Skip to main content
Log in

Lagrangians, SO(3)-Instantons and Mixed Equation

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript


The mixed equation, defined as a combination of the anti-self-duality 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Figure 1
Figure 2

Similar content being viewed by others


  1. Here we diverge from our convention that connections on 3-manifolds are denoted by the letter B because soon we will focus on the case that Y=S1×Σ and β0 is the pullback of a connection on Σ.


  1. Aebischer, B., Borer, M., Kälin, M., Leuenberger, C., Reimann, H.M.: Symplectic Geometry. Progress in Mathematics, vol. 124. Birkhäuser, Basel (1994). An introduction based on the seminar in Bern, 1992

    Book  Google Scholar 

  2. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  Google Scholar 

  3. Atiyah, M.: New invariants of 3- and 4-dimensional manifolds. In: The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987), pp. 285–299 (1988)

    Chapter  Google Scholar 

  4. Bourguignon, J.-P., Blaine Lawson, H. Jr.: Stability and isolation phenomena for Yang-Mills fields. Commun. Math. Phys. 79(2), 189–230 (1981)

    Article  MathSciNet  Google Scholar 

  5. Daemi, A., Fukaya, K., Lipyanskiy, M.: Lagrangians, SO(3)-instantons and the Atiyah-Floer conjecture. arXiv preprint (2021)

  6. Donaldson, S., Kronheimer, P.: The Geometry of Four-Manifolds. Oxford University Press, London (1990)

    Book  Google Scholar 

  7. Donaldson, S.K.: Floer Homology Groups in Yang-Mills Theory. Cambridge Tracts in Mathematics, vol. 147. Cambridge University Press, Cambridge (2002). With the assistance of M. Furuta and D. Kotschick

    Book  Google Scholar 

  8. Donaldson, S.K.: Polynomial invariants for smooth four-manifolds. Topology 29(3), 257–315 (1990)

    Article  MathSciNet  Google Scholar 

  9. Dostoglou, S., Salamon, D.A.: Self-dual instantons and holomorphic curves. Ann. Math. (2) 139(3), 581–640 (1994)

    Article  MathSciNet  Google Scholar 

  10. Floer, A.: An instanton-invariant for 3-manifolds. Commun. Math. Phys. 118(2), 215–240 (1988)

    Article  MathSciNet  Google Scholar 

  11. Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. 28(3), 513–547 (1988)

    Article  MathSciNet  Google Scholar 

  12. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I. AMS/IP Studies in Advanced Mathematics, vol. 46. Am. Math. Soc., Providence; International Press, Somerville (2009)

    Google Scholar 

  13. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part II. AMS/IP Studies in Advanced Mathematics, vol. 46. Am. Math. Soc., Providence; International Press, Somerville (2009)

    Google Scholar 

  14. Fukaya, K.: Anti-self-dual equation on 4-manifolds with degenerate metric. Geom. Funct. Anal. 8(3), 466–528 (1998)

    Article  MathSciNet  Google Scholar 

  15. Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)

    Article  MathSciNet  Google Scholar 

  16. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2013)

    Google Scholar 

  17. Hang, F., Lin, F.: A Liouville type theorem for minimizing maps. Methods Appl. Anal. 9(3), 407–424 (2002). Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday

    Article  MathSciNet  Google Scholar 

  18. Lipyanskiy, M.: Gromov-Uhlenbeck Compactness. arXiv preprint (2014). arXiv:1409.1129

  19. McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology, 2nd edn. American Mathematical Society Colloquium Publications, vol. 52. Am. Math. Soc., Providence (2012)

    Google Scholar 

  20. Oh, Y.-G.: Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. Commun. Pure Appl. Math. 46(7), 949–993 (1993)

    Article  MathSciNet  Google Scholar 

  21. Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27(1), 1–33 (1995)

