The Kapustin--Witten equations on ALE and ALF gravitational instantons

TL;DR

This paper investigates solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons, providing asymptotic estimates and a vanishing theorem for specific manifolds and structure groups.

Contribution

It offers new asymptotic estimates for the Higgs field and establishes a vanishing theorem on certain gravitational instantons with SU(2) structure group.

Findings

Derived asymptotic estimates for the Higgs field on ALE and ALF spaces.

Proved a vanishing theorem for solutions on R^4 and R^3 x S^1 with SU(2) group.

Abstract

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$.

Equations

$$\overline{\ast}\mathopen{}\mathclose{{}\left(e^{i\theta}F_{\nabla^{\mathbb{C}}}}\right)=e^{i\theta}F_{\nabla^{\mathbb{C}}}.$$

$$\displaystyle\cos(\theta)\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right)^{-}-\sin(\theta)\mathrm{d}_{\nabla}^{-}\Phi$$

$$\displaystyle\sin(\theta)\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right)^{+}+\cos(\theta)\mathrm{d}_{\nabla}^{+}\Phi$$

$${\mathrm{d}}_{\nabla }^{\ast }\Phi$$

$$F_{\nabla }=\ast {\mathrm{d}}_{\nabla }\Phi +\frac{1}{2}[\Phi \wedge \Phi ].$$

$$\mathcal{E}_{\scriptscriptstyle\mathrm{KW}}(\nabla,\Phi)=\int\limits_{M}\mathopen{}\mathclose{{}\left(|F_{\nabla}|^{2}+|\nabla\Phi|^{2}+\tfrac{1}{4}|[\Phi\wedge\Phi]|^{2}}\right)\mathrm{vol}.$$

$$\lim\limits_{R\rightarrow\infty}R^{j}\mathopen{}\mathclose{{}\left\|\mathopen{}\mathclose{{}\left(\nabla^{\mathrm{LC}}}\right)^{j}\mathopen{}\mathclose{{}\left(g-\phi^{*}\mathopen{}\mathclose{{}\left(dR^{2}+g_{\mathbb{T}^{k}}+R^{2}\pi_{Y}^{*}(g_{B})}\right)}\right)}\right\|_{L^{\infty}(S_{R})}=0.$$

$$\underset{R\to \infty }{\lim }\underset{S_R}{\inf }|\Phi |=\underset{R\to \infty }{\lim }\underset{S_R}{\sup }|\Phi |=\underset{R\to \infty }{\lim }\underset{M_R}{\sup }|\Phi |=c.$$

$$j_{\Phi}\coloneqq\sum_{i=1}^{4}[\nabla\Phi_{i},\Phi_{i}]\in\Gamma\mathopen{}\mathclose{{}\left(\Lambda^{1}\otimes\mathfrak{g}_{P}}\right).$$

$${\nabla }^{\ast }\nabla \Phi$$

$${\mathrm{d}}_{\nabla }^{\ast }{F}_{\nabla }$$

$$\nabla^{*}\nabla\Phi=\mathrm{d}_{\nabla}^{*}\mathrm{d}_{\nabla}\Phi-\ast\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\ast F_{\nabla}}\right)\wedge\Phi}\right]-\mathrm{Ric}_{g}(\Phi).$$

$$\mathrm{d}_{\nabla}^{\pm}\Phi=\mp t^{\pm 1}\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right)^{\pm}.$$

$$\mathrm{d}_{\nabla}\Phi=\tfrac{t^{-1}-t}{2}\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right)-\tfrac{t^{-1}+t}{2}\ast\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right).$$

$$\mathrm{d}_{\nabla}\Phi=\tfrac{t-t^{-1}}{t+t^{-1}}\ast\mathrm{d}_{\nabla}\Phi-\tfrac{2}{t+t^{-1}}\ast\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right),$$

$$F_{\nabla }=\frac{1}{2}[\Phi \wedge \Phi ]+\frac{t-t^{-1}}{2}{\mathrm{d}}_{\nabla }\Phi -\frac{t+t^{-1}}{2}\ast {\mathrm{d}}_{\nabla }\Phi .$$

