Abstract
The mixed equation, defined as a combination of the antiselfduality equation in gauge theory and Cauchy–Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah–Floer conjecture.
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Notes
Here we diverge from our convention that connections on 3manifolds are denoted by the letter B because soon we will focus on the case that Y=S^{1}×Σ and β_{0} is the pullback of a connection on Σ.
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Acknowledgements
The third author would like to thank Tomasz Mrowka, Dusa McDuff and Dennis Sullivan for useful conversations. The authors are also grateful to the Simons Center for Geometry and Physics for providing a stimulus environment where part of this paper and the sequel one were being completed at different stages.
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Appendices
Appendix A: Elliptic regularity of bundlevalued 1forms
In this appendix, first we review some wellknown results about regularity of the Laplace–Beltrrami operator. Then we consider slight variations to the case of bundle valued maps. Throughout this section, M denotes a compact Riemannian manifold possibly with boundary. In this appendix, for any Riemannian manifold M and differential kforms α and β on M, we slightly diverge from our notation in (1.20), and denote the inner product of α and β by
For any real number r>1, we also write r^{∗} for the conjugate of r which satisfies
The following lemma is a standard fact about the Laplace–Beltrrami operator (see [GT13, Theorems 9.14 and 9.15], [ADN59, Theorem 15.2] and [Weh042, Chaps. 3 and Appendix D].)
Lemma A.1
Let k be a nonnegative integer and p>1 be a real number. Let u be an \(L^{p}_{k}\) function on M.

(i)
If k≥1, suppose there is an \(L^{p}_{k1}\) function F on M such that for any smooth function φ with φ_{∂M}=0, we have
$$ \int _{M} \langle u, \Delta \varphi \rangle = \int _{M} \langle F, \varphi \rangle . $$(A.2)Then u is in \(L^{p}_{k+1}(M)\), and there is a constant C, independent of u, such that
$$ \!u\!_{L^{p}_{k+1}(M)}\leq C(\!F\!_{L^{p}_{k1}(M)}+\!u\!_{L^{p}(M)}). $$(A.3)In the case that k=0, the assumption (A.2) has to be replaced with
$$ \int _{M} \langle u, \Delta \varphi \rangle \leq \kappa \! \varphi \!_{L^{p^{*}}_{1}(M)}, $$(A.4)and the conclusion (A.3) has to be modified to:
$$ \!u\!_{L^{p}_{1}(M)}\leq C(\kappa +\!u\!_{L^{p}(M)}). $$(A.5) 
(ii)
If k≥1, suppose there are functions F and G on M such that for any smooth function φ with ∂_{ν}φ_{∂M}=0 we have:
$$ \int _{M} \langle u, \Delta \varphi \rangle = \int _{M} \langle F, \varphi \rangle +\int _{\partial M} \langle G, \varphi \rangle . $$(A.6)If F and G are respectively in \(L^{p}_{k1}(M)\) and \(L^{p}_{k}(M)\), then u is in \(L^{p}_{k+1}(M)\). Furthermore, there is a constant C, independent of u, such that
$$ \!u\!_{L^{p}_{k+1}(M)}\leq C(\!F\!_{L^{p}_{k1}(M)}+\!G\!_{L^{p}_{k}(M)}+ \!u\!_{L^{p}(M)}). $$(A.7)In the case that k=0, the assumption (A.6) has to be replaced with:
$$ \int _{M} \langle u, \Delta \varphi \rangle \leq \kappa \! \varphi \!_{L^{p^{*}}_{1}(M)}, $$(A.8)and the conclusion (A.9) has to be modified to:
$$ \!u\!_{L^{p}_{1}(M)}\leq C(\kappa +\!u\!_{L^{p}(M)}). $$(A.9)
We recall the following definition from Sect. 3.1 about some functions spaces associated to the sections of a vector bundle.
Definition A.10
Suppose U is a (possibly noncompact) manifold with boundary and E is a vector bundle over U. Then the space of smooth sections of E with compact support are denoted by Γ_{c}(U,E). The space of compactly supported sections of E, which vanish on the boundary of E, are denoted by Γ_{τ}(U,E). Suppose a connection A_{0} is fixed on E. Then Γ_{ν}(U,E) is the space of all compactly supported sections s of E such that the covariant derivative of s in the normal directions to the boundary of U vanish.
The following Lemma is a slightly generalized version of [Weh051, Lemma A.2].
Lemma A.11
Let k be a positive integer, and r>1 be a real number. Let M be a compact nmanifold with boundary and a Riemannian metric g, U be an open subset of M, and K be an open subspace of U whose closure in U is compact. Let E be an SO(3)vector bundle over M equipped with a smooth connection A_{0}. Let σ be a smooth vector field on U. Let Γ_{∘}(U,E) be one of the spaces Γ_{τ}(U,E) or Γ_{ν}(U,E), where Γ_{ν}(U,E) is defined using A_{0}. Then there is a constant C such that the following holds. Let
and for any ϕ∈Γ_{c}(U,E), ψ∈Γ_{∘}(U,E) we have
Then α(σ) is an element of \(L^{r}_{k+1}(K)\) and we have:
Proof
Without loss of generality, we may assume that U is a precompact open subset of the halfspace
E is trivialized over U and the connection A_{0} is given by a 1form with values in R^{3}. We will denote this 1form with A_{0}, too. We may pick this trivialization in a way that the normal covariant derivative with respect to the connection A_{0} agrees with the ordinary derivative. That is to say, Γ_{ν}(U,R^{3}) defined with respect to A_{0} is the space of all compactly supported sections η of R^{3} such that ∂_{ν}η vanishes along the boundary.
Fix a function ρ:M→R which is supported in U and is equal to 1 on K. Then we show that there are compactly supported maps F and G from U to R^{3} such that for any η∈Γ_{∘}(U,R^{3}) we have
and F, G respectively have finite \(L^{r}_{k1}\), \(L^{r}_{k}\) norms.
First we claim that
where B_{g} is defined by firstly taking the Lie derivative \(\mathcal {L}_{\sigma }g\) of the Riemannian metric g and then requiring B_{g} to satisfy the following identity for any pair of 1forms β and β′:
To see (A.15), we pick a sequence {α_{i}}_{i∈N} of smooth 1forms on U with values in R^{3} such that α_{i} vanishes in a neighborhood of \(U\cap \partial {\mathbb{H}}^{n}\) and the sequence {α_{i}} is L^{r}convergent to α. Then the left hand side of (A.15) is equal to
Now by taking the limit in (A.16) we obtain the desired identity.
The assumption (A.12) can be used to rewire the first term in the right hand side of (A.15) as
where ∗_{n−1} in the last line denotes the Hodge operator on ∂M.
We rewrite the second term in the right hand side of (A.15) as
Therefore, we can use (A.13), to write
Finally, the last three terms of (A.15) are equal to
By applying further integration by parts to the expressions in (A.18), we can find F and G satisfying (A.14), which are respectively in \(L^{r}_{k1}\) and \(L^{r}_{k}\), and satisfy
for some constant C′ depending only on A_{0}, g, σ, U and K. Therefore, Lemma A.1 (part (i) or (ii) depending on whether ∘=τ or ν) implies that
This inequality proves the desired claim. □
The following lemma is an extension of the previous lemma to the case that k=0.
Lemma A.20
Let r, M, K, U, σ, E and A_{0} be as in Lemma A.11. Let ∘ be either τ and ν. There is a constant C such that the following holds. Let α be an L^{r} section of Λ^{1}⊗E over the open subset U of M such that for any ϕ∈Γ_{c}(U,E) and ψ∈Γ_{∘}(U,E):
Then α(σ) belongs to \(L^{r}_{1}(K)\) and
Proof
In the following C is a constant independent of α which might increase from each line to the next one. As in the proof of Lemma A.11, we can show that α satisfies (A.15). In particular, we have
The first term on the left hand side of the above inequality can be estimated as in (A.17):
To. obtain the second inequality, we use the first assumption in (A.21). Next, we find an upper bound for the second term in (A.23) using the second inequality in (A.21) following an argument similar to the previous lemma:
It is straightforward to bound the remaining three terms in (A.23) with \(C\! \alpha \!_{L^{r}(U)}\!\eta \!_{L^{r^{*}}_{1}(U)}\). Consequently, Lemma A.1 implies that α(σ) is in \(L^{r}_{1}(K)\) and (A.22) holds. □
Lemma A.26
Let k be a nonnegative integer and r>1 is a real number. Suppose M is a Riemannian manifold possibly with boundary. Suppose Σ is a closed surface and F is an SO(3)bundle over Σ. Suppose β={β_{x}}_{x∈M} is a smooth family of connections on F parametrized by M. Suppose f is an \(L^{r}_{k}\) section of the bundle T^{∗}Σ⊗F over Σ×M. If k≥1, suppose there are \(L^{r}_{k}\) sections ζ_{1} and ζ_{2} of the pullback of F over Σ×M such that for any smooth section ξ of the pullback of F over Σ×M, we have
where d_{β}ξ denotes the section of T^{∗}Σ⊗F over Σ×M given by the exterior derivatives of ξ in the Σ direction with respect to the family of connections β. Then \(\nabla ^{\beta}_{\Sigma }f\), the covariant derivative of f in the Σ direction with respect to β, is in \(L^{r}_{k}\), and there is a constant C, independent of f, such that:
In the case that k=0, the assumption (A.27) has to be replaced with
In this case, \(\nabla ^{\beta}_{\Sigma }f\) belongs to L^{r}(X×Σ) and
Lemma A.26 can be regarded as the family version of A.1 where we also replace the degree two elliptic operator Δ with the degree one operator \(d_{\beta}\oplus d_{\beta}^{*}\). This proposition in the case that F is the trivial bundle and β is the trivial family of connections is proved in [Weh051, Lemma 2.9]. Clearly, this implies the lemma for the case that F is trivial and B is arbitrary. The proof in the case that F is nontrivial is similar.
Appendix B: Regularity of holomorphic curves in a Banach space
Suppose B is a Banach space and M is a compact Riemannian manifold. In this appendix, we are interested in maps from M to B. For 1<p<∞ and any nonnegative integer k, we can define the Sobolev norm \(\!\cdot \!_{L^{p}_{k}}\) on the space of such maps in the usual way. The completion of space of smooth maps from M to B with respect to this Sobolev norm is denoted by \(L^{p}_{k}(M,B)\). As an example, let B=L^{p}(N) for a compact manifold N. Any function in C^{∞}(M×N), determines an element of L^{p}(M,B). In fact, the space of smooth functions on M×N is dense in L^{p}(M,B) (see [Weh041] and [Lip14]). This gives us the following identifications of Sobolev spaces:
More generally, C^{∞}(M×N) is dense in \(L^{p}_{k}(M,L^{p}(\Sigma ))\) for any nonnegative integer k, and we have (see [Weh041, Lip14]):
For the rest of this appendix, we fix B_{p} to be a Banach space that can be identified with a closed subspace of the space L^{p}(N) for a closed manifold N. In particular, the intersection B_{q}:=B_{p}∩L^{q}(N) with q>p determines a closed subspace of L^{q}(N). For q<p, B_{q} is the closure of B_{p} in L^{q}(N).
Lemma B.2
Suppose M is a Riemannian manifold with boundary. Let k be a nonnegative integer and p>1 be a real number. Let \(u\in L^{p}_{k}(M,B_{p})\). Then the same claims as in parts (i) and (ii) of Lemma A.1 hold if we assume that F, G and φ are B_{p}valued.
Sketch of the Proof.
Without loss of generality, we can assume that B_{p}=L^{p}(N). Using the identifications in (B.1), we can regard u as an L^{p} map from N to the Banach space \(L^{p}_{k}(M)\). Next, we can apply the properties of the Laplacian operator acting on \(L^{p}_{k}(M)\) to obtain the desired conclusions. For more details, we refer the reader to [Weh041, Lemma 2.1] and [Lip14, Sect. 3.3]. □
The proof of the following proposition about regularity of Banach valued Cauchy–Riemann equation can be found in [Weh041, Theorem 1.2] and [Lip14, Lemmas 27 and 28]. In this proposition, B_{p} denotes the direct sum B_{p}⊕B_{p}. This space admits an obvious complex structure J_{0} given by
The subspace \(\mathcal {L}:=0\oplus B_{p}\) defines a completely real subspace of B_{p} with respect to \(\mathcal {J}_{0}\).
Proposition B.4
Suppose U is a bounded open subspace of
and U_{∂} denotes the intersection \({\mathbb{H}}^{2}\cap U\). Suppose \(\mathcal {J}:\mathbf{B}_{p}\to {\mathrm{End}}(\mathbf{B}_{p},\mathbf{B}_{p})\) is a smooth family of complex structures such that \(\mathcal {J}(x)=\mathcal {J}_{0}\) for \(x\in \mathcal {L}\). For p>2 and k≥2, suppose u:U→B_{p} is an \(L^{p}_{k}\) map that satisfies
and the boundary condition
Then for any open subspace K⊂U, whose closure in U is compact, the map u is in \(L^{p}_{k+1}(K)\). Moreover, there is a constant C, depending only on K, such that
If u_{i}:U→B_{p} is a sequence of \(L^{p}_{k}\) map that satisfies
such that u_{i} and z_{i} are respectively \(L^{p}_{k}\)convergent to u and z, then u_{i} restricted to K is \(L^{p}_{k+1}\)convergent to the restriction of u to K. In the case that k=1, similar results hold if we replace \(L^{p}_{k+1}\) with \(L^{p/2}_{k+1}\).
Sketch of the proof
For k≥2, suppose u is a map that satisfies (B.5) and (B.6). We apply \(\partial _{\theta}+\mathcal {J}(u)\partial _{s}\) to (B.5). Then we have:
Using the assumptions k≥2, \(u\in L^{p}_{k}\) and \(z\in L^{p}_{k}\), we can conclude that the left hand side of the above identity is an element of \(L^{p}_{k1}\). The maps u and z can be written as (u_{0},u_{1}) and (z_{0},z_{1}) with respect to the decomposition of B_{p}. The boundary condition (B.6) implies that \(u_{0}_{U_{\partial}}=0\) and \(\partial _{s}u_{1}_{U_{\partial}}=z_{0}_{U_{\partial}}\). Therefore, we can invoke Lemma B.2 to verify the claim. To be a bit more detailed, we use the assumption k≥2 to conclude that the products of two \(L^{p}_{k1}\) functions are still in \(L^{p}_{k1}\). In the case that k=1, the products of two L^{p}(U,B_{p}) functions is in L^{p/2}(U,B_{p}), which in turn is a subspace of L^{p/2}(U,B_{p/2}). That allows us to use the same argument to prove the claim in this case. The sequential versions of these claims can be also treated similarly. □
We need a slight improvement of Proposition B.4 to the case k=0 [Lip14, Lemma 29].
Proposition B.10
Suppose U is given as in Proposition B.4. Suppose \(\mathcal {J}:\mathbf{B}_{p}\to {\mathrm{End}}(\mathbf{B}_{p},\mathbf{B}_{p})\) is a smooth family of complex structures such that \(\mathcal {J}(x)=\mathcal {J}_{0}\) for \(x\in \mathcal {L}\) and for any x∈B_{p}, the space \(\mathcal {L}\) is totally real with respect to \(\mathcal {J}(x)\), i.e., \(\mathbf{B}_{p}=\mathcal {L}\oplus \mathcal {J}(x)\mathcal {L}\). For p>2, let u:U→B_{p} be in \(L^{p}_{1}\). Suppose q>p and u is also an L^{q} map from U to B_{q}. Suppose u satisfies
and the boundary condition (B.6). Then u is an \(L^{q}_{1}\) map from U to B_{q} and
Moreover, if u_{i}:U→B_{p} are \(L^{q}_{1}\) solutions of
such that u_{i} is convergent to u in \(L^{p}_{1}\cap L^{q}\) and z_{i} is convergent to z in L^{q}, then u_{i} is convergent to u in \(L^{q}_{1}\).
Proof
Given p>2 and any bounded domain Ω in R^{2} with smooth boundary, let \(L^{p}_{1}(\Omega ,\mathbf{B}_{p})_{\partial}\) be the space of \(L^{p}_{1}\) maps u:Ω→B_{p} such that the restriction of u to the boundary is in \(\mathcal {L}\). Then the Cauchy–Riemann operator
is a surjective bounded operator with kernel being constant maps to \(\mathcal {L}\). This can be seen in the same way as in Lemma B.2.
Now suppose x∈∂U and \(D_{r}(x)=B_{r}(x)\cap {\mathbb{H}}^{2}\) is contained in U. Suppose Ω_{r} is the region given by rounding the corners of D_{r}(x) such that it is contained in D_{r}(x) and it contains D_{r/2}(x). Since \(\mathcal {J}(u):U \to {\mathrm{End}}(\mathbf{B}_{p},\mathbf{B}_{p})\) is continuous and \(\mathcal {J}(x)=J_{0}\), the operator \(\partial _{\theta }\mathcal {J}(u)\partial _{s}:L^{p}_{1}(\Omega _{r}, \mathbf{B}_{p})_{\partial }\to L^{p}(\Omega _{r},\mathbf{B}_{p})\) is surjective with kernel being constant maps to \(\mathcal {L}\) if r is small enough. This holds because the operator \(\partial _{\theta }\mathcal {J}(u)\partial _{s}\) is a deformation of the operator in (B.14) by a bounded operator of small norm for small values of r. We assume that r is chosen such that the same claim holds if we replace q with p. Now let ρ:Ω_{r}→R be a smooth bump function that vanishes on the complement of D_{r/2}(x) and equals 1 on D_{r/3}(x). Then our assumption implies that ρu is an element of \(L^{p}_{1}(\Omega _{r},\mathbf{B}_{p})_{\partial}\) and
is in L^{q}. Thus there is \(u'\in L^{q}_{1}(\Omega ,\mathbf{B}_{q})_{\partial}\) such that
This implies that u′−ρu is a constant map to \(\mathcal {L}\). In particular, the restriction of u to D_{r/3}(x) is in \(L^{q}_{1}(\Omega ,\mathbf{B}_{q})_{\partial}\). For an interior point x, we may apply a similar argument to show that the restriction of u to a neighborhood of x in is \(L^{q}_{1}(\Omega ,\mathbf{B}_{q})_{\partial}\). The only new point that we need is that we can find an isomorphism T:B_{p}→B_{p} such that \(T^{1} \mathcal {J}(x) T=\mathcal {J}_{0}\). In fact, we may take T to be the linear map that sends (v_{0},v_{1})∈B_{p}⊕B_{p} to \((v_{0},0)+\mathcal {J}(x)(v_{1},0)\). Since \(\mathcal {L}\) is totally with respect to \(\mathcal {J}(x)\), T is an isomorphism. □
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Daemi, A., Fukaya, K. & Lipyanskiy, M. Lagrangians, SO(3)Instantons and Mixed Equation. Geom. Funct. Anal. 34, 659–732 (2024). https://doi.org/10.1007/s00039024006778
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DOI: https://doi.org/10.1007/s00039024006778