Gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincar\'e inequality

TL;DR

This paper establishes gap theorems in Yang-Mills theory for complete four-dimensional manifolds satisfying a weighted Poincaré inequality, and applies these results to various examples, including a uniqueness theorem for the basic instanton.

Contribution

It introduces new gap theorems in Yang-Mills theory specifically for four-dimensional manifolds with weighted Poincaré inequalities, and proves a uniqueness result for the basic instanton.

Findings

Proved gap theorems for Yang-Mills connections on certain manifolds.

Applied the theorems to multiple examples of manifolds.

Established a uniqueness theorem for the basic instanton.

Abstract

In this paper we prove gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincar\'e inequality. We apply the theorems to many examples of manifolds. We also prove a uniqueness theorem for the basic instanton.

Equations

$$YM\left(A\right)={\int }_M{\left|F_A\right|}^2.$$

$${\int }_{M}q{\varphi }^2\le {\int }_M{\left|\nabla \varphi \right|}^2.$$

$$\left|F_A^{+}\right|\le \frac{1}{\gamma }\left(2q+\frac{1}{3}scal-2{\lambda }_{W^{+}}\right)$$

$$\left|F_A^{+}\right|\le \frac{1}{\gamma }\left(\frac{1}{p}\left(2-\frac{1}{2p}\right)q+\frac{1}{3}scal-2{\lambda }_{W^{+}}\right)$$

$$\left|F_A^{\pm }\right|\le \frac{1}{\gamma }\left\{\frac{1}{p}\left(2-\frac{1}{2p}\right)\left(\frac{9}{4}+\frac{1}{4r^2}+\frac{3}{4{\left(\sinh r\right)}^2}\right)-4\right\}$$

$$\left|F_A^{-}\right|\le \frac{2}{\gamma }\left(\frac{1}{4r^2}+\frac{1}{{\left(\sinh r\right)}^2}-\frac{1}{{\left(\sinh \left(2r\right)\right)}^2}\right)$$

$$\left|F_A\right|\le \frac{4}{\gamma }$$

$$r\left(x\right)=dis{t}_M\left(x,x_0\right)$$

$$B_R=\left\{xinM:r\left(x\right)<R\right\}$$

$$R{m}_{ijkl}=-⟨{\nabla }_{e_i}{\nabla }_{e_j}{e}_k-{\nabla }_{e_j}{\nabla }_{e_i}{e}_k-{\nabla }_{\left[e_i,e_j\right]}{e}_k,e_l⟩,$$

$$Ri{c}_{ij}=\sum _{k}R{m}_{ikjk},$$

$$scal=\sum _{i}Ri{c}_{ii}$$

$$W_{ijkl}$$

$$+\frac{1}{\left(n-1\right)\left(n-2\right)}scal\left(g_{ik}{g}_{jl}-g_{il}{g}_{jk}\right)$$

$$W^{+}=\frac{1}{2}\left(W+\ast W\right)$$

$${\lambda }_{W^{+}}={\lambda }_3^{+}$$

$$0\le {\lambda }_{W^{+}}\le \sqrt{\frac{2}{3}}\left|W^{+}\right|$$

$$d_A^{\ast }{F}_A=0$$

$$F_A^{+}=\frac{1}{2}\left(F_A+\ast F_A\right)$$

$$F_A^{+}=0.$$

$${⟨a,b⟩}_{{\Omega }^p\left(M,V\right)}=\sum _{i_1<\cdots <i_p}{⟨a_{i_1\dots i_p},b_{i_1\dots i_p}⟩}_V$$

$${⟨A,B⟩}_{so\left(N\right)}=-c\cdot tr\left(AB\right)$$

$$\gamma =\{\begin{array}{ll}\frac{2}{\sqrt{3c}}& ifN=3\\ \frac{2\sqrt{2}}{\sqrt{3c}}& ifN\ge 4\end{array}.$$

$${\left|F_A^{+}\right|}^p\Delta {\left|F_A^{+}\right|}^p$$

$$-2p{\lambda }_{W^{+}}{\left|F_A^{+}\right|}^{2p}-p\gamma {\left|F_A^{+}\right|}^{2p+1}$$