    Article  MathSciNet  Google Scholar 

  22. Salamon, D., Wehrheim, K.: Instanton Floer homology with Lagrangian boundary conditions. Geom. Topol. 12(2), 747–918 (2008)

    Article  MathSciNet  Google Scholar 

  23. Uhlenbeck, K.K.: Connections with Lp bounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982)

    Article  Google Scholar 

  24. Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys. 83(1), 11–29 (1982)

    Article  MathSciNet  Google Scholar 

  25. Wehrheim, K.: Banach space valued Cauchy-Riemann equations with totally real boundary conditions. Commun. Contemp. Math. 6(4), 601–635 (2004)

    Article  MathSciNet  Google Scholar 

  26. Wehrheim, K.: Uhlenbeck Compactness. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2004)

    Book  Google Scholar 

  27. Wehrheim, K.: Anti-self-dual instantons with Lagrangian boundary conditions. I. Elliptic theory. Commun. Math. Phys. 254(1), 45–89 (2005)

    Article  MathSciNet  Google Scholar 

  28. Wehrheim, K.: Anti-self-dual instantons with Lagrangian boundary conditions II: bubbling. Commun. Math. Phys. 258(2), 275–315 (2005)

    Article  MathSciNet  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Appendix A: Elliptic regularity of bundle-valued 1-forms

In this appendix, first we review some well-known 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 k-forms α and β on M, we slightly diverge from our notation in (1.20), and denote the inner product of α and β by

$$ \int _{M}\langle \alpha ,\beta \rangle . $$

For any real number r>1, we also write r for the conjugate of r which satisfies

$$ \frac{1}{r}+\frac{1}{r^{*}}=1. $$

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 non-negative integer and p>1 be a real number. Let u be an \(L^{p}_{k}\) function on M.

  1. (i)

    If k≥1, suppose there is an \(L^{p}_{k-1}\) function F on M such that for any smooth function φ with φ|∂M=0, we have

    $$ \int _{M} \langle u, \Delta \varphi \rangle = \int _{M} \langle F, \varphi \rangle . $$

    Then u is in \(L^{p}_{k+1}(M)\), and there is a constant C, independent of u, such that

    $$ |\!|u|\!|_{L^{p}_{k+1}(M)}\leq C(|\!|F|\!|_{L^{p}_{k-1}(M)}+|\!|u|\!|_{L^{p}(M)}). $$

    In the case that k=0, the assumption (A.2) has to be replaced with

    $$ |\int _{M} \langle u, \Delta \varphi \rangle |\leq \kappa |\!| \varphi |\!|_{L^{p^{*}}_{1}(M)}, $$

    and the conclusion (A.3) has to be modified to:

    $$ |\!|u|\!|_{L^{p}_{1}(M)}\leq C(\kappa +|\!|u|\!|_{L^{p}(M)}). $$
  2. (ii)

    If k≥1, suppose there are functions F and G on M such that for any smooth function φ with νφ|∂M=0 we have:

    $$ \int _{M} \langle u, \Delta \varphi \rangle = \int _{M} \langle F, \varphi \rangle +\int _{\partial M} \langle G, \varphi \rangle . $$

    If F and G are respectively in \(L^{p}_{k-1}(M)\) 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

    $$ |\!|u|\!|_{L^{p}_{k+1}(M)}\leq C(|\!|F|\!|_{L^{p}_{k-1}(M)}+|\!|G|\!|_{L^{p}_{k}(M)}+| \!|u|\!|_{L^{p}(M)}). $$