$${\mathrm{d}}_{\nabla }^{\ast }{\mathrm{d}}_{\nabla }\Phi$$

$$=-\frac{t-t^{-1}}{t+t^{-1}}\ast [F_{\nabla }\wedge \Phi ]-\frac{2}{t+t^{-1}}\ast [{\mathrm{d}}_{\nabla }\Phi \wedge \Phi ]$$

$$\displaystyle=\ast\mathopen{}\mathclose{{}\left[\ast\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right)\wedge\Phi}\right],$$

$${\mathrm{d}}_{\nabla }^{\ast }{F}_{\nabla }$$

$$\displaystyle=j_{\Phi}+\ast[(\ast\mathrm{d}_{\nabla}\Phi)\wedge\Phi]+\tfrac{t-t^{-1}}{2}\ast\mathopen{}\mathclose{{}\left[\ast\mathopen{}\mathclose{{}\left(F_{\nabla}-\tfrac{1}{2}[\Phi\wedge\Phi]}\right)\wedge\Phi}\right]+\tfrac{t+t^{-1}}{2}\ast[F_{\nabla}\wedge\Phi]$$

$$\displaystyle=j_{\Phi}+\ast[(\ast\mathrm{d}_{\nabla}\Phi)\wedge\Phi]+\tfrac{t-t^{-1}}{2}\ast\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\tfrac{t-t^{-1}}{2}\ast\mathrm{d}_{\nabla}\Phi-\tfrac{t+t^{-1}}{2}\mathrm{d}_{\nabla}\Phi}\right)\wedge\Phi}\right]$$

$$\displaystyle\quad+\tfrac{t+t^{-1}}{2}\ast\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\tfrac{1}{2}[\Phi\wedge\Phi]+\tfrac{t-t^{-1}}{2}\mathrm{d}_{\nabla}\Phi-\tfrac{t+t^{-1}}{2}\ast\mathrm{d}_{\nabla}\Phi}\right)\wedge\Phi}\right]$$

$$\displaystyle=j_{\Phi}+\ast[(\ast\mathrm{d}_{\nabla}\Phi)\wedge\Phi]+\tfrac{t-t^{-1}}{2}\ast\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\tfrac{t-t^{-1}}{2}\ast\mathrm{d}_{\nabla}\Phi-\tfrac{t+t^{-1}}{2}\mathrm{d}_{\nabla}\Phi}\right)\wedge\Phi}\right]$$

$$\displaystyle\quad+\tfrac{t+t^{-1}}{2}\ast\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\tfrac{t-t^{-1}}{2}\mathrm{d}_{\nabla}\Phi-\tfrac{t+t^{-1}}{2}\ast\mathrm{d}_{\nabla}\Phi}\right)\wedge\Phi}\right]$$

$$=j_{\Phi },$$

$$e_{\mathrm{KW}}=|F_{\nabla }{|}^2+|\nabla \Phi {|}^2+\frac{1}{4}|[\Phi \wedge \Phi ]{|}^2⩾0.$$

$$\nabla^{*}\nabla(\nabla\Phi)=\nabla(\nabla^{*}\nabla\Phi)+\sum\limits_{i,j=1}^{4}\mathopen{}\mathclose{{}\left(2[F_{ik},\nabla_{k}\Phi_{j}]+[(\mathrm{d}_{\nabla}^{*}F_{\nabla})_{i},\Phi_{j}]}\right)\ \mathrm{d}x^{i}\otimes\mathrm{d}x^{j}.$$

$$\Delta f⩽C(f^2+f)$$

$$f=\sqrt{e_{\mathrm{KW}}},b=f+1,q=2+\Vert e_{\mathrm{KW}}{\Vert }_{L^{\infty }(B_{2\mathrm{dist}(x,x_0)}(x))}^{-1},n=4,\gamma =1,a_0=r_0,\text{mbox}anda=\frac{1}{2}{a}_0,$$

$$e_{\mathrm{KW}}(x)⩽C^{\prime }\Vert e_{\mathrm{KW}}{\Vert }_{L^1(B_{r_0}(x),g)}.$$

