Generalized & G₂ flows

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.

Let $S$ be a Kato surface and $D$ its maximal reduced divisor of rational curves. On $S\setminus D$ 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 $0<\mu <2$, 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 $2\pi b_2(S)$; in the intermediate case the length of the limiting circle is determined by the Green and leafwise monodromies.

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 $m>d\ge 1$. 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 ${\mathfrak{m}}_2(n)$, $n\ge 5$, admits a generalized nilsoliton if and only if $5\le n\le 8$.

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 $S^3/\Gamma$ or $S^3\times S^1/\Gamma$. We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion $G_2$, strong torsion $\mathrm{Spin}(7)$-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.

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.

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 $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-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 $S^1$.

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.

In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed $G_2$-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 $C^{1,\alpha }$ convergence. Our principal result shows that this $C^{1,\alpha }$ 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 $G_2$-solitons. Since the available $C^{1,\alpha }$ control does not directly close an elliptic bootstrap for the $G_2$-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy $\epsilon$-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed $C^{1,\alpha }$ 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 $G_2$-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 $L^{\frac{7}{2}}$ -curvature bound then yields pointed $C^{\infty }$ compactness.

We construct an explicit two-parameter family of complete, non-compact, three-dimensional, smooth steady gradient generalized Ricci solitons with $\mathrm{SO}(2)\times \mathbb{R}$ 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 $\mathcal{C}\ge 0$. For $\mathcal{C}=0$, the asymptotic geometry exhibits power-law decay; for $\mathcal{C}>0$, the metric converges exponentially fast to a flat cylinder of finite radius.

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 $n\ge 0$. 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.

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.

We establish a general result ensuring a $C^1$ 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.

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.

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.

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.

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.

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, $SU(3)$, $G_2$, and $\mathrm{Spin}(7)$ geometry. We also show that the transverse geometry is always conformally co-closed, with the conformal factor given by the associated soliton potential.

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.

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 ${\mathrm{G}}_2$-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 ${\mathrm{G}}_2$-structures. Finally, we consider geometric flows of ${\mathrm{G}}_2$-structures. We review the basics of ${\mathrm{G}}_2$-geometry and survey several different geometric flows of ${\mathrm{G}}_2$-structures. In particular, we clarify in what sense STE results for the ${\mathrm{G}}_2$ 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 ${\mathrm{G}}_2$-structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.

In this paper, we establish Li-Yau-type and Hamilton-type estimates for positive solutions to the heat equation associated with the generalized Ricci flow, under a less stringent curvature condition. Compared with [25] and [35], these estimates generalize the results in Ricci flow to this new flow under the weaker Ricci curvature bounded assumption. As an application, we derive the Harnack-type inequalities in spacetime and find the monotonicity of one parabolic frequency for positive solutions of the heat equation under bounded Ricci curvature.

In this paper, we explore a new class of integrable sigma models, which we refer to as the "dual regime" of Yang-Baxter (YB) deformed $\mathrm{O}(2N)$ sigma models. This dual regime manifests itself in the conformal perturbation approach. Namely, it is well known that conventional YB-deformed $\mathrm{O}(N)$ sigma models are described in the UV by a collection of free bosonic fields perturbed by some relevant operators. The holomorphic parts of these operators play the role of screening operators which define certain integrable systems in the free theory. All of these integrable systems depend on a continuous parameter $b$, which parametrizes the central charge, and are known to possess the duality under $b^2⟷-1-b^2$. Although $\mathrm{O}(2N+1)$ integrable systems are self-dual, $\mathrm{O}(2N)$ systems are not. In particular, the $\mathrm{O}(2N)$ integrable systems provide new perturbations of the sigma model type. We identify the corresponding one-loop metric and $B-$field and show that they solve the generalized Ricci flow equation.

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.

In this work, we discuss the stability of the pluriclosed flow and generalized Ricci flow. We proved that if the second variation of generalized Einstein–Hilbert functional is nonpositive and the infinitesimal deformations are integrable, the flow is dynamically stable. Moreover, we prove that the pluriclosed steady solitons are dynamically stable when the first Chern class vanishes.