    In the case that k=0, the assumption (A.6) has to be replaced with:

    $$ |\int _{M} \langle u, \Delta \varphi \rangle |\leq \kappa |\!| \varphi |\!|_{L^{p^{*}}_{1}(M)}, $$

    and the conclusion (A.9) has to be modified to:

    $$ |\!|u|\!|_{L^{p}_{1}(M)}\leq C(\kappa +|\!|u|\!|_{L^{p}(M)}). $$

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 non-compact) 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 Γc(U,E). The space of compactly supported sections of E, which vanish on the boundary of E, are denoted by Γτ(U,E). Suppose a connection A0 is fixed on E. Then Γν(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 n-manifold 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 A0. Let σ be a smooth vector field on U. Let Γ(U,E) be one of the spaces Γτ(U,E) or Γν(U,E), where Γν(U,E) is defined using A0. Then there is a constant C such that the following holds. Let

$$ \begin{aligned} &f\in L^{r}_{k}(U,E), \hspace{.5cm} \alpha ,\xi \in L^{r}_{k}(U,\Lambda ^{1}(M)\otimes E), \\ &\zeta \in L^{r}_{k-1}(U,\Lambda ^{1}(M)\otimes E), \hspace{.5cm} \omega \in L^{r}_{k}(U,\Lambda ^{2}(M)\otimes E), \end{aligned} $$

and for any ϕ∈Γc(U,E), ψ∈Γ(U,E) we have

$$ \int _{M}\langle \alpha , d_{A_{0}} \phi \rangle =\int _{M} \langle f, \phi \rangle , $$
$$ \int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}(\psi \cdot \iota _{ \sigma }g) \rangle = \int _{M}\langle \omega , d_{A_{0}} (\psi \cdot \iota _{\sigma }g)\rangle +\int _{M}\langle \zeta , \psi \cdot \iota _{ \sigma }g \rangle +\int _{\partial M}\langle \xi , \psi \cdot \iota _{ \sigma }g \rangle . $$

Then α(σ) is an element of \(L^{r}_{k+1}(K)\) and we have:

$$ |\!| \alpha (\sigma )|\!|_{L^{r}_{k+1}(K)}\leq C (|\!|f|\!|_{L_{k}^{r}(U)}+| \!|\xi |\!|_{L_{k}^{r}(U)} +|\!|\zeta |\!|_{L_{k-1}^{r}(U)}+|\!| \omega |\!|_{L_{k}^{r}(U)}+ |\!| \alpha |\!|_{L_{k}^{r}(U)}). $$


Without loss of generality, we may assume that U is a precompact open subset of the half-space

$$ \mathbb{H}^{n}:=\{(x_{1},\dots ,x_{n})\in \mathbf{R}^{n}\mid x_{1} \geq 0\}, $$

E is trivialized over U and the connection A0 is given by a 1-form with values in R3. We will denote this 1-form with A0, too. We may pick this trivialization in a way that the normal covariant derivative with respect to the connection A0 agrees with the ordinary derivative. That is to say, Γν(U,R3) defined with respect to A0 is the space of all compactly supported sections η of R3 such that νη vanishes along the boundary.

Fix a function ρ:MR 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 R3 such that for any η∈Γ(U,R3) we have

$$ \int _{M} \langle \rho \alpha (\sigma ), \Delta \eta \rangle = \int _{M} \langle F, \eta \rangle +\int _{\partial M} \langle G, \eta \rangle $$

and F, G respectively have finite \(L^{r}_{k-1}\), \(L^{r}_{k}\) norms.

First we claim that

$$\begin{aligned} \int _{M} \langle \rho \alpha (\sigma ), \Delta \eta \rangle =&-\int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle - \int _{M} \langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta )\rangle - \int _{M}\rho \operatorname{div}(\sigma ) \langle \alpha , d\eta \rangle \\ &-\int _{M} \rho \langle B_{g}\alpha , d\eta \rangle -\int _{M} \langle \iota _{\sigma }(d\rho \wedge \alpha ), d\eta \rangle , \end{aligned}$$

where Bg is defined by firstly taking the Lie derivative \(\mathcal {L}_{\sigma }g\) of the Riemannian metric g and then requiring Bg to satisfy the following identity for any pair of 1-forms β and β′:

$$ \mathcal {L}_{\sigma }(g)(\beta ,\beta ')=\langle B_{g} \beta ,\beta ' \rangle. $$