$$\left|h\right|\Delta \left|h\right|$$

$$-2\sum _{i<j;k<l}{W}_{ijkl}⟨h_{ij},h_{kl}⟩+\sum _{i,j,k}⟨h_{ij},\left[F_{ik},h_{jk}\right]⟩.$$

$${\left(\Delta h\right)}_{ij}$$

$$+\sum _k\left[F_{ik},h_{jk}\right]-\sum _k\left[F_{jk},h_{ik}\right].$$

$$R{m}_{ijkl}$$

$$-\frac{1}{6}scal\left(g_{ik}{g}_{jl}-g_{il}{g}_{jk}\right)$$

$${\left(\Delta h\right)}_{ij}=\frac{1}{3}scal{h}_{ij}-\sum _{k,l}{W}_{ijkl}{h}_{kl}+\sum _k\left[F_{ik},h_{jk}\right]-\sum _k\left[F_{jk},h_{ik}\right].$$

$$⟨\Delta h,h⟩$$

Full text

1 Introduction

Let $M$ be a $n$-dimensional Riemannian manifold, let $E$ be a Riemannian vector bundle on $M$ with structure group $G\subset O\left(N\right)$, let $A$ be a connection in $E$ and let $F_{A}$ be the curvature of $A$. We define the Yang-Mills energy of $A$ by

[TABLE]

We say that $A$ is a Yang-Mills connection if $A$ is a critical point of this functional. For $n=4$ we say that $A$ is an anti-self-dual instanton if $F_{A}^{+}=0$. In Section 2 we explain these definitions in more detail.

Any instanton is a Yang-Mills connection but there are Yang-Mills connections which are not instantons. In [32] Sibner-Sibner-Uhlenbeck proved the existence of non-instanton Yang-Mills connections in the four-dimensional sphere with $G=SU\left(2\right)$ (see also [6], [28] and [30]). In addition we can easily find non-instanton Yang-Mills connections with a large structure group (for $G=SO\left(4\right)$ and $G=SU\left(N\right)$ with $N\geq 4$ see Section 2.1 in [33]). In this paper we find sufficient conditions for a Yang-Mills connection to be an instanton.

By a gap theorem for $F_{A}^{+}$ we mean the following: under certain assumptions on $M$ if $A$ is a Yang-Mills connection and $\left|F_{A}^{+}\right|_{L^{2}}$ or $\left|F_{A}^{+}\right|_{L^{\infty}}$ is small enough then $F_{A}^{+}=0$. In [7] and [8] Bourguignon-Lawson-Simons proved a $L^{\infty}$ gap of $F_{A}^{+}$ for compact four-dimensional manifolds with a certain positive curvature (for the extension to complete manifolds see [31]). In [24] MinOo proved a $L^{2}$ gap of $F_{A}^{+}$ for these manifolds (see also [27] and for the extension to complete manifolds see [14]). In [16] Feehan proved a $L^{2}$ gap of $F_{A}^{+}$ for compact four-dimensional manifolds with a good metric using results of [18]. We remark that the extension of this gap theorem to general metrics is an open problem. In [20] Gursky-Kelleher-Streets proved a $L^{2}$ gap of $F_{A}^{+}$ for compact four-dimensional manifolds with positive Yamabe invariant.

By a gap theorem for $F_{A}$ we mean the following: under certain assumptions on $M$ if $A$ is a Yang-Mills connection and $\left|F_{A}\right|_{L^{\frac{n}{2}}}$ or $\left|F_{A}\right|_{L^{\infty}}$ is small enough then $F_{A}=0$ where $n=\dim M$ (see [17], [19], [29], [36] and the above references). We remark that for $n=4$ gap theorems for $F_{A}^{+}$ and $F_{A}^{-}$ are more important than gap theorems for $F_{A}$ because they can be applied to non-flat connections.

In this paper we extend the results of Bourguignon-Lawson-Simons to complete four-dimensional manifolds with a weighted Poincaré inequality (Theorem 1 and Theorem 2). Using our main results we prove a gap theorem for the model spaces with constant sectional curvature and the model spaces with constant holomorphic sectional curvature (Theorem 3) and we prove a uniqueness theorem for the basic instanton (Theorem 4).