$$\forall p\in [1,\infty )\cup \{\infty \}:\Vert e_{\mathrm{KW}}{\Vert }_{L^p(M,g)}⩽C\Vert e_{\mathrm{KW}}{\Vert }_{L^1(M,g)}.$$

$$\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{p}(M,g)}\leqslant\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{\infty}(M)}^{(p-1)/p}\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{1}(M,g)}^{1/p}\leqslant(C^{\prime})^{(p-1)/p}\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{1}(M,g)}\leqslant\max\mathopen{}\mathclose{{}\left(\{C^{\prime},1\}}\right)\>\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{1}(M,g)},$$

$$m\coloneqq\tfrac{\exp_{x}^{*}\mathopen{}\mathclose{{}\left(\mathrm{vol}_{g}}\right)}{\mathrm{d}\rho\wedge\mathrm{vol}_{\mathbb{S}^{3}}}.$$

$${\partial }_{\rho }({\rho }^{-3}m)⩽0,$$

$$\forall x\in M:\quad\frac{\mathrm{dist}(x,\cdot)^{2}}{1+\mathrm{dist}(x,\cdot)^{k}}\ G(x,\cdot)\in L^{\infty}\mathopen{}\mathclose{{}\left(M-\{x\}}\right).$$

Full text

1. Introduction

Background

Let us fix a smooth, oriented, Riemannian 4-manifold, $(M,g)$. Let $\Lambda_{\mathbb{C}}^{*}M=\Lambda^{*}M\otimes\mathbb{C}$ be the complexified exterior algebra bundle. Let $\rm{G}$ be a compact Lie group and $P\rightarrow M$ a smooth principal $\rm{G}$-bundle over $M$. Let $\rm{G}_{\mathbb{C}}$ be the complex form of $\rm{G}$. We then have a principal $\rm{G}_{\mathbb{C}}$-bundle $P_{\mathbb{C}}=P\times_{\rm{G}}\rm{G}_{\mathbb{C}}$, defined via the adjoint action of $\rm{G}$ on $\rm{G}_{\mathbb{C}}$. The Hodge star operator $\ast$ can be extended in two inequivalent ways to $\Lambda_{\mathbb{C}}^{*}M\otimes\mathfrak{g}_{P}\simeq\Lambda^{*}M\otimes\mathfrak{g}_{P_{\mathbb{C}}}$, either as a complex linear operator or as a conjugate linear operator. In this paper we consider complexified instantons using the conjugate linear extension. We investigated the complex linear extension in [NO20].

Let us denote the conjugate linear extension of the Hodge star operator by $\overline{\ast}$. The Kapustin–Witten equations can be viewed as complexified self-duality equations as follows: Let $(\nabla,\Phi)$ be a pair consisting of a connection on $P_{\mathbb{C}}$ and a section of $\Lambda^{1}\otimes\mathfrak{g}_{P_{\mathbb{C}}}$ (the Higgs field). Then $\nabla^{\mathbb{C}}\coloneqq\nabla+i\Phi$ is a connection on $P_{\mathbb{C}}$. In [KW07], Kapustin and Witten introduced a family of complexified self-duality equations, parametrized by $\theta\in\mathbb{R}$, as

[TABLE]

We recommend [GU12] for an introduction to the Kapustin–Witten equations.

Note that when $\Phi=0$ everywhere and $2\theta\equiv 0\>(\textnormal{mod}\>\pi)$, equation 1.1 reduces to the classical self-duality equation on $P$. As is standard in complex gauge theory, we break the structure group down from $\rm{G}_{\mathbb{C}}$ to $\rm{G}$ by adding the Coulomb type equation $\mathrm{d}_{\nabla}^{*}\Phi=0$. Since $\mathfrak{g}_{P_{\mathbb{C}}}$ has a canonical real structure, one can separate the real and imaginary parts of equation 1.1 and get the following system of equations:

[TABLE]

The equations 1.2a, 1.2b, and 1.2c are called the $\theta$-Kapustin–Witten equations. When $\theta=\tfrac{\pi}{4}$, equations 1.2a and 1.2b can be rewritten as the following single equation:

[TABLE]

There is a Yang–Mills–Higgs type energy functional corresponding to the Kapustin–Witten equation 1.3. This functional, which we call the Kapustin–Witten energy is

[TABLE]

Similarly to instantons, solutions to the Kapustin–Witten equation 1.3 with finite Kapustin–Witten energy are, at least formally, absolute minimizers of \tagform@1.4. When $M$ is closed and $2\theta\not\equiv 0\>(\textnormal{mod}\>\pi)$, then all solutions to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c satisfy that $\nabla$ is flat, $\Phi$ is $\nabla$-parallel, and $[\Phi\wedge\Phi]$ vanishes identically; cf. [GU12]*Corollary 3.3.