To see (A.15), we pick a sequence {αi}iN of smooth 1-forms on U with values in R3 such that αi vanishes in a neighborhood of \(U\cap \partial {\mathbb{H}}^{n}\) and the sequence {αi} is Lr-convergent to α. Then the left hand side of (A.15) is equal to

$$\begin{aligned} &\lim _{i\to \infty}\int _{M} \langle \rho \alpha _{i}(\sigma ),d^{*}d \eta \rangle \\ &\qquad =\lim _{i\to \infty} \int _{M} \langle d\iota _{\sigma }( \rho \alpha _{i}),d\eta \rangle =\lim _{i\to \infty}\left [ \int _{M} \langle \mathcal {L}_{\sigma }(\rho \alpha _{i}),d \eta \rangle - \langle \mathcal {\iota}_{\sigma }d(\rho \alpha _{i}),d \eta \rangle \right ] \\ &\qquad =\lim _{i\to \infty}\left [-\int _{M} \langle \rho \alpha _{i}, \mathcal {L}_{\sigma }d \eta \rangle - \int _{M}\operatorname{div}( \sigma ) \langle \rho \alpha _{i},d \eta \rangle -\int _{M}\mathcal {L}_{ \sigma}(g)(\rho \alpha _{i}, d \eta )\right . \\ &\quad \qquad{} \left . -\int _{M}\langle d\alpha _{i}, \rho \iota _{\sigma }g \wedge d\eta \rangle -\int _{M}\langle \iota _{\sigma }(d\rho \wedge \alpha _{i}), d\eta \rangle \right ] \\ &\qquad =\lim _{i\to \infty}\left [-\int _{M} \rho \langle \alpha _{i},d \iota _{\sigma }d\eta \rangle - \int _{M} \rho \operatorname{div}( \sigma ) \langle \alpha _{i}, d\eta \rangle -\int _{M} \rho \langle B_{g} \alpha _{i}, d\eta \rangle \right . \\ &\quad \qquad {}\left . -\int _{M}\langle \alpha _{i}, d^{*}(\rho \iota _{\sigma }g \wedge d\eta )\rangle -\int _{M}\langle \iota _{\sigma }(d\rho \wedge \alpha _{i}), d\eta \rangle \right ]. \end{aligned}$$

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

$$\begin{aligned} \int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle =& \int _{M} \langle \alpha ,d_{A_{0}}(\rho \iota _{\sigma }d\eta ) \rangle -\int _{M} \langle \alpha ,\rho [A_{0},\iota _{\sigma }d \eta ] \rangle - \int _{M} \langle \alpha ,\mathcal {(}d\rho )\cdot ( \iota _{\sigma }d\eta ) \rangle \\ =&\int _{M} \langle \rho f-*[\rho \alpha ,*A_{0}]-*(\alpha \wedge *d \rho ), \iota _{\sigma }d\eta \rangle \\ =&\int _{M} \langle \left (\rho f-*[\rho \alpha ,*A_{0}]-*(\alpha \wedge *d\rho )\right )\iota _{\sigma }g, d\eta \rangle \\ =&\int _{M} \langle d^{*}\mathopen{}\left ( \left (\rho f-*[\rho \alpha ,*A_{0}]-*(\alpha \wedge *d\rho )\right )\iota _{\sigma }g \right )\mathclose{},\eta \rangle \\ &+\int _{\partial M} \langle *_{n-1}*(\left (\rho f-*[\rho \alpha ,*A_{0}]-*( \alpha \wedge *d\rho )\right )\iota _{\sigma }g), \eta \rangle , \end{aligned}$$