Let $M$ be a Riemannian manifold and let $q$ be an almost everywhere continuous function in $M$. We say that $M$ satisfies a weighted Poincaré inequality with weight function $q$ if for any $\phi$ in $C^{\infty}\left(M\right)$ with compact support we have

[TABLE]

Weighted Poincaré inequalities (Hardy type inequalities) were studied for the first time in geometry by Carron [10] (for submanifolds) and Li-Wang [23] (for Riemannian manifolds). In Section 4 we give many examples of weighted Poincaré inequalities.

In what follows $r$ is the distance function, $B_{R}$ is the ball of radius $R$, $\lambda_{W^{+}}$ is the largest eigenvalue of $W^{+}$ and $\gamma$ is a constant depending only on $G$. In Section 2 we explain these notations in more detail.

Assuming that $vol\left(B_{R}\right)$ grows slower than $R^{p}$ and $\left|F_{A}^{+}\right|$ grows slower than $r^{2-p}$ we prove a $L^{\infty}$ type gap of $F_{A}^{+}$ without any assumption on the integral of $F_{A}^{+}$. This is very different from previous works.

Theorem 1**.**

Let $M$ be a complete four-dimensional Riemannian manifold. Suppose that $M$ satisfies a weighted Poincaré inequality with weight function $q$ and $vol\left(B_{R}\right)\leq C_{1}R^{p}$ for $R$ sufficiently large. Let $E$ be a Riemannian vector bundle on $M$ with structure group $G\subset O\left(N\right)$ and let $A$ be a Yang-Mills connection in $E$. Suppose that $\left|F_{A}^{+}\right|\leq C_{2}r^{2-p}$ for $r$ sufficiently large. If

[TABLE]

then either (1) is an equality or $F_{A}^{+}=0$.

Here $C_{1}>0$, $C_{2}>0$, $p>0$ and we assume that the function in the right hand side of (1) is non-negative and non-zero.

Assuming that $q$ grows slower than $r^{2}$ and $F_{A}^{+}$ is in $L^{2p}$ we prove a $L^{\infty}$ type gap of $F_{A}^{+}$ without any assumption on the volume of $B_{R}$.

Theorem 2**.**

Let $M$ be a complete four-dimensional Riemannian manifold. Suppose that $M$ satisfies a weighted Poincaré inequality with weight function $q$ and $q\leq Cr^{2}$ for $r$ sufficiently large. Let $E$ be a Riemannian vector bundle on $M$ with structure group $G\subset O\left(N\right)$ and let $A$ be a Yang-Mills connection in $E$. Suppose that $F_{A}^{+}$ is in $L^{2p}$. If

[TABLE]

then either (2) is an equality or $F_{A}^{+}=0$.

Here $C>0$, $p>\frac{1}{4}$ and we assume that the function in the right hand side of (2) is non-negative and non-zero.

In [8] Bourguignon-Lawson proved a $L^{\infty}$ gap of $F_{A}^{\pm}$ for $S^{4}$. We improve this result by a factor $\sqrt{2}$ for $G=SO\left(3\right)$. They also proved a $L^{\infty}$ gap of $F_{A}^{-}$ for $CP^{2}$ (see Theorem 5.26 in [8]). We improve this result by a factor $\sqrt{2}$ (resp. $2$) for $G\subset SO\left(N\right)$ with $N\geq 4$ (resp. $G=SO\left(3\right)$). In [14] Dodziuk-MinOo proved a $L^{2}$ gap of $F_{A}^{\pm}$ for $R^{4}$. We prove a $L^{\infty}$ type gap of $F_{A}^{\pm}$ for $R^{4}$ without any assumption on the integral of $F_{A}^{\pm}$. We remark that for $R^{4}$ the assumption that $\left|F_{A}^{\pm}\right|$ grows slower than $r^{-2}$ does not imply that $F_{A}^{\pm}$ is in $L^{2}$. In [29] Price proved a $L^{2}$ type gap of $F_{A}$ for $H^{n}$ with $n\geq 5$. In [36] Zhou proved a $L^{\frac{n}{2}}$ type gap of $F_{A}$ for $H^{n}$ with $n\geq 16$. We prove a $L^{\infty}$ type gap of $F_{A}^{\pm}$ for $H^{4}$. We also prove a $L^{\infty}$ type gap of $F_{A}^{-}$ for $CH^{2}$ and a $L^{\infty}$ gap of $F_{A}^{\pm}$ for $S^{3}\times R$ without any assumption on the integral of $F_{A}^{\pm}$.