Witten conjectures that the moduli spaces of equation 1.3 have applications to low dimensional topology; cf. [W12, W18]. Related work has been done recently by, for example, Taubes [T18, T19], Mazzeo and Witten [MW14, MW17], He and Mazzeo [HM17], and He and Walpuski [HW19].

In this paper, we consider finite energy solutions to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c on certain noncompact, complete, Ricci-flat, Riemannian 4-manifolds, called ALE and ALF gravitational instantons.

Let us, briefly, introduce these classes spaces: Let $(M,g)$ be a (noncompact), smooth, and oriented Riemannian 4-manifold with Levi-Civita connection denoted by $\nabla^{\mathrm{LC}}$. For all $R>0$ and $x_{0}\in M$, let $B_{R}(x_{0})\subset M$ be the (closed) geodesic ball of radius $R$ around $x_{0}$, and let $S_{R}(x_{0})\coloneqq\partial B_{R}(x_{0})$.

Definition 1.1** (ALE and ALF gravitational instantons).**

Let $(M,g)$ be as above and fix $x_{0}\in M$. Let $Y$ be a compact 3-manifold, which is a $\mathbb{T}^{k}$-fibration over a closed base $B$ with projection $\pi_{Y}$, where $k=0$, or $1$, together with a connection on $Y$ when $k=1$, and a metric $g_{B}$ on $B$. Assume that the end of $X$ is modeled on $Y\times\mathbb{R}_{+}$, that is, there exists $R_{0}>0$, such that $B_{R_{0}}(x_{0})$ is a smooth, compact manifold with boundary and $M-B_{R_{0}}(x_{0})$ is diffeomorphic to $Y\times(R_{0},\infty)$. Moreover, there exists a diffeomorphism $\phi:M-B_{R_{0}}(x_{0})\rightarrow Y\times(R_{0},\infty)$, such that for $j=0,1$, and $2$, we have

[TABLE]

We call $(M,g)$ Asymptotically Locally Euclidean (ALE), if $k=0$, and Asymptotically Locally Flat (ALF), if $k=1$. An ALE or ALF 4-manifold is called a gravitational instanton, if it is Ricci-flat.

Remark 1.2**.**

Note that we do not require $(M,g)$ to be hyperkähler, or even complex. For example, the Euclidean–Schwarzschild manifold can be considered.

The prototypical example of an ALE gravitational instanton is $\mathbb{R}^{4}$ with its canonical flat metric. Other examples are given by the construction of Kronheimer [Kronheimer1989].

The prototypical example of an ALF gravitational instanton is $\mathbb{R}^{3}\times\mathbb{S}^{1}$ with its canonical flat metric. Other important examples include the Euclidean–Schwarzschild, the multi-Taub–NUT, and the Atyah–Hitchin manifolds. Many more (hyperkähler) examples are given via the Gibbons–Hawking construction [GH78].

Main results

Our first main theorem is an asymptotic bound on the Higgs field, $\Phi$, when the underlying manifold, $(M,g)$ is an ALE or ALF gravitational instanton. The proof uses ideas of [JT80]*Theorem 10.3 adapted to the 4-dimensional setting and to curved geometries.

Main Theorem 1**.**

Let $(\nabla,\Phi)$ be a finite energy solution to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c, with $2\theta\not\equiv 0\>(\textnormal{mod}\>\pi)$, on an ALE or ALF gravitational instanton $(M,g)$. Then there is a constant $c\geqslant 0$, such that

[TABLE]

Furthermore, if $c=0$, then $\Phi=0$ everywhere.

Combining Main Theorem 1 with [T17]*Theorem 1.1, we prove the following result.