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

$$\begin{aligned} \int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle =&\int _{M}\langle \alpha , d^{*}( \eta d(\rho \iota _{ \sigma }g))\rangle -\int _{M}\langle \alpha , d^{*}d(\rho \iota _{ \sigma }g \eta )\rangle \\ =&\int _{M}\langle d \alpha , \eta d(\rho \iota _{\sigma }g) \rangle - \int _{\partial M}\langle *_{n-1}(\alpha \wedge *d(\rho \iota _{ \sigma }g)), \eta \rangle \\ &-\int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}(\iota _{\sigma }g \rho \eta )\rangle + (-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*d( \iota _{\sigma }g \rho \eta )]\rangle \\ &+\int _{M}\langle \alpha , d^{*}[A_{0},\iota _{\sigma }g \rho \eta ] \rangle +(-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*[A_{0},\iota _{ \sigma }g \rho \eta ]]\rangle . \end{aligned}$$

Therefore, we can use (A.13), to write

$$\begin{aligned} \int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle =&\int _{M}\langle *(d \alpha \wedge *d(\rho \iota _{\sigma }g)), \eta \rangle -\int _{\partial M}\langle *_{n-1}(\alpha \wedge *d( \rho \iota _{\sigma }g)), \eta \rangle \\ &-\int _{M}\langle \omega ,d_{A_{0}}(\iota _{\sigma }g \rho \eta ) \rangle -\int _{M}\langle \zeta ,\iota _{\sigma }g \rho \eta \rangle - \int _{\partial M}\langle \xi ,\iota _{\sigma }g \rho \eta \rangle \\ &+(-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*d(\iota _{\sigma }g \rho \eta )]\rangle +\int _{M}\langle \alpha , d^{*}[A_{0},\iota _{ \sigma }g \rho \eta ]\rangle \\ &+(-1)^{n-1}\int _{M} \langle \alpha , *[A_{0},*[A_{0},\iota _{ \sigma }g \rho \eta ]]\rangle . \end{aligned}$$

Finally, the last three terms of (A.15) are equal to

$$\begin{aligned} &{-}\int _{M}\langle d^{*}\left [(B_{g}+\operatorname{div}(\sigma )) \rho \alpha -\iota _{\sigma }(\alpha \wedge d\rho )\right ], \eta \rangle \\ &\qquad -\int _{\partial M}\langle *_{n-1}*[(B_{g}+\operatorname{div}( \sigma ))\rho \alpha -\iota _{\sigma }(\alpha \wedge d\rho )],\eta \rangle. \end{aligned}$$

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 \(L^{r}_{k-1}\) and \(L^{r}_{k}\), and satisfy

$$ |\!|F|\!|_{L^{r}_{k-1}}+|\!|G|\!|_{L^{r}_{k}(U)}\leq C'(|\!|f|\!|_{L_{k}^{r}(U)}+| \!|\omega |\!|_{L_{k}^{r}(U)}+ |\!|\zeta |\!|_{L_{k-1}^{r}(U)}+|\!| \xi |\!|_{L_{k}^{r}(U)}+ |\!| \alpha |\!|_{L_{k}^{r}(U)}) $$

for some constant C′ depending only on A0, g, σ, U and K. Therefore, Lemma A.1 (part (i) or (ii) depending on whether ∘=τ or ν) implies that

$$ |\!| \rho \alpha (\sigma )|\!|_{L^{r}_{k+1}(U)} \leq C (|\!|f|\!|_{L_{k}^{r}(U)}+|\!|\omega |\!|_{L_{k}^{r}(U)} +|\!| \zeta |\!|_{L_{k-1}^{r}(U)}+|\!|\xi |\!|_{L_{k}^{r}(U)}+ |\!| \alpha | \!|_{L_{k}^{r}(U)}). $$

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 A0 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 Lr section of Λ1E over the open subset U of M such that for any ϕ∈Γc(U,E) and ψ∈Γ(U,E):

$$ | \int _{M}\langle \alpha , d_{A_{0}} \phi \rangle |\leq C_{1} |\!| \phi |\!|_{L^{r^{*}}(U)}, \hspace{1cm} | \int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}(\psi \cdot \iota _{ \sigma }g) \rangle |\leq C_{2} |\!|\psi |\!|_{L_{1}^{r^{*}}(U)}. $$

