions of the generalized Ricci flow by means of families of automorphisms of the underlying abelian bundle gerbe covering families of diffeomorphisms isotopic to the identity.

Positive Hermitian curvature flow on 2-step nilpotent Lie groups

Thu, 09 Oct 2025 00:00:00 GMT

Ettore Lo Giudice — We study the positive Hermitian curvature flow for left-invariant metrics on 2-step nilpotent Lie groups with a left-invariant complex structure J. We describe the long-time behavior of the flow under the assumption that J[g, g] is contained in the center of g. We show that under our assumption the flow gₜ exists for all positive t and (G,(1+t)⁻¹gₜ) converges, in the Cheeger-Gromov topology, to a 2-step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.

BRIDGES Lectures: Flows of geometric structures, especially G₂-structures

Fri, 15 Aug 2025 00:00:00 GMT

Spiro Karigiannis — The BRIDGES meeting in gauge theory, extremal structures, and stability was held June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Spiro Karigiannis on "Flows of geometric structures, especially G₂-structures". Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information. The main theme is short time existence (STE) and uniqueness for geometric flows. We first introduce geometric structures on manifolds and geometric flows of such structures. We discuss some qualitative features of geometric flows, and consider the notions of strong and weak parabolicity. We focus on the Ricci flow, explaining carefully the DeTurck trick to establish short-time existence and uniqueness, an argument which we then extend to a general class of geometric flows of Riemannian metrics, previewing similar ideas for flows of G₂-structures. Finally, we consider geometric flows of G₂-structures. We review the basics of G₂-geometry and survey several different geometric flows of G₂-structures. In particular, we clarify in what sense STE results for the G₂ Laplacian flow differ from STE results for other geometric flows. We conclude with a summary of some recent results by the author with Dwivedi and Gianniotis, including a classification of all possible heat-type flows of G₂-structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.