Theorem 3**.**

Let $M$ be a four-dimensional Riemannian manifold. Let $E$ be a Riemannian vector bundle on $M$ with structure group $G\subset O\left(N\right)$ and let $A$ be a Yang-Mills connection in $E$.

(i) Let $M=S^{4}$. If $\left|F_{A}^{\pm}\right|\leq\frac{4}{\gamma}$ then either $F_{A}^{\pm}=0$ or $\left|F_{A}^{\pm}\right|=\frac{4}{\gamma}$.

(ii) Let $M=R^{4}$. If $\left|F_{A}^{\pm}\right|\leq\frac{2}{\gamma r^{2}}$ then $F_{A}^{\pm}=0$.

(iii) Let $M=H^{4}$. If $F_{A}^{\pm}$ is in $L^{2p}$ for some $\frac{3}{8}\leq p\leq\frac{3}{4}$ and

[TABLE]

then $F_{A}^{\pm}=0$.

(iv) Let $M=CP^{2}$. If $\left|F_{A}^{-}\right|\leq\frac{8}{\gamma}$ then either $F_{A}^{-}=0$ or $\left|F_{A}^{-}\right|=\frac{8}{\gamma}$.

(v) Let $M=CH^{2}$. If $F_{A}^{-}$ is in $L^{1}$ and

[TABLE]

then $F_{A}^{-}=0$.

(vi) Let $M=S^{3}\times R$. If $\left|F_{A}^{\pm}\right|\leq\frac{2}{\gamma}$ then either $F_{A}^{\pm}=0$ or $\left|F_{A}^{\pm}\right|=\frac{2}{\gamma}$.

Here $S^{4}$ is the sphere with constant sectional curvature $1$, $R^{4}$ is the Euclidean space, $H^{4}$ is the hyperbolic space with constant sectional curvature $-1$, $CP^{2}$ is the complex projective space with constant holomorphic sectional curvature $4$, $CH^{2}$ is the complex hyperbolic space with constant holomorphic sectional curvature $-4$ and $S^{3}\times R$ is a cylinder with the product metric.

In [8] Bourguignon-Lawson proved that if $E$ is a bundle on $S^{4}$ with structure group $G\subset SO\left(4\right)$ and $A$ is a Yang-Mills connection in $E$ with $\left|F_{A}\right|\leq\frac{4}{\gamma}$ then either $A$ is flat or $E$ is one of the four-dimensional bundles of tangent spinors with the canonical Riemannian connection. We improve this result for $G=SU\left(2\right)$.

Theorem 4**.**

Let $E$ be a Riemannian vector bundle on $S^{4}$ with structure group $G=SU\left(2\right)$ and let $A$ be a Yang-Mills connection in $E$. If

[TABLE]

then either $A$ is flat or the corresponding instanton in $R^{4}$ (using a stereographic projection of $S^{4}\setminus\left\{p\right\}$) is the ADHM-BPST basic instanton.

This paper is organized as follows. In Section 2 we give the basic definitions, notations and conventions. In Section 3 we prove Theorem 1 and Theorem 2. In Section 4 we give examples of weighted Poincaré inequalities. In Section 5 we prove Theorem 3 and Theorem 4.

We would like to thank Detang Zhou, Gonçalo Oliveira and Alex Waldron for the support.

2 Preliminaries

In this section we give the basic definitions, notations and conventions.

Let $\left(M,g\right)$ be a $n$-dimensional Riemannian manifold. We define the distance function (to $x_{0}$) by

[TABLE]

and the ball of radius $R$ (and center $x_{0}$) by

[TABLE]

where $x_{0}$ is a fixed point in $M$. We define the Riemann curvature by

[TABLE]

the Ricci curvature by

[TABLE]

the scalar curvature by

[TABLE]

and the Weyl curvature by

[TABLE]

where $\left\{e_{i}\right\}$ is a local orthonormal basis of $TM$. We have that $W$ is a self-adjoint operator in $\Omega^{2}\left(M\right)$. Now suppose that $n=4$. We define the self-dual part of $W$ by

[TABLE]

where $*$ is the Hodge star. We have that $W^{+}$ is a self-adjoint operator in $\Omega_{+}^{2}\left(M\right)$. We define the largest eigenvalue of $W^{+}$ by

[TABLE]

where $\lambda_{1}^{+}\leq\lambda_{2}^{+}\leq\lambda_{3}^{+}$ are the eigenvalues of $W^{+}$. We remark that

[TABLE]

where $\left|W^{+}\right|=\sqrt{\left(\lambda_{1}^{+}\right)^{2}+\left(\lambda_{2}^{+}\right)^{2}+\left(\lambda_{3}^{+}\right)^{2}}$ (see Proposition 1 in [9]). We can define $\lambda_{W^{-}}$ in a similar way using $W^{-}=\frac{1}{2}\left(W-*W\right)$.

Let $M$ be a $n$-dimensional Riemannian manifold, let $E$ be a Riemannian vector bundle on $M$, let $A$ be a connection in $E$ and let $F_{A}$ be the curvature of $A$. We say that $A$ is a Yang-Mills connection if

[TABLE]

where $d_{A}^{*}$ is the adjoint of $d_{A}$. Now suppose that $n=4$. We define the self-dual part of $F_{A}$ by

[TABLE]

where $*$ is the Hodge star. We say that $A$ is an anti-self-dual instanton if

[TABLE]

We can define self-dual instantons in a similar way using $F_{A}^{-}=\frac{1}{2}\left(F_{A}-*F_{A}\right)$. For more information about Yang-Mills theory see [8] and [35].

Let $M$ be a Riemannian manifold and let $V$ be a Riemannian vector bundle on $M$. We define the inner product of $\Omega^{p}\left(M,V\right)$ (the set of $p$-forms in $M$ with values in $V$) by

[TABLE]

where $\left\{e_{i}\right\}$ is a local orthonormal basis of $TM$.

Let $G$ be a Lie group embedded in $O\left(N\right)$ (the orthogonal group). We define the inner product of $so\left(N\right)$ (the orthogonal Lie algebra) by

[TABLE]

where $c>0$. We define

[TABLE]

We remark that (i) taking $c=1$ we get the Frobenius inner product of $so\left(N\right)$, (ii) taking $c=N-2$ we get minus the Killing form of $so\left(N\right)$ and (iii) taking $c=\frac{1}{2}$ we have that the standard basis of $so\left(N\right)$ is orthonormal.

3 Proof of Theorem 1 and Theorem 2

In this section we prove Theorem 1 and Theorem 2.

First we find a formula for the Laplacian of $\left|F_{A}^{+}\right|^{p}$. We define $\Delta=\sum_{i}\nabla_{i}\nabla_{i}$.

Lemma 5**.**

Let $M$ be a four-dimensional Riemannian manifold and let $E$ be a Riemannian vector bundle on $M$ with structure group $G\subset O\left(N\right)$. If $A$ is a Yang-Mills connection in $E$ then

[TABLE]

where $p>0$.

Proof.

Let $F=F_{A}$ and let $\left\{e_{i}\right\}$ be a local orthonormal basis of $TM$.

Claim 1. Suppose that $h$ is in $\Omega^{2}\left(M,so\left(E\right)\right)$. If $d_{A}h=0$ and $d_{A}^{*}h=0$ then

[TABLE]

Proof. Using Theorem 3.10 in [8] we have

[TABLE]

Using the fact that

[TABLE]

we get

[TABLE]

Taking the inner product with $h_{ij}$ and renaming the indices we get

[TABLE]

Using the fact that

[TABLE]

we get Claim 1.