Then α(σ) belongs to \(L^{r}_{1}(K)\) and

$$ |\!| \alpha (\sigma )|\!|_{L^{r}_{1}(K)}\leq C(C_{1}+C_{2}+|\!| \alpha |\!|_{L^{r}(U)}). $$


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

$$\begin{aligned} |\int _{M} \langle \rho \alpha (\sigma ), \Delta \eta \rangle |\leq & \hspace{2mm} |\int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle |+ | \int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle |+ |\int _{M}\rho \operatorname{div}(\sigma ) \langle \alpha , d\eta \rangle | \\ &+|\int _{M} \rho \langle B_{g}\alpha , d\eta \rangle |+ |\int _{M} \langle \iota _{\sigma }(d(\rho )\wedge \alpha ), d\eta \rangle |. \end{aligned}$$

The first term on the left hand side of the above inequality can be estimated as in (A.17):

$$\begin{aligned} |\int _{M} \rho \langle \alpha ,d \iota _{\sigma }d\eta \rangle | \leq &|\int _{M} \langle \alpha ,d_{A_{0}}(\rho \iota _{\sigma }d \eta ) \rangle |+|\int _{M} \langle \alpha ,\rho [A_{0},\iota _{ \sigma }d \eta ] \rangle |+ |\int _{M} \langle \alpha ,\mathcal {(}d \rho )\cdot (\iota _{\sigma }d\eta ) \rangle | \\ \leq &C(C_{1}+|\!| \alpha |\!|_{L^{r}(U)})|\!|\eta |\!|_{L^{r^{*}}_{1}(U)}. \end{aligned}$$

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:

$$\begin{aligned} |\int _{M}\langle \alpha , d^{*}(\rho \iota _{\sigma }g \wedge d\eta ) \rangle | \leq &|\int _{M}\langle \alpha , d^{*}( \eta d(\rho \iota _{ \sigma }g))\rangle |+ |\int _{M}\langle \alpha , d_{A_{0}}^{*}d_{A_{0}}( \iota _{\sigma }g \rho \eta )\rangle | \\ &+|\int _{M} \langle \alpha , *[{A_{0}},*d(\iota _{\sigma }g \rho \eta )]\rangle |+ |\int _{M}\langle \alpha , d^{*}[{A_{0}},\iota _{ \sigma }g \rho \eta ]\rangle | \\ &+|\int _{M} \langle \alpha , *[{A_{0}},*[{A_{0}},\iota _{\sigma }g \rho \eta ]]\rangle | \\ \leq &C(C_{2}+|\!| \alpha |\!|_{L^{r}(U)})|\!|\eta |\!|_{L^{r^{*}}_{1}(U)}. \end{aligned}$$

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 non-negative 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={βx}xM is a smooth family of connections on F parametrized by M. Suppose f is an \(L^{r}_{k}\) section of the bundle TΣ⊗F over Σ×M. If k≥1, suppose there are \(L^{r}_{k}\) sections ζ1 and ζ2 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

$$ \int _{M\times \Sigma} \langle f, d_{\beta} \xi \rangle = \int _{M \times \Sigma} \langle \zeta _{1}, \xi \rangle , \hspace{1cm} \int _{M\times \Sigma} \langle f, *_{\Sigma }d_{\beta }\xi \rangle = \int _{M\times \Sigma} \langle \zeta _{2}, \xi \rangle , $$

where dβξ denotes the section of TΣ⊗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:

$$ |\!|\nabla ^{\beta}_{\Sigma }f|\!|_{L^{r}_{k}(M\times \Sigma )}\leq C(| \!|\xi _{1}|\!|_{L^{r}_{k}(M\times \Sigma )}+|\!|\xi _{2}|\!|_{L^{r}_{k}(M \times \Sigma )}+|\!|f|\!|_{L^{r}_{k}(M\times \Sigma )}). $$