Corollary 1.3**.**

Let $(\nabla,\Phi)$ be a finite energy solutions to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c, with $2\theta\not\equiv 0\>(\textnormal{mod}\>\pi)$, on $M=\mathbb{R}^{4}$ or $\mathbb{R}^{3}\times\mathbb{S}^{1}$ with its flat metric, and let $\rm{G}=\mathrm{SU}(2)$.

Then $\nabla$ is flat, then $\Phi$ is $\nabla$-parallel, and $[\Phi\wedge\Phi]=0$.

We conjecture that Corollary 1.3 holds on an arbitrary ALE or ALF gravitational instanton.

Organization of the paper

In Section 2, we compute second order equations that are satisfied by solutions to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c, with $2\theta\not\equiv 0\>(\textnormal{mod}\>\pi)$. While these equations are known in the literature, we include their proof for clarity and completeness. These are used in the proofs of Main Theorem 1. In Section 3, we study the analytic properties of the Kapustin–Witten energy density. In Section 4, we recall a few useful properties of ALE and ALF gravitational instantons. Finally, in Section 5 we present the proofs of Main Theorems 1 and 1.3.

Acknowledgment**.**

The authors are grateful to Mark Stern for many helpful conversations about gauge theory. We also thank Bera Gorapada for his valuable comments on the results of this paper.

Ákos Nagy would like to thank Siqi He for useful discussions about the Kapustin–Witten equations, and he also thanks the Universidade Federal Fluminense and IMPA for their hospitality during the final stages of the preparation of this paper.

Finally, we thank

We also thank the anonymous referees for their comments and suggestions.

Gonçalo Oliveira is supported by Fundação Serrapilheira 1812-27395, by CNPq grants 428959/2018-0 and 307475/2018-2, and FAPERJ through the program Jovem Cientista do Nosso Estado E-26/202.793/2019.

2. The second order Kapustin–Witten equations

For the next lemma, let $(x_{1},x_{2},x_{3},x_{4})$ be a local, normal chart on $M$ at an arbitrary point, and let

[TABLE]

be the supercurrent generated by $\Phi$.

Lemma 2.1**.**

Let $(M,g)$ be any Riemannian 4-manifold, $P\to M$ a principal $\rm{G}$-bundle, and regard the Ricci tensor of $(M,g)$, $\mathrm{Ric}_{g}$, as an endomorphism of $\Lambda^{1}\otimes\mathfrak{g}_{P}$. If $(\nabla,\Phi)$ is a solution to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c, with $2\theta\not\equiv 0\>(\textnormal{mod}\>\pi)$, on $P\to M$, it also satisfies the following system of second order equation:

[TABLE]

Remark 2.2**.**

When $(M,g)$ is Ricci-flat, then equations 2.2a and 2.2b are the Euler–Lagrange equations of the Kapustin–Witten energy \tagform@1.4.

Remark 2.3**.**

In Main Theorem 1, one can replace the condition that \mathopen{}\mathclose{{}\left(\nabla,\Phi}\right) is a solution to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c with the assumption that it only solves the second order Kapustin–Witten equations 2.2a and 2.2b, and the conclusions still hold, while in Corollary 1.3, one can now conclude that $\nabla\Phi=[\Phi\wedge\Phi]=0$ and $\nabla$ is a Yang–Mills connection.

Proof.

Using the Weitzenböck formula and $\mathrm{d}_{\nabla}^{*}\Phi=0$, we get

[TABLE]

Since $2\theta\not\equiv 0\>(\textnormal{mod}\>\pi)$, the number $t=\tan(\theta)$ is defined and nonzero. In this case we may rewrite equations 1.2a and 1.2b as

[TABLE]

Then, writing $\mathrm{d}_{\nabla}^{\pm}\Phi=\tfrac{1}{2}(\mathrm{d}_{\nabla}\Phi\pm\ast\mathrm{d}_{\nabla}\Phi)$, and adding these two equations we find

[TABLE]

On the other hand, multiplying equation 2.4 by $t^{\mp 1}$ and adding up the resulting equations yields (after dividing by $t+t^{-1}$)

[TABLE]

which by rearranging can also be read as

[TABLE]

Thus, using equations 2.6 and 2.7, together with the Bianchi identity $\mathrm{d}_{\nabla}^{*}\ast F_{\nabla}=0$, we get

[TABLE]

where in the last equality we replaced $F_{\nabla}$ using equation 2.7 and the Jacobi identity $[[\Phi\wedge\Phi]\wedge\Phi]=0$. Combining the above equation with equation 2.3 concludes the proof of equation 2.2a.

