Geometric Flows: Analysis and Applications

• Geometric flows are evolution equations that update geometric structures on manifolds based on curvature, torsion, and other differential invariants.
• They use techniques like the DeTurck trick to introduce strict parabolicity, ensuring short-time existence and uniqueness of the solutions.
• Applications include G₂-structure flows where Laplacian and Dirichlet energy methods analyze torsion, energy gaps, and singularity formation.

A geometric flow is a one-parameter (typically parabolic) evolution equation for a geometric structure on a smooth manifold. Such flows act on objects including metrics, G-structures, or tensor fields, and are driven by curvature, torsion, or other differential invariants. They are central in geometric analysis, enabling canonical metric constructions, study of topological invariants, deformation theory of special holonomy, and in applications to mathematical physics.

1. Geometric Structures and the General Notion of Geometric Flow

Let $M$ be a smooth manifold of dimension $n$, with frame bundle $FM\to M$. An $H$-structure is a reduction of $FM$ to a closed, connected subgroup $H\subset SO(n)$, encoded as a global section of the associated bundle $FM/H$. Examples include Riemannian metrics (corresponding to $O(n)$-structures), almost complex structures ($GL(m,\mathbb{C})$), almost Hermitian structures ($U(m)$), and $G_2$-structures (a reduction to $G_2\subset SO(7)$ specified by a positive 3-form).

A geometric flow is an evolution PDE: $\partial_t \alpha = P(\alpha)$ where $\alpha$ is a geometric structure (e.g., a metric $g$ or tensor $\xi$), and $P$ is a differential operator built from invariants of $\alpha$. This framework captures flows for metrics (Ricci flow), $G_2$-structures (Laplacian flow), $H$-structures, and more (Fadel et al., 2022, Karigiannis, 15 Aug 2025).

2. Parabolicity and the DeTurck Trick

The analytic behavior of geometric flows is dictated by the principal symbol $\sigma_2(P)(\xi)$ of $P$. A flow is strictly parabolic if for all nonzero covectors $\xi$ and $v$ in the relevant fiber,

$$\langle \sigma_2(P)(\xi) v, v \rangle \geq C |\xi|^2 |v|^2$$

for some $C>0$. Strict parabolicity guarantees short-time existence and uniqueness via classical quasilinear parabolic theory.

However, many geometric flows are only weakly parabolic due to invariance under diffeomorphisms—for instance, the Ricci flow: $\partial_t g = -2\,\operatorname{Ric}(g)$ In harmonic coordinates, Ricci flow approximates a heat equation modulo a Lie derivative in the direction of a vector field $W(g)$. The DeTurck trick introduces a gauge-fixing term $\mathcal{L}_W g$: $$\partial_t g = -2\,\mathrm{Ric}(g) + \mathcal{L}_{W(g)} g$$ yielding a strictly parabolic system (Karigiannis, 15 Aug 2025).

An analogous approach applies to $G_2$-structure flows and other flows on tensor fields, where a suitable gauge is chosen to restore parabolicity and thus STE (short-time existence) and uniqueness.

3. Flows of G$_2$-Structures: Key Flows and Analytical Results

A $G_2$-structure on a 7-manifold is determined by a positive 3-form $\phi$; the intrinsic torsion $T$ measures deviation from holonomy $G_2$. Several main flows arise (Karigiannis, 15 Aug 2025):

• Laplacian Flow: $\partial_t \phi = \Delta_\phi \phi$, often in the closed case $d\phi=0$. Bryant–Xu established STE and uniqueness for closed initial $\phi$ using the identification $\phi = \phi_0 + d\sigma$, yielding a quasilinear parabolic flow for $\sigma$.
• Modified Laplacian Coflow (Grigorian): $$\partial_t \psi = \Delta_\phi \psi + 2d(\dotsc \psi)$$ for the dual 4-form $\psi = *_{\phi}\phi$, admitting STE/uniqueness for coclosed initial data.
• Dirichlet Energy Flow: Negative gradient of torsion norm, $$\partial_t \phi = -(\operatorname{div} T) \lrcorner \phi$$, which is strictly parabolic on the space of metrics.
• General Heat-type Flows: Any second-order invariant can be written (up to lower order) as

$$\partial_t\phi = (\mathbf{c}_2\,\operatorname{Rc} + \mathbf{c}_3\,F + \mathbf{c}_4\,\mathcal{L}_{T_7}g)\lrcorner\phi + (\mathbf{c}_5\,\operatorname{div}T + \mathbf{c}_6\,\operatorname{div}T^t)\lrcorner\phi + \dotsc$$

for suitable coefficients, where $F$ is a contraction of curvature and $T_7$ a torsion component (Karigiannis, 15 Aug 2025).

A key analytical advance is a classification of all such flows and a criterion for STE/uniqueness by adding a suitable Lie derivative term (a modified DeTurck trick) (Karigiannis, 15 Aug 2025).

4. Torsion, Dirichlet Energy, and Harmonic Flows of H-Structures

For a general $H$-structure (reduction of $SO(n)$ to $H$), the intrinsic torsion $T$ fully encodes its deformation class. The $L^2$-Dirichlet energy

$$\mathcal{E}(\xi) = \frac{1}{2}\int_M |T|^2\,\mathrm{vol}_g$$

defines a natural functional on the space of $H$-structures. The negative gradient flow (“harmonic flow”) is

$$\partial_t \xi = \operatorname{Div} T \lrcorner \xi$$

General evolution equations for $T$ can be written in terms of its commutator with the skew part of the deformation operator, and Bianchi-type identities relate torsion and curvature tensors (Fadel et al., 2022).

Key analytical features include:

• Almost-monotonicity formulae for localized energy quantities, leading to $\varepsilon$-regularity theorems and energy gap results: below a threshold energy, only torsion-free structures exist.
• Long-time existence and singularity formation dichotomy: small initial energy implies global existence, whereas absence of a torsion-free structure in the homotopy class enforces finite-time blow-up, as demonstrated for $G_2$ and $\mathrm{Spin}(7)$ structures on flat tori (Fadel et al., 2022).

5. Comparison of Flows: Ricci, Metric, and G-Structure Evolutions

Geometric flows acting on Riemannian metrics form the foundational class. The Ricci flow and its generalizations are quintessential: $$\partial_t g = -2\,\operatorname{Ric}(g) + b\,R(g)\,g$$ For such flows, the DeTurck gauge ensures strong parabolicity. Moreover, the analytical mechanism for short-time existence and uniqueness extends naturally to a broad class of metric flows and to flows of induced geometric structures (e.g., $G_2$-structures) via appropriate gauge-fixing (Karigiannis, 15 Aug 2025).

STE and uniqueness are determined by the nature of the principal symbol and the choice of gauge; this underpins both the traditional metric flows and modern advances in $G$-structure flows. Analogous proofs apply, e.g., to torsion flows, $G_2$ Dirichlet flow, and higher-order geometric structures.

6. Open Directions and Structural Insights

The structure theory developed for flows of geometric $H$-structures and $G_2$-geometry suggests several further research lines:

• Broader heat-type flows beyond Laplacian evolution, especially flows designed so that their fixed points coincide with desired geometric structures (e.g., Ricci solitons, torsion-free $G_2$-structures).
• Sharp maximum-principle and energy estimates for torsion along flows, paralleling Perelman-type entropy identities in Ricci flow.
• Existence results without restrictive cohomological (e.g., closedness) assumptions: key problems include STE/uniqueness for the full Laplacian flow $\partial_t \phi = \Delta \phi$ without $d\phi = 0$, and construction of flows with prescribed limit sets.
• Convergence analyses, soliton classification, and singularity formation criteria, especially in relation to topological invariants.

The harmonic flow of $H$-structures and spinorial energy flows are particularly active topics, tightly linking analysis, topology, and canonical metric theory. The unifying framework and classification results mark a