In the case that k=0, the assumption (A.27) has to be replaced with

$$ |\int _{\Sigma \times X} \langle f, d_{\beta} \xi \rangle |+ |\int _{ \Sigma \times X} \langle f,*_{\Sigma }d_{\beta}\xi \rangle | \leq \kappa |\!|\xi |\!|_{L^{r^{*}}(\Sigma \times X)}. $$

In this case, \(\nabla ^{\beta}_{\Sigma }f\) belongs to Lr(X×Σ) and

$$ |\!|\nabla ^{\beta}_{\Sigma }f|\!|_{L^{r}(X\times \Sigma )}\leq C( \kappa +|\!|f|\!|_{L^{r}(X\times \Sigma )}). $$

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 non-trivial 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 non-negative 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=Lp(N) for a compact manifold N. Any function in C(M×N), determines an element of Lp(M,B). In fact, the space of smooth functions on M×N is dense in Lp(M,B) (see [Weh041] and [Lip14]). This gives us the following identifications of Sobolev spaces:

$$ L^{p}(M,L^{p}(N))=L^{p}(N,L^{p}(M))=L^{p}(M\times N). $$

More generally, C(M×N) is dense in \(L^{p}_{k}(M,L^{p}(\Sigma ))\) for any non-negative integer k, and we have (see [Weh041, Lip14]):

$$ \begin{aligned} &L^{p}_{k}(M\times N)=L^{p}_{k}(M,L^{p}(N))\cap L^{p}_{k}(N,L^{p}(M)), \\ &L^{p}_{k}(M, L^{p}(N))=L^{p}(N,L^{p}_{k}(M)). \end{aligned} $$

For the rest of this appendix, we fix Bp to be a Banach space that can be identified with a closed subspace of the space Lp(N) for a closed manifold N. In particular, the intersection Bq:=BpLq(N) with q>p determines a closed subspace of Lq(N). For q<p, Bq is the closure of Bp in Lq(N).

Lemma B.2

[Weh041] and [Lip14]

Suppose M is a Riemannian manifold with boundary. Let k be a non-negative 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 Bp-valued.

Sketch of the Proof.

Without loss of generality, we can assume that Bp=Lp(N). Using the identifications in (B.1), we can regard u as an Lp 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, Bp denotes the direct sum BpBp. This space admits an obvious complex structure J0 given by

$$ \mathcal {J}_{0}(v_{0},v_{1})=(-v_{1},v_{0}). $$

The subspace \(\mathcal {L}:=0\oplus B_{p}\) defines a completely real subspace of Bp with respect to \(\mathcal {J}_{0}\).

Proposition B.4

Suppose U is a bounded open subspace of

$$ {\mathbb{H}}^{2}:=\{(s,\theta )\in \mathbf{R}^{2}\mid s \geq 0\}, $$

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:UBp is an \(L^{p}_{k}\) map that satisfies

$$ \partial _{\theta }u-\mathcal {J}(u)\partial _{s} u=z\in L^{p}_{k}(U, \mathbf{B}_{p}), $$

and the boundary condition

$$ u|_{U_{\partial}}\subset \mathcal {L}. $$

Then for any open subspace KU, 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

$$ |\!|u|\!|_{L^{p}_{k+1}(K)}\leq C(|\!|z|\!|_{L^{p}_{k}(U)}+|\!|u|\!|_{L^{p}}(U)). $$