Now we prove equation 2.2b:

[TABLE]

which completes the proof. ∎

3. The Kapustin–Witten energy density

The Kapustin–Witten energy \tagform@1.4 is the integral of

[TABLE]

We call $e_{\scriptscriptstyle\mathrm{KW}}$ the Kapustin–Witten energy density. First we prove a decay result for $e_{\scriptscriptstyle\mathrm{KW}}$.

Proposition 3.1**.**

Then there is a positive number $C=C(M,g,\rm{G})$, such that, if $(\nabla,\Phi)$ is a smooth solution to second order equations 2.2a and 2.2b with finite energy, then $e_{\scriptscriptstyle\mathrm{KW}}$ decays uniformly to zero at infinity.

Proof.

Let r_{0}=\min\mathopen{}\mathclose{{}\left(\{\mathrm{inj}(M,g)/2,1\}}\right). Note that $r_{0}>0$, since $(M,g)$ is either ALE or ALF. Let $f=\sqrt{e_{\scriptscriptstyle\mathrm{KW}}}\in L_{1}^{2}(B_{2r_{0}}(x))$. By the Weitzenböck formula and the Ricci-flatness of $(M,g)$, we have (in normal coordinates)

[TABLE]

Now further using equations 2.2a and 2.2b, we get that $\Delta|\nabla\Phi|^{2}\leqslant C(f^{3}+f)-|\nabla^{2}\Phi|$. Similar computations for $|F_{\nabla}|^{2}$ and $|[\Phi\wedge\Phi]|^{2}$, and Kato’s inequality yields that $f$ weakly satisfies (after maybe redefining $C$) the following inequality:

[TABLE]

Hence, by [Uhlenbeck1982]*Theorem 3.2, using the notations of the reference

[TABLE]

we get that, for some other positive number $C^{\prime}=C^{\prime}(M,g)$:

[TABLE]

By the finiteness of the energy, we get that the integral of $e_{\scriptscriptstyle\mathrm{KW}}$ on $B_{r_{0}}(x)$ decays uniformly to zero as $\mathrm{dist}(x,x_{0})\rightarrow\infty$, and thus so does $e_{\scriptscriptstyle\mathrm{KW}}(x)$, which concludes the proof. ∎

Corollary 3.2**.**

There is a positive number $C=C(M,g,\rm{G})$, such that

[TABLE]

Proof.

By equation 3.5, we get that $e_{\scriptscriptstyle\mathrm{KW}}\in L^{\infty}(M)$ and, in fact, $\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{\infty}(M)}\leqslant C^{\prime}\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{1}(M,g)}$. Hence, for any $p\geqslant 1$, using Hölder’s inequality, we get

[TABLE]

thus $\|e_{\scriptscriptstyle\mathrm{KW}}\|_{L^{p}(M,g)}$ is finite, and can be bounded by a constant independent of $p$. ∎

4. On the geometry of ALE and ALF gravitational instantons

In this section we recall a few geometric properties of ALE and ALF gravitational instantons that will be used in the proof of Main Theorem 1.

For quantities $A,B$ the relation $A\lesssim B$ is equivalent to $A=O(B)$, while $A\sim B$ is equivalent to $A\lesssim B$ and $B\lesssim A$. Furthermore, let $k=0$, if $(M,g)$ is ALE, and $k=1$, if $(M,g)$ is ALF.

Let $x\in M$, $\rho$ be the radial coordinate on $T_{x}M$ and define $m\in C^{\infty}(T_{x}M;\mathbb{R}_{+})$ via

[TABLE]

From the Laplacian Comparison Theorem—see, for example [Walpuski]*Proposition 20.7—we have

[TABLE]

away from the cut locus. Let us now recall the local and global versions of the Gromov–Bishop Theorem, applied to the case of ALE and ALF gravitational instantons.

Theorem 4.1** (The Local Bishop–Gromov Theorem for $(M,g)$).**

As $\rho\to 0$, the quantity $\rho^{-3}m$ converges to a constant, and as $\rho\to\infty$, we have $m\lesssim\rho^{3-k}$.

Theorem 4.2** (The Global Bishop–Gromov Theorem for $(M,g)$).**

The quotient, $r^{-4}\mathrm{Vol}(B_{r}(x))$, is a nonincreasing function of $r$ and converges to a constant as $r\to 0$. As $r\to\infty$, by the Definition 1.1, we have $\mathrm{Vol}(B_{r}(x_{0}))\sim r^{4-k}$.

Theorem 4.2 and [LY86]*Theorem 5.2 yield the following Lemma.

Lemma 4.3**.**

There is a smooth, positive Green’s function, $G$ on $(M,g)$, and

[TABLE]

Finally, we present (and prove) the appropriate Hölder–Sobolev Embedding.

Lemma 4.4** (The Hölder–Sobolev Embedding Theorem on $(M,g)$).**

For all $p\in(4,\infty)$, there are positive numbers, $C_{\textnormal{HS}}$ and $R_{0}$, such that for any $f$ with $|\nabla f|\in L^{p}(M,g)$ and $x,y\in M$ with $\mathrm{dist}(x,y)\geqslant R_{0}$, we have

[TABLE]

Proof.

We follow the proof as given in, for example, [JT80]*Corollary 2.7. Let $f$ be as in the statement, $x,y\in M$ arbitrary, let $w$ be the midpoint of a geodesic connecting $x$ and $y$, and $R\coloneqq\tfrac{1}{4}\mathrm{dist}(x,y)$. Then for any $z\in M$, we have $|f(x)-f(y)|\leqslant|f(x)-f(z)|+|f(z)-f(y)|$. If we integrate both sides of this inequality with respect to $z\in B_{R}(w)$, then we get

[TABLE]

and using $\gamma_{x,z}$ to denote the arc-length parametrized geodesic connecting $x$ to $z$, so $|\dot{\gamma}_{x,z}(t)|=1$, then

[TABLE]

where we have used the triangle inequality to get that $\mathrm{dist}(x,z)\leqslant 3R$. Now we write

[TABLE]

and use equation 4.2 to deduce that

[TABLE]

This, together with the fact that $\gamma_{x,z}(t)\in B_{3R}(x)$ for $z\in B_{R}(w)$ and all $t\in[0,3R]$ yields

[TABLE]

where we have used Hölder’s inequality with conjugate exponents $p$ and $q$. Now, for $\mathrm{dist}(x,\cdot)^{-3}$ to be in $L_{loc}^{q}(M,g)$ we must have $p>4$ in which case, for $R\gg 1$ we find, using Theorem 4.1, that

[TABLE]

Using a similar trick to control the integral of $|f(z)-f(y)|$ and inserting into equation 4.5 we find that for $R\gg 1$

[TABLE]

where we have used Theorem 4.2 to bound the volume of the balls. ∎

5. The proofs of Main Theorems 1 and 1.3

Let $(\nabla,\Phi)$ be a finite energy solution to the second order equations 2.2a and 2.2b on an ALE of ALF gravitational instanton, $(M,g)$.

First of all, $\tfrac{1}{2}|\Phi|^{2}$ is subharmonic, because

[TABLE]

Let $G$ be the Green’s function of $(M,g)$ from Lemma 4.3. We then define a nonnegative function

[TABLE]

We now can prove Main Theorem 1.

Proof of Main Theorem 1.

The proof below is inspired by [JT80]*Theorem 10.3.

Let $r\coloneqq\tfrac{1}{2}\mathrm{dist}(x,x_{0})$. Using that $|\nabla\Phi|^{2}+\tfrac{1}{4}|[\Phi\wedge\Phi]|^{2}\leqslant e_{\scriptscriptstyle\mathrm{KW}}$, together with Lemma 4.3 and Corollary 3.2, we have for all $x\in M$:

[TABLE]

Thus $w$ is bounded, and furthermore, $w(x)$ converges uniformly to zero as $\mathrm{dist}(x,x_{0})\rightarrow\infty$. The same argument shows that the integrand in equation 5.2 is absolutely convergent, and thus the smoothness of $w$ also follows. Now, by the construction of $w$, the function $h\coloneqq w+\tfrac{1}{2}|\Phi|^{2}$ is harmonic. Next, we show that $h=o(\mathrm{dist}(\cdot,x_{0}))$. For each $R>0$, define

[TABLE]

Since $|\Phi|^{2}$ is subharmonic, the supremum is achieved at some point $\tilde{x}$, with $\mathrm{dist}(\tilde{x},x_{0})=R$, that is $m(R)=|\Phi(\tilde{x})|^{2}$. Furthermore, by Kato’s inequality, $|\mathrm{d}(|\Phi|^{2})|\leqslant 2|\nabla\Phi||\Phi|\leqslant 2e_{\scriptscriptstyle\mathrm{KW}}^{1/2}|\Phi|$, and thus, using Lemma 4.4 on $B_{R}(x_{0})$ with $R\gg 1$, we get

[TABLE]

Since $k=0$, or $1$, we can chose, for example, $p=\frac{3(4-k)}{2}>4$, and find that

[TABLE]

and thus (for $R$ large enough) m(R)=O\mathopen{}\mathclose{{}\left(R^{2/3}}\right), which shows that $|\Phi|^{2}$ grows strictly slower than linearly, and thus $h=w+\tfrac{1}{2}|\Phi|^{2}$ is harmonic and $o(\mathrm{dist}(\cdot,x_{0}))$. Therefore, it must be constant by the gradient estimate of Cheng and Yau in [Cheng]*Section 4, which is nicely summarized in the form we need in [Li1995]*Lemma 1.5. Let this constant be $c$. Clearly, $c\geqslant 0$ and $|\Phi|^{2}=2c-w$. Since $w$ is nonnegative and converges uniformly to zero at infinity, this proves equation 1.6. Finally, since $w$ is nonnegative, if $h=c$ is zero, then so is $\Phi$, which completes the proof of Main Theorem 1. ∎

Remark 5.1**.**

*In the cases of the gravitational instantons of type ALG and ALH the proof of Main Theorem 1 given above does not work, because those manifolds do not satisfy the conditions of [LY86]Theorem 5.2 and hence do not have positive Green’s functions.

Finally, we prove Corollary 1.3. This is a vanishing result for finite energy Kapustin–Witten fields on $M=\mathbb{R}^{4}$ or $\mathbb{R}^{3}\times\mathbb{S}^{1}$, equipped with their flat metrics. The proof is a combination of Main Theorem 1 and [T17]*Theorem 1.2.

Proof of Corollary 1.3.

Let $M$ be either $\mathbb{R}^{4}$ or $\mathbb{R}^{3}\times\mathbb{S}^{1}$, and equip it with its standard (flat) metric. Let $(\nabla,\Phi)$ be a finite energy solution to the $\theta$-Kapustin–Witten equations 1.2a, 1.2b, and 1.2c (or even just the second order equations 2.2a and 2.2b) with structure group $\rm{G}=\mathrm{SU}(2)$.

If $M=\mathbb{R}^{3}\times\mathbb{S}^{1}$, then let us pull back $(\nabla,\Phi)$ to $\mathbb{R}^{4}$. In both cases, we get a smooth solution to equations 2.2a and 2.2b on $\mathbb{R}^{4}$ with bounded $\Phi$. In particular, $|\Phi|$ has bounded average over spheres, and thus by [T17]*Theorem 1.2 we get that both $\nabla\Phi$ and $[\Phi\wedge\Phi]$ vanish identically, which yields the claims of Corollary 1.3 immediately. ∎

Remark 5.2**.**

*The only time $M=\mathbb{R}^{4}$ or $\mathbb{R}^{3}\times\mathbb{S}^{1}$, and $\rm{G}=\mathrm{SU}(2)$ were needed in the proof of Corollary 1.3 is when we used [T17]Theorem 1.2. Thus generalizations of this theorem would immediately provide generalizations of Corollary 1.3.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.