In this paper, we study the $t$-Gauduchon Ricci-flat condition under the Chern-Ricci flow. In this setting, we provide examples of Chern-Ricci flow on compact non-Kähler Calabi-Yau manifolds which do not preserve the $t$-Gauduchon Ricci-flat condition for $t<1$. The approach presented generalizes some previous constructions on Hopf manifolds. Also, we provide non-trivial new examples of balanced non-pluriclosed solution to the pluriclosed flow on non-Kähler manifolds. Further, we describe the limiting behavior, in the Gromov-Hausdorff sense, of geometric flows of Hermitian metrics (including the Chern-Ricci flow and the pluriclosed flow) on certain principal torus bundles over flag manifolds. In this last setting, we describe explicitly the Gromov-Hausdorff limit of the pluriclosed flow on principal $T^2$-bundles over the Fano threefold $\mathbb{P}(T_{{\mathbb{P}}^2})$.

In this survey, we consider various analytic problems related to the geometry of the Chern connection on Hermitian manifolds, such as the existence of metrics with constant Chern-scalar curvature, generalizations of the Kähler-Einstein condition to the non-Kähler setting, and the convergence of the Chern-Ricci flow on compact complex surfaces.

A ${\mathrm{G}}_2$-structure on a $7$-manifold $M$ is called a ${\mathrm{G}}_{2}T$-structure if $M$ admits a ${\mathrm{G}}_2$-connection ${\nabla }^T$ with totally skew-symmetric torsion $T_{\phi }$. If furthermore, $T_{\phi }$ is closed then it is called a strong ${\mathrm{G}}_{2}T$-structure. In this paper we investigate the geometry of (strong) ${\mathrm{G}}_{2}T$-manifolds in relation to its curvature, $S^1$ action and almost Hermitian structures. In particular, we study the Ricci flatness condition of ${\nabla }^T$ and give an equivalent characterisation in terms of geometric properties of the ${\mathrm{G}}_2$ Lee form. Analogous results are also obtained for almost Hermitian $6$-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the $S^1$ reduction by the dual of the ${\mathrm{G}}_2$ Lee form, we show that Ricci-flat strong ${\mathrm{G}}_{2}T$-structures correspond to solutions of the $\mathrm{SU}(3)$ heterotic system on certain almost Hermitian half-flat $6$-manifolds. Many explicit examples are described and in particular, we construct the first examples of strong ${\mathrm{G}}_{2}T$-structures with ${\nabla }^T$ not Ricci flat. Lastly, we classify ${\mathrm{G}}_2$-flows inducing gauge fixed solutions to the generalised Ricci flow akin to the pluriclosed flow in complex geometry. The approach is this paper is based on the representation theoretic methods due to Bryant.

The notion of Courant algebroid relation is used to introduce a definition of relation between divergence operators on Courant algebroids. By introducing invariant divergence operators, a notion of generalised T-duality between divergences is presented through an existence and uniqueness result for related divergence operators on T-dual pairs of exact Courant algebroids, which naturally incorporates the dilaton shift. When combined with the notion of generalised isometry, this establishes circumstances under which generalised Ricci tensors are related, proving that T-duality is compatible with generalised string background equations. This enables an analysis of the compatibility between T-duality and generalised Ricci flow, showing that the T-dual of a solution of generalised Ricci flow is also a solution of generalised Ricci flow. Our constructions are illustrated through many explicit examples.

The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart from the purely geometric paradigm. This fact requires an extension of the Swampland conjectures to scalar field spaces with non-trivial potentials, rather than just moduli spaces. To address this task, we utilise geometric flows, in particular generalised Ricci flow, to probe the associated scalar field spaces. Considering internal spaces supported by three-form fluxes, we first show that the distance defined in terms of the Perelman entropy functional needs to be refined in order to encompass fluxes. Doing so, we extend the Ricci Flow Conjecture to include Kalb-Ramond flux besides the metric and the dilaton field. This allows us to probe infinite-distance points within these scalar field spaces in a purely geometric way. We subsequently construct a geometric flow for internal manifolds supported by Ramond-Ramond fluxes and discuss its role in the Ricci Flow Conjecture. Our analysis suggests that in the presence of fluxes the Distance Conjecture might be better characterised in terms of a cost function on the space of metrics, rather than a genuine distance.

We extend some results known for the Kähler-Ricci flow to the Chern-Ricci flow regarding the independence of singularity types for long-time solutions. Specifically, we show that if a solution to the Chern-Ricci flow exists with uniformly bounded torsion and curvature, then any other solution starting from an initial metric of the same $\partial \bar{\partial }$ class will also exhibit uniform bounds on torsion and curvature.

We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding our main analytical results, we prove a priori $C^{\infty }$ estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's $C^3$ estimate for the complex Monge-Ampère equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.

We study the flow of Hermitian metrics governed by the second Chern-Ricci form on a compact complex manifolds. The flow belongs to the family of Hermitian curvature flows introduced by Streets and Tian and it was considered by Lee in order to study compact Hermitian manifolds with almost negative Chern bisectional curvature. We show a regularity result on compact complex parallelizable manifolds and we prove that Chern-flat metrics are dynamically stable.

We explore three versions of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi–Yau 3-folds, interpreting their dimensional reductions to the Kähler geometry of the base. Precisely, we reduce Ansätze for the Laplacian coflow, modified or not by de Turck's trick, both on trivial products $C{Y}^3\times S^1$ and on contact Calabi–Yau 7-manifolds, obtaining in each case a natural modification of the Kähler–Ricci flow.

We develop a framework inspired by Lauret's "bracket flow" to study the generalized Ricci flow, as introduced by Streets, on discrete quotients of Lie groups. As a first application, we establish global existence on solvmanifolds in arbitrary dimensions, a result which is new even for the pluriclosed flow. We also define a notion of generalized Ricci soliton on exact Courant algebroids that is geometrically meaningful and allows for non-trivial expanding examples. On nilmanifolds, we show that these solitons arise as rescaled limits of the generalized Ricci flow, provided the initial metrics have "harmonic torsion", and we classify them in low dimensions. Finally, we provide a new formula for the generalized Ricci curvature of invariant generalized metrics in terms of a moment map for the action of a non-reductive real Lie group.

We show that every gradient shrinking soliton of the generalized Ricci flow on compact manifold is a Ricci soliton. And we prove that the pluriclosed soliton is gradient Kahler-Ricci soliton under a broad cohomological condition. Moreover, we construct the first example of non-trivial shrinking generalized soliton, which can serve as a singularity model of the generalized Ricci flow.

We establish a transcendental generalization of Nakamaye's theorem to compact complex manifolds when the form is not assumed to be closed. We apply the recent analytic technique developed by Collins and Tosatti to show that the non-Hermitian locus of a nef and big $(1,1)$-form, which is not necessarily closed, on a compact complex manifold equals the union of all positive-dimensional analytic subvarieties where the restriction of the form is not big (null locus). As an application, we can give an alternative proof of the Nakai–Moishezon criterion of Buchdahl and Lamari for complex surfaces and generalize this result in higher dimensions Finally, we investigate finite time non-collapsing singularities of the Chern–Ricci flow, partially answering a question raised by Tosatti and Weinkove.

We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short time existence and uniqueness of solutions on compact manifolds, in turn showing that the Courant isometry group is preserved by the flow. We show a scalar curvature monotonicity formula and prove that generalized Ricci flow is a gradient flow, extending fundamental works of Hamilton and Perelman. Using these we show a convergence result for certain nonsingular solutions to generalized Ricci flow.

We prove the existence of a one-parameter family of pairwise non-isometric, complete, positively curved, steady generalized Ricci solitons of gradient type on ${\mathbb{R}}^3$ that are invariant under the natural cohomogeneity one action of SO(3). In the context of generalized Ricci flow, this result represents the analogue of Bryant's construction of the complete rotationally invariant steady soliton for the Ricci flow.

The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and $\mathrm{Rc}(g)=\frac{1}{4}{H}_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each $G_i$ is a compact simple Lie group and $K$ is a closed subgroup of them holding some extra assumption, we consider $M=G_1\times G_2/\Delta K$. Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on $M$ among a subset of $G$-invariant metrics and, if $G_1=G_2$, then it is globally stable.

We provide a derivative estimate for the pluriclosed flow, controlling higher order derivatives of Chern curvature and torsion using the Chern curvature. Moreover, we derive an estimate for torsion tensor using Chern Ricci curvature in dimension two. And in the Hermitian-symplectic case, we find a monotonic quantity and use it to prove that all Hermitian-symplectic solitons are Kähler Ricci solitons.

It is shown that on compact $Spin(7)$–manifold with exterior derivative of the Lee form lying in the Lie algebra $spin(7)$ the curvature $R$ of the $Spin(7)$–torsion connection $R\in S^2{\Lambda }^2$ with vanishing Ricci tensor if and only if the $3$-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the $3$-form torsion is parallel with respect to the Levi-Civita and to the $Spin(7)$–torsion connections simultaneously. Precise conditions for a compact $Spin(7)$–manifold to has closed torsion are given in terms of the Ricci tensor of the $Spin(7)$–torsion connection. It is shown that a compact $Spin(7)$–manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $Spin(7)$–manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $Spin(7)$–structure.

Curvature properties of the characteristic connection on an integrable $G_2$ manifold are investigated. We consider integrable $G_2$ manifold of constant type, i.e. the scalar product of the exterior derivative of the $G_2$ form with its Hodge dual is a constant. We show that on an integrable $G_2$ manifold of constant type with $G_2$-instanton characteristic curvature and vanishing Ricci tensor the torsion 3-form is harmonic. Consequently, we prove that the characteristic curvature is symmetric in exchange the first and the second pair and Ricci flat if and only if the three-form torsion is parallel with respect to the Levi-Civita and to the characteristic connection simultaneously and this is equivalent to the condition that the characteristic curvature satisfies the Riemannian first Bianchi identity. We find that the Hull connection is a $G_2$-instanton exactly when the torsion is closed. We observe that any compact integrable $G_2$ manifold with closed torsion is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the characteristic connection. In particular, this vector field is an infinitesimal automorphism of the $G_2$ structure and preserves the torsion three form.

It is observed that on a compact almost complex Calabi-Yau with torsion 6-manifold the Nijenhuis tensor is parallel with respect to the torsion connection. If the torsion is closed then the space is a compact generalized gradient Ricci soliton. In this case, the torsion connection is Ricci-flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. On a compact almost complex Calabi-Yau with torsion 6-manifold it is shown that the curvature of the torsion connection is symmetric on exchange of the first and the second pairs and has vanishing Ricci tensor if and only if it satisfies the Riemannian first Bianchi identity.

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $dT=\delta T=0$ or the curvature of the torsion connection $R\in S^2{\Lambda }^2$ then the scalar curvature of a $\nabla$-Einstein manifold is determined by the norm of the torsion up to a constant. It is proved that a compact generalized gradient Ricci soliton with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature are constants. In this case the torsion 3-form is harmonic and the gradient function has to be constant. Necessary and sufficient conditions a metric connection with skew torsion to satisfy the Riemannian first Bianchi identity as well as the contracted Riemannian second Binachi identity are presented. It is shown that if the torsion connection satisfies the Riemannian first Bianchi identity then it satisfies the contracted Riemannian second Bianchi identity. It is also proved that a metric connection with skew torsion satisfying the curvature identity $R(X,Y,Z,V)=R(Z,Y,X,V)$ must be flat.