If ui:UBp is a sequence of \(L^{p}_{k}\) map that satisfies

$$ \partial _{\theta }u_{i}-\mathcal {J}(u_{i})\partial _{s} u_{i}=z_{i} \in L^{p}_{k}(U,\mathbf{B}_{p}), $$

such that ui and zi are respectively \(L^{p}_{k}\)-convergent to u and z, then ui 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:

$$ \partial _{s}^{2} u+\partial _{\theta}^{2} u=\mathcal {J}(u)\partial _{s}( \mathcal {J}(u))\partial _{s}u+ \partial _{\theta}(\mathcal {J}(u)) \partial _{s}u +\partial _{\theta }z+\mathcal {J}(u)\partial _{s}z. $$

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 \(L^{p}_{k-1}\). The maps u and z can be written as (u0,u1) and (z0,z1) with respect to the decomposition of Bp. 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 \(L^{p}_{k-1}\) functions are still in \(L^{p}_{k-1}\). In the case that k=1, the products of two Lp(U,Bp) functions is in Lp/2(U,Bp), which in turn is a subspace of Lp/2(U,Bp/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 xBp, 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:UBp be in \(L^{p}_{1}\). Suppose q>p and u is also an Lq map from U to Bq. Suppose u satisfies

$$ \partial _{\theta }u-\mathcal {J}(u)\partial _{s} u=z\in L^{q}(U, \mathbf{B}_{q}), $$

and the boundary condition (B.6). Then u is an \(L^{q}_{1}\) map from U to Bq and

$$ |\!|u|\!|_{L^{q}_{1}}\leq C(|\!|z|\!|_{L^{q}}+|\!|u|\!|_{L^{q}}). $$

Moreover, if ui:UBp are \(L^{q}_{1}\) solutions of

$$ \partial _{\theta }u_{i}-\mathcal {J}(u_{i})\partial _{s} u_{i}=z_{i} \in L^{q}(U,\mathbf{B}_{q}), $$

such that ui is convergent to u in \(L^{p}_{1}\cap L^{q}\) and zi is convergent to z in Lq, then ui is convergent to u in \(L^{q}_{1}\).


Given p>2 and any bounded domain Ω in R2 with smooth boundary, let \(L^{p}_{1}(\Omega ,\mathbf{B}_{p})_{\partial}\) be the space of \(L^{p}_{1}\) maps u:Ω→Bp such that the restriction of u to the boundary is in \(\mathcal {L}\). Then the Cauchy–Riemann operator

$$ \partial _{\theta }-\mathcal {J}_{0}\partial _{s} :L^{p}_{1}(\Omega , \mathbf{B}_{p})_{\partial }\to L^{p}(\Omega ,\mathbf{B}_{p}) $$

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 Ωr is the region given by rounding the corners of Dr(x) such that it is contained in Dr(x) and it contains Dr/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 ρrR be a smooth bump function that vanishes on the complement of Dr/2(x) and equals 1 on Dr/3(x). Then our assumption implies that ρu is an element of \(L^{p}_{1}(\Omega _{r},\mathbf{B}_{p})_{\partial}\) and

$$ \partial _{\theta}(\rho u) -\mathcal {J}(u)\partial _{s}(\rho u)=\rho z+ \partial _{\theta}(\rho ) u -\mathcal {J}(u)\partial _{s}(\rho ) u $$

is in Lq. Thus there is \(u'\in L^{q}_{1}(\Omega ,\mathbf{B}_{q})_{\partial}\) such that

$$ \partial _{\theta }u' -\mathcal {J}(u)\partial _{s}u'=\rho z+\partial _{ \theta}(\rho ) u -\mathcal {J}(u)\partial _{s}(\rho ) u. $$

This implies that u′−ρu is a constant map to \(\mathcal {L}\). In particular, the restriction of u to Dr/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:BpBp such that \(T^{-1} \mathcal {J}(x) T=\mathcal {J}_{0}\). In fact, we may take T to be the linear map that sends (v0,v1)∈BpBp 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. □

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Daemi, A., Fukaya, K. & Lipyanskiy, M. Lagrangians, SO(3)-Instantons and Mixed Equation. Geom. Funct. Anal. 34, 659–732 (2024).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: