Regularity of the Donaldson Geometric Flow

We prove a regularity theorem for the solutions of the Donaldson geometric flow equation on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The minimal initial conditions lay in the Besov space B21,p(M,Λ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{1,p}_{2}(M, {\varLambda }^{2})$\end{document} for p > 4. The Donaldson geometric flow was introduced by Simon Donaldson in Donaldson (Asian J. Math.3, 1–16 1999). For a detailed exposition see Krom and Salamon (J. Symplectic Geom.17, 381–417 2019).


Introduction
Let M be a smooth closed Riemannian four-manifold. Denote by g the Riemannian metric, denote by dvol ∈ Ω 4 (M) the volume form of g and let * : Ω k (M) → Ω n−k (M) be the Hodge * -operator associated to the metric and orientation. Let ω be a symplectic form on M compatible with the metric and let S a be the space of symplectic forms representing the cohomology class a = [w] ∈ H 2 (M; R). This is formally an infinite dimensional manifold and the tangent space at any element ρ ∈ S a is the space of exact two-forms. The Donaldson geometric flow on S a is given by the evolution equation where Θ ρ and * ρ are defined pointwise on M by the equations for all one-forms λ ∈ Ω 1 (M). The operator * ρ is the Hodge star operator for the metric g ρ which is uniquely determined by the conditions for all ω ∈ Λ 2 T * M. A detailed explanation of the construction of the metric g ρ and the operator * ρ is given in [5]. Each ρ ∈ Ω 2 with ρ 2 > 0 determines an inner product ·, · ρ on the space of exact two-forms defined by These inner products determine a metric on the infinite dimensional space S a called the Donaldson metric. The Donaldson geometric flow is the negative gradient flow with respect to the Donaldson metric of the energy functional E : S a → R defined by The Donaldson flow equation has a beautiful geometric origin laid out in Donaldson's paper [1]. The key idea is that the space of diffeomorphisms of a hyperKähler surface has the structure of an infinite dimensional hyperKähler manifold. The group of symplectomorphisms with respect to a preferred symplectic structure ω then acts by composition on the right and this group action is generated by a hyperKähler moment map. In analogy with the finite dimensional case, one then studies the negative gradient flow to the moment map square functional with respect to the L 2 -inner product. If we push the preferred symplectic structure ω forward by the diffeomorpisms of M to the space of symplectic structures in a fixed cohomology class, we obtain the Donaldson flow equation (1). If we push the moment map square forward, we obtain the energy functional E : S a → R in (3) and if we push the L 2 -metric forward, we obtain the Donaldson metric (2). The Donaldson flow remains the negative gradient flow to this energy functional with respect to the Donaldson metric (2) on the space of symplectic structures in a fixed cohomology class. The Donaldson flow remains well defined for a general symplectic 4-manifold (M, ω) equipped with a compatible Riemannian metric g. The motivation to study the Donaldson flow on the space of symplectic structures comes from the longstanding open uniqueness problem for symplectic structures on closed fourmanifolds (see [9] for an exposition). The hope is that the Donaldson flow provides a tool to settle this question at least in some favorable cases, such as the hyperKähler surface and the complex projective plane CP 2 . This hope is strengthened by the observation that the preferred symplectic structure ω is the unique absolute minimum and that the Hessian of the energy functional E is positive definite at the absolute minimum (see [5]). In the case of M = CP 2 , we can further show that the Fubini-Study form is the only critical point of the flow. Thus, if longtime existence and convergence of the Donaldson flow can be established, the Donaldson flow would provide a proof for the uniqueness of the symplectic structures on CP 2 of a given cohomology class up to isotopy. In the case of the hyperKähler surface, Donaldson [1] proved that the higher critical points are not strictly stable. The idea would then be to perturb the flow near the critical points such that eventually the perturbed flow converges towards the unique absolute mimimum.
The main result of this paper is the regularity for critical points and the regularity for the flow (Section 4). Local existence and the semiflow property of the Donaldson flow have been established in the Ph.D. thesis of the author [6]. These results lay the foundation for future studies focusing on longtime existence and the problem of solutions escaping to infinity. Section 2 contains the main geometric ideas, in particular a result by Donaldson [2] that shows that the map ρ → ρ + u from the space of symplectic forms representing a cohomology class in a fixed affine space in H 2 (M; R) with dimension equal to b + 2 to the space of self-dual two-forms is a local Banach space diffeomorphism. In Section 3 we then study the evolution of ρ + u if ρ evolves by the Donaldson flow equation. This evolution equation is the key to prove the regularity theorem for solutions of the Donaldson flow equation in Section 4.

Notation and Conventions
Let (M, g) be a closed Riemannian oriented four-manifold. Let π : E → M be a natural k-dimensional vector bundle over M. We denote by W ,p (M, E) the Sobolev completion of the space of sections Ω 0 (M, E). If the bundle in question is clear from the context, we will just write W ,p instead of W ,p (M, E). We will suppress the constants that solely depend on the parameters dim(M), vol(M), k, p from the notations when we make estimates in Sobolev norms. We denote by S the space of smooth symplectic structures on M compatible with the orientation. We write S a for ρ ∈ S that represent a fixed cohomology class a ∈ H 2 (M; R). Let L ∈ H 2 (M; R) be an affine subspace. Then, S L denotes the subset of S such that the symplectic structures represent cohomology classes in L. We define A cohomology class or an affine subspace of cohomology classes as subscript has the same meaning as in the smooth counterpart. where the sum runs over all multi-indices α = (α 1 , . . . , α ) ∈ N 0 with |α| = i=1 α i ≤ m and a α ≥ 0 for all α.

The Map K
In this section, we study the map from the space of symplectic forms to the space of self-dual two-forms. We apply a theory developed by Donaldson [2] on elliptic problems for closed two-forms on four dimensional closed manifolds that fullfill a pointwise constraint with 'negative tangents'. The main insight is that this map is a local Banach space diffeomorphism between appropriately choosen spaces and the regularity of ρ + u determines the regularity of ρ. The precise statement is given in Theorem 1.
We will use the construction of a Riemannian metric determined by a nondegenerate two-form and a volume form explained in [5]. Here is a brief overview. Fix a nondegenerate two-form ρ ∈ Ω 2 (M) such that ρ ∧ ρ > 0. There exists a Riemannian metric g ρ such that its volume form agrees with dvol of (M, g). The associated Hodge star operator * ρ : If X ∈ Vect(M) is a vector field then * ρ ρ(X, ·) = −ρ ∧ g(X, ·).
The map is an involution that preserves the exterior product, acts as the identity on the orthogonal complement of ρ with respect to the exterior product and it sends ρ to −ρ. Moreover, it maps Ω + to Ω + ρ , the self-dual forms with respect to the metric g ρ . The Hodge star operator * ρ : Ω 2 (M) → Ω 2 (M) associated to g ρ is given by * ρ ω = R ρ * R ρ ω.
Let ω ∈ Ω 2 (M) be a self-dual two-form and let J : T M → T M be an almost complex structure such that g = ω(·, J ·). We define the almost complex structure J ρ by ρ(J ρ ·, ·) := ρ(·, J ·) and a self-dual two-form ω ρ with respect to g ρ by ω ρ := R ρ ω. Then, and * ρ (λ ∧ ω ρ ) = λ • J ρ (5) for all one-forms λ ∈ Ω 1 (M). Now, fix a symplectic form ρ ∈ S a . Let S a+H ρ be the space of symplectic forms representing a cohomology class in the affine space a + H ρ ⊂ H 2 (M; R). Define the map K : S a+H ρ → Ω + by Denote the extension of this map to the Sobolev space S k,p a+H ρ also by K.
(i) For every ρ ∈ S k,p a+H ρ , there exists a W k,p -neighborhood of ρ such that K restricted to this neighborhood is a diffeomorphism of Banach spaces. (iii) There exist polynomials p 1 , p 2 with positive coefficients with the following significance. If ρ ∈ S k,p and K(ρ) ∈ W k+1,p (M, Λ + ) with 1 u ≤ C < ∞, then ρ ∈ S k+1,p and Proof See page 7.
We will need the following three lemmas to prove Theorem 1.

Remark 1 The set
for a fixed volume form dvol and a self-dual two-form θ was considered by Donaldson in [2] in the context of the following problem: How many symplectic structures ρ exist, such that ρ ∧ ρ = dvol for a prescribed volume form dvol, ρ is compatible with a prescribed almost complex structure and ρ lays in a given positive affine subspace of H 2 (M, R)? The answer is that it is unique. The set P dvol,θ is a three-dimensional submanifold of Λ 2 V with two components. The key property of this manifold established in Lemma 1 is called 'negative chords'.

Lemma 2 (The Linearization of K) Let ρ s be a path of nondegenerate 2-forms and ρ
In particular, For the last equality, we used that the linear map R ρ : for a 4-dimensional real vector space V and it maps Λ + to Λ + ρ and vice versa, where Λ + ρ is the space of self-dual 2-forms for the metric g ρ (see [5] for a proof of these facts).
Lemma 3 (Lie Derivative) Let X be a vector field on M and ρ a nondegenerate two-form. Then, Proof Let ψ s , s ≥ 0 be the family of diffeomorphisms on M generated by the vector field X. Then, . Thus, Using Lemma 2, we then compute This proves the lemma.
Proof of Theorem 1 We prove (i). By Lemma 2 the linearization of K is given by We claim that this is an isomorphism. Then, (i) follows from the inverse function theorem for Banach spaces. By Hodge's theory for the operator d * ρ d + dd * ρ every σ ∈ S k,p a+H ρ can be written as a sum σ = ρ + dλ + h for a unique λ ∈ W k+1,p (M, T * M) and a h ∈ H ρ such that M dλ ∧ * ρ h = 0 and d * ρ λ = 0. Hence, Since h is closed, dd (+ ρ ) λ = 0, and thus Together with d * ρ λ = 0, this implies that λ is harmonic and h = 0. This shows that K is injective. Now, let η ∈ W k,p (M, Λ + ). Since W k,p is closed under products and composition with smooth functions for k − 4 p > 0, u 0 R ρ η is in W k,p (M, Λ + ρ ) and by Hodge's theory there exists a unique one-form λ ∈ W k+1,p (M, T * M) and a harmonic two-form h which is self-dual with respect to * ρ such that u 0 R ρ η = dλ + h. Then, K (dλ + h) = K (u 0 R ρ η) = η and this shows that K is surjective. This proves (i).
We prove (iii). Let ρ ∈ S k,p and let Lρ be a Lie derivative of ρ in an arbitrary direction. By Cartan's formula the Lie derivative of ρ is exact and by Lemma 3 The right hand side is a term of the form for polynomials P 1 , P 2 , P 3 with smooth coefficient functions in the indicated variables. It follows from elliptic regularity theory and product estimates for Sobolev spaces (see, e.g. [6,Lemma A.2]) that Lρ ∈ W k,p and that there exist polynomials p 1 , p 2 with positive coefficients independent of ρ with the following significance . Since the Lie derivative Lρ was arbitrary, the result follows.

The Evolution Equation for K(ρ)
In view of Theorem 1, the Donaldson flow has an equivalent description on the space of selfdual two-forms given by the evolution equation for K(ρ) = ρ + u . This equation exposes the parabolic nature of the Donaldson flow equation and it is the key for the regularity theorems which we will prove in the later sections.
To obtain a global formula, we introduce the operator S ρ , where λ ∈ Ω 1 , R ρ is defined by (4) and ρ(X λ , ·) := λ. We say ω 1 , ω 2 , ω 3 ∈ Ω + form a standard local frame of Ω + if and only if locally Theorem 2 (The Evolution of ρ + u ) (i) Suppose ρ is a smooth solution to the Donaldson flow equation. Then, the evolution of the 2-form ρ + u is given by the equation Here, R ρ is defined by (4).
(iii) Let ω 1 , ω 2 , ω 3 form a local standard frame of Ω + . Then, the evolution of the functions K i := ρ∧ω i dvol ρ is given by Here, X H denotes the Hamiltonian vector field of the function H . The bracket {·, ·} ρ denotes the Poisson bracket with respect to the symplectic structure ρ. E ρ ω and E ρ ω are error terms depending on the frame ω 1 , ω 2 , ω 3 that vanish whenever ∇ω i = 0 and are given by (iv) Assume the hyperKähler case. Then, the evolution of the functions K i = ρ∧ω i dvol ρ is given by Proof See page 13.
We need the following lemma on the properties of S ρ and its adjoint (S ρ ) * ρ .
Proof We prove (i). To compute the adjoint of S ρ let λ be a 1-form and ξ ∈ Ω + , then Here, we used the identity * ρ ι(X)ρ = −ρ ∧ g(X, ·) in the last equation. Using this identity again together with the definition of the Hodge star operator * ρ , we have Thus, we have proved that for all self-dual two-forms ξ ∈ Ω + (M) and one-forms λ ∈ Ω 1 (M). This proves (i). Part (ii) follows from the computation where we used that * ρ R ρ 2ρ + u = R ρ 2ρ + u in the first equation. This proves (ii). We prove (iii). Let ω ∈ Ω + . Observe that Now suppose ∇ω = 0. Then, we have dω = 0 and it follows from the last equation that (S ρ ) * ρ ω = 0. This proves (iii). We prove (iv). Let ξ ∈ Ω + and f ∈ Ω 0 (M, R). Then, This proves (iv). We prove (v). It follows from (iv) that where the last equality follows from identity (5). This proves (v).
We prove (vii). It follows from (v) that in a standard frame ω 1 , ω 2 , ω 3 for Ω + , S ρ is given by

This proves (vii).
We end the section with a proof of Theorem 2.

Regularity
In this section, we prove that a solution to the Donaldson flow equation is as smooth as its initial condition allows, given that it is an element of L 2 (I, W 2,p ) ∩ W 1,2 (I, L p ) for p > 4. In particular, it is smooth if its initial conditions are smooth. The proof combines two insights. First, the regularity of ρ + u determines the regularity of ρ. This is the content of Theorem 1 (iii). Second, the evolution of ρ + u is given by a parabolic operator, where the right hand side of the equation is essentially a product of two derivatives. This is the content of Theorem 2 (iii). This allows bootstrapping. The details are given in the next two theorems. The first theorem illustrates the ideas in the simpler case of a critical point. The next lemma will be needed in the bootstrapping process.
Proof Let Lρ be a Lie derivative of ρ in an arbitrary direction. Let ρ 0 be a smooth nondegenerate form such that ρ − ρ 0 L ∞ < δ for a small δ > 0. Then, there exists a unique 1-form λ ∈ W 1,p such that dλ = Lρ and d * ρ 0 λ = 0. By elliptic regularity theory for the operator d + ρ 0 + d * ρ 0 there exists a constant c > 0 such that for a polynomial p with positive coefficients vanishing at zero, we have Therefore, for polynomials with positive coefficient p 1 and p 2 , and Since this is true for an arbitrary Lie derivative of ρ and small enough δ > 0, it follows that This proves the lemma.
We will need the following lemma on elliptic regularity of the operator d * ρ d u : in the case that ρ and thus the coefficients are not smooth.
Lemma 6 (Elliptic Regularity) Let p > 4, q > 1, k ≥ 0 and let ρ ∈ W k+1,p (M, Λ 2 ) such that ρ ∧ ρ > 0. Assume there exists a constant c 0 > 0 such that for all v, w ∈ C ∞ (M) and all ε > 0 we can estimate Let v ∈ W 1,q (M) and f ∈ W k,q (M) such that Proof We only prove the case k = 0, the general case follows by induction over k. Choose coordinate charts for M and a subordinate partition of unity of M. Let ψ ∈ C ∞ 0 (M) be a cutoff function. Then, we have Let B ⊂ R 4 be a ball around zero. Let be the Hodge laplacian on R 4 with respect to the standard metric. We know from ellpitic regularity if v ∈ W . We choose a coordinate chart such that the image of the support of ψ is contained in B and the push-forward of ρ equals the standard symplectic structure at 0 ∈ B. We can always achieve this by a change of coordinates. Let us denote the push-forward of ρ under this coordinate by ρ α . Furthermore, we denote by ρ α the operator d * ρ d u expressed in this chart and by v α , f α the push-forward of ψv, f , respectively. By estimate (10), there exists a polynomial with positive coefficents p such that By interpolating ∂v α L q ≤ c v α 1 2 L q v α 1 2 W 2,q with the Galgliardo-Nirenberg interpolation inequality, we find If v α is a smooth solution to ρ α v α = f α , By choosing ε and the ball B small, we see that there exits a constant c 4 = c 4 (q, ρ W 1,p ) such that v W 2,q ≤ c 4 ( f L q + v L q ) .
then we approximate f and ρ by smooth functions f k , respectively, smooth nondegenerate two-forms ρ k , such that For each pair (f k , ρ k ) we find by standard L 2 -theory for elliptic operators a smooth function v k that solves and M v k dvol = M vdvol. The constant c 4 in (12) can be choosen uniformly in k for big k and it follows that {v k } k∈N has a weakly convergent subsequence with limitv ∈ W 2,q that satisfies the estimate (12) and d * ρ d uv = f . Hence, for all ϕ ∈ C 1 (M). By choosing a sequence ϕ k ∈ C 1 (M) such that ϕ k converges to (v −v) in W 1,2 (M) we see thatv = v. This proves the lemma.

Proof of Theorem 3
If ρ is a critical point of the Donaldson flow then it follows from Theorem 2 (iii) that for a local standard frame ω 1 , ω 2 , ω 3 and for cyclic permutations of i, j, k. The right hand side of this equation consists of products of derivatives of the functions K i times a polynomial in the ρ and 1 u variables plus lower order terms in the functions K i times another polynomial of the same form. Thus, schematically we may write Since 1 − 4 p > 0, ρ ∈ C 0 and since we assume that ρ is a symplectic structure, we have sup M 1 u < ∞. It follows that the L ∞ -norms of P 1 , P 2 are bounded. Using Hölder's inequality we see that the right hand side is an element of L p 2 . For two functions v, w ∈ C ∞ (M), we have the estimate where we used Hölder's inequality in the first inequality and the Gagliardo-Nirenberg interpolation inequality in the second. It follows from Lemma 6 that K is in W 2, p 2 . By Rellich's embedding theorem W 1, p 2 → L p where For 4 < p < 8, we have p > p.
Thus, K ∈ W 1,p and by Lemma 5 ρ ∈ W 1,p . If we repeat this argument with p replaced by p , we find that K ∈ W 1, p 2 , p > p and Hence, eventually we find that K ∈ W 2,q and ρ ∈ W 1,q for all q ≥ p. Now, we can use Theorem 1 to see that ρ ∈ W 2,q as well. This implies that the right hand side of (13) is in W 1,q and elliptic regularity gives us that K ∈ W 3,q . Now, an obvious iteration of these arguments using elliptic regularity and Theorem 1 deduces the regularity of ρ from the regularity of K, proving the theorem.
Theorem 4 (Flow Lines) Let ρ ∈ W 1,2 (I, L p ) ∩ L 2 (I, W 2,p ) be a solution to the Donaldson flow equation for p > 4 with initial condition ρ(t = 0, ·) = ρ 0 . For every integer k ≥ 1 and all 4 < p < p the following are equivalent: The proof of this theorem uses parabolic regularity theory. In particular, we use the 'maximal regularity' property of parabolic operators in divergence form. We refer to [7] for these results. The maximal regularity property is usually formulated for operators with time independent smooth coefficients in divergence form, the Hodge laplacian being the archetypal example. The next lemma assures that the operator has the maximal regularity property as well, even though its coefficients depend on time and are non-smooth in our used case.
be a path of nondegenerate forms. Assume that there exists a constant c 0 > 0 such that for all v, w ∈ C ∞ (M) and all ε > 0 we can estimate Then for all v ∈ C ∞ 0 (R, C ∞ 0 (M)), there exists a constant c(q, ρ L ∞ (R,W 1,p ) ) > 0 such that Proof Choose coordinate charts for R × M and a subordinate partition of unity. Let ψ ∈ C ∞ (R×M) be a cutoff function. Let I ×B ∈ R×R 4 be a ball around (0, 0). Choose a coordinate chart such that the support of ψ is mapped into I × B and such that the pushforward of ρ under this coordinate chart at (0, 0) equals the standard symplectic structure on R 4 . We can always achieve this by a change of coordinates. Let us denote the pushforward of ρ under this coordinate chart by ρ α and the operator d * ρ d u expressed in this chart by ρ α . Further we denote by v α the function ψv expressed in this chart. From maximal regularity for the standard Laplace operator on R × R 4 there exists a constant c 1 = c 1 (q) > 0 such that ∂ t v α L 2 (R,L q ) ≤ c 1 ∂ t v α + v α L 2 (R,L q ) ≤ c 1 ∂ t v α + ρ α v α L 2 (R,L q ) + ( − ρ α )v α L 2 (R,L q ) .
By choosing the partition of unity such that ψ has small enough support and using the estimate (11), we find that there exists a constant c 2 = c 2 (q, ρ L ∞ (R,W 1,p ) ) such that ∂ t v α L 2 (R,L q ) ≤ c 2 ∂ t v α + ρ α v α L 2 (R,L q ) + v α L 2 (R,L q ) .
From this the global estimate, follows by choosing a partition of unity such that the previous estimates hold for all chart domains and by estimating additional first order terms appearing from the multiplication of v with the cutoff functions with the Gagliardo-Nirenberg interpolation inequality. This proves the lemma.
Since W 1,p ⊂ C 0 for p > 4, we have The last term on the right handside of (15) is in L 2 (I, W 1,p ), hence the right hand side of (15) is in L 2 (I, W 1, p 2 ). As in the critical point case, the estimate (14) is valid for any two functions v, w ∈ C ∞ (M). Then by the ellipitic regularity Lemma 6 and the maximal regularity Lemma 7, we have