Claim 2. We have

[TABLE]

Proof. We have that $F^{+}$ is in $\Omega^{2}\left(M,so\left(E\right)\right)$. Using the Bianchi identity and the fact that $A$ is a Yang-Mills connection we have $d_{A}F^{+}=0$ and $d_{A}^{*}F^{+}=0$. Using Claim 1 and the fact that $F^{+}$ is self-dual we get Claim 2.

Claim 3. We have

[TABLE]

Proof. Using Proposition 2.8 in [20] we have

[TABLE]

Using the fact that $\lambda_{W^{+}}$ is the largest eigenvalue of $W^{+}$ we have

[TABLE]

Using Lemma 2.6 in [20] we have

[TABLE]

We give an alternative proof of this inequality. Scaling Lemma 2.30 and Note 2.31 in [8] with respect to our definition of the inner product of $so\left(N\right)$ (see Section 2) we have

[TABLE]

where

[TABLE]

We see that

[TABLE]

To get the second line we write the sum in $i,j,k\left(distinct\right)$ as $i<j<k$, $i<k<j$, $j<i<k$, $j<k<i$, $k<i<j$, $k<j<i$ and we rename the indices. We see that

[TABLE]

To get the first and third line we use the fact that $F^{+}$ is self-dual. To get the second line we use the inequality of arithmetic and geometric means. To get the fourth line we use the fact that $\gamma=\frac{2\sqrt{2}b}{\sqrt{3}}$ (see Section 2). Using Claim 2 and the above inequalities we get Claim 3.

Using Claim 3 and a direct calculation we get the lemma. ∎

Now we prove Theorem 1.

Proof.

Let $\phi$ be the cut-off function in $M$ with

[TABLE]

Claim 1. We have

[TABLE]

Proof. Using Lemma 5 we have

[TABLE]

Multiplying by $\phi^{2}$ and integrating by parts we get

[TABLE]

Putting $\left|F^{+}\right|^{\frac{1}{2}}\phi$ in the weighted Poincaré inequality (see Section 4) we have

[TABLE]

We see that

[TABLE]

Rearranging terms we get Claim 1.

Claim 2. We have

[TABLE]

Proof. We have

[TABLE]

By assumption we can find $R_{0}>0$ such that $vol\left(B_{R}\right)\leq C_{1}R^{p}$ for $R\geq R_{0}$ and $\left|F^{+}\right|\leq C_{2}r^{2-p}$ in $M\setminus B_{R_{0}}$. We see that

[TABLE]

Taking $C_{3}=C_{1}C_{2}e^{p}$ we get Claim 2.

Using Claim 1 and Claim 2 we have

[TABLE]

By assumption we have that the term in the parentheses is non-negative. Taking the limit as $R\to\infty$ and using the monotone convergence theorem we get

[TABLE]

We see that

[TABLE]

Suppose that (1) is not an equality. In this case we have that the term in the parentheses is positive in some point of $M$. By continuity we have $F^{+}=0$ in an open subset of $M$. Using the unique continuation principle we conclude that $F^{+}=0$. ∎

Now we prove Theorem 2.

Proof.

Let $\phi$ be the cut-off function in $M$ with

[TABLE]

Using the fact that $p>\frac{1}{4}$ we have $2-\frac{1}{2p}-\frac{\epsilon}{R^{2}}>0$ for $R$ sufficiently large and $0<\epsilon\leq 1$.

Claim 1. We have

[TABLE]

Proof. Using Lemma 5 we have

[TABLE]

Multiplying by $\phi^{2}$ and integrating by parts we get

[TABLE]

Using the fact that

[TABLE]

we get Claim 1.

Claim 2. We have

[TABLE]

Proof. Putting $\left|F^{+}\right|^{p}\phi$ in the weighted Poincaré inequality (see Section 4) we have

[TABLE]

Multiplying by $\left(2-\frac{1}{2p}-\frac{\epsilon}{R^{2}}\right)\left(1+\frac{\epsilon}{R^{2}}\right)^{-1}$ we get

[TABLE]

Using the fact that

[TABLE]

we get Claim 2.

Using Claim 1 and Claim 2 and rearranging terms we have

[TABLE]

We have

[TABLE]

By assumption we can find $R_{0}>0$ such that $q\leq Cr^{2}$ in $M\setminus B_{R_{0}}$. We see that

[TABLE]

By assumption we have that the term in the parentheses in the left hand side is non-negative and $\left|F^{+}\right|_{L^{2p}}<\infty$. Taking the limit as $R\to\infty$ and using the monotone convergence theorem we get

[TABLE]

Taking the limit as $\epsilon\to 0$ we get

[TABLE]

We see that

[TABLE]

Suppose that (2) is not an equality. In this case we have that the term in the parentheses is positive in some point of $M$. By continuity we have $F^{+}=0$ in an open subset of $M$. Using the unique continuation principle we conclude that $F^{+}=0$. ∎

4 Weighted Poincaré inequalities

In this section we give examples of weighted Poincaré inequalities.

First we recall the definition of a weighted Poincaré inequality.

Let $M$ be a Riemannian manifold and let $q$ be an almost everywhere continuous function in $M$. We say that $M$ satisfies a weighted Poincaré inequality with weight function $q$ if for any $\phi$ in $C^{\infty}\left(M\right)$ with compact support we have

[TABLE]

Now we give the weighted Poincaré inequalities that we use in the proof of Theorem 3.

For $R^{4}$ we use Hardy’s inequality.

Lemma 6**.**

For any $\phi$ in $C^{\infty}\left(R^{n}\right)$ with compact support we have

[TABLE]

where $r\left(x\right)=\left|x\right|$. In other words $R^{n}$ satisfies a weighted Poincaré inequality with weight function

[TABLE]

For $H^{4}$ we use Theorem 2.1 in [5].

Lemma 7**.**

For any $\phi$ in $C^{\infty}\left(H^{n}\right)$ with compact support we have

[TABLE]

where $r\left(x\right)=dist_{H^{n}}\left(x,x_{0}\right)$. In other words $H^{n}$ satisfies a weighted Poincaré inequality with weight function

[TABLE]

For $CP^{2}$ we find a new weighted Poincaré inequality. We remark that this inequality improves the bottom of the spectrum of the Laplacian of $CH^{n}$ because $q>n^{2}$ (see [22] and [25]). We would like to know whether the result is sharp in the sense of [5].

Lemma 8**.**

For any $\phi$ in $C^{\infty}\left(CH^{n}\right)$ with compact support we have

[TABLE]

where $r\left(x\right)=dist_{CH^{n}}\left(x,x_{0}\right)$. In other words $CH^{n}$ satisfies a weighted Poincaré inequality with weight function

[TABLE]

Proof.

Claim 1. For any $\phi$ in $C^{\infty}\left(CH^{n}\right)$ with compact support we have

[TABLE]

Proof. Let $M$ be a complete Riemannian manifold with a pole. Using Theorem 1.1 in [1] for any $\phi$ in $C^{\infty}\left(M\right)$ with compact support we have

[TABLE]

where $r\left(x\right)=dist_{M}\left(x,x_{0}\right)$ and $x_{0}$ is the pole. Using Bochner’s formula and the fact that $\left|\nabla r\right|^{2}=1$ we have

[TABLE]

Using the above formulas we get Claim 1.

Claim 2. We have

[TABLE]

Proof. Using the fact that

[TABLE]

(see for example Theorem 1.6 in [22]) we have

[TABLE]

and

[TABLE]

Using the above formulas we get Claim 2.

Using Claim 1 and Claim 2 we get the lemma. ∎

Now we give more examples of weighted Poincaré inequalities for completeness.

Example 9**.**

Let $M$ be a Riemannian manifold. We define the bottom of the spectrum of the Laplacian by

[TABLE]

where the infimum is taken over all $\phi$ in $C^{\infty}\left(M\right)$ with compact support. We see that for any $\phi$ in $C^{\infty}\left(M\right)$ with compact support we have

[TABLE]

In other words $M$ satisfies a weighted Poincaré inequality with weight function

[TABLE]

Example 10**.**

Let $\bar{M}$ be a Riemannian manifold and let $M$ be a hypersurface in $\bar{M}$. We say that $M$ is stable if the second variation of the area functional is non-negative for any normal variation with compact support. We have that $M$ is stable if and only if for any $\phi$ in $C^{\infty}\left(M\right)$ with compact support we have

[TABLE]

where $A$ is the second fundamental form, $\bar{R}ic$ is the Ricci curvature of $\bar{M}$ and $N$ is a unit vector in $TM^{\perp}$. In other words $M$ is stable if and only if $M$ satisfies a weighted Poincaré inequality with weight function

[TABLE]

Example 11**.**

Let $M$ be a minimal hypersurface in $R^{n+1}$. Using Corollary 0.2 in [10] for any $\phi$ in $C^{\infty}\left(M\right)$ with compact support we have

[TABLE]

where $r\left(x\right)=dist_{R^{n+1}}\left(x,x_{0}\right)|M$. In other words $M$ satisfies a weighted Poincaré inequality with weight function

[TABLE]

For more examples of weighted Poincaré inequalities and their applications in the geometry and topology of Riemannian manifolds and hypersurfaces see [10], [11], [21], [23] and [34].

We would like to find weighted Poincaré inequalities for certain ALE and ALF spaces leading to new applications of Theorem 1. For example for the Euclidean Schwarzschild manifold (see [26]).

For different applications of weighted Poincaré inequalities in gauge theory see [12] and [13].

5 Proof of Theorem 3 and Theorem 4

In this section we prove Theorem 3 and Theorem 4.

First we prove Theorem 3.

Proof.

For $S^{4}$ we have $scal=12$, $W=0$ and $vol\left(B_{R}\right)\leq C$. For $CP^{2}$ we have $scal=24$, $W^{-}=0$ and $vol\left(B_{R}\right)\leq C$. For $S^{3}\times R$ we have $scal=6$, $W=0$ and $vol\left(B_{R}\right)\leq CR$. Taking $p=1$ and $q=0$ and using Theorem 1 we get the results for $S^{4}$, $CP^{2}$ and $S^{3}\times R$.

For $R^{4}$ and $H^{4}$ (resp. $CH^{2}$) we have that the inequality of $\left|F^{\pm}\right|$ (resp. $\left|F^{-}\right|$) in the assumption is not an equality because the term in the right hand side goes to infinity as $x$ goes to $x_{0}$ (the pole).

For $R^{4}$ we have $scal=0$, $W=0$ and $vol\left(B_{R}\right)\leq CR^{4}$. Taking $p=4$ and $q$ as in Lemma 6 and using Theorem 1 we get the result for $R^{4}$.

For $H^{4}$ we have $scal=-12$ and $W=0$. Using the fact that $\frac{3}{8}\leq p\leq\frac{3}{4}$ we have

[TABLE]

Taking $q$ as in Lemma 7 and using Theorem 2 we get the result for $H^{4}$.

For $CH^{2}$ we have $scal=-24$ and $W^{-}=0$. Taking $p=\frac{1}{2}$ and $q$ as in Lemma 8 and using Theorem 2 we get the result for $CH^{2}$. ∎

Now we prove Theorem 4.

Proof.

Without loss of generality we define the inner product of $su\left(N\right)$ by

[TABLE]

We have that $su\left(N\right)$ is canonically embedded in $so\left(2N\right)$. We define the inner product of $so\left(2N\right)$ by

[TABLE]

In this way the embedding is an isometry.

Claim 1. If $A$ is not flat then $A$ is an instanton in $S^{4}$ with $\left|F_{A}\right|_{g_{S^{4}}}=\sqrt{3}$ and $charge\left(E\right)=1$.

Proof. Taking $c=\frac{1}{2}$ and $N=4$ in the definition of $\gamma$ (see Section 2) and using (3) we have $\left|F_{A}\right|\leq\sqrt{3}$. Using the fact that

[TABLE]

we get $\left|F_{A}^{+}\right|\leq\sqrt{3}$ and $\left|F_{A}^{-}\right|\leq\sqrt{3}$. Applying Theorem 3 to $F_{A}^{+}$ we have $\left|F_{A}^{+}\right|=\sqrt{3}$ or $F_{A}^{+}=0$. Applying Theorem 3 to $F_{A}^{-}$ we have $\left|F_{A}^{-}\right|=\sqrt{3}$ or $F_{A}^{-}=0$. Using the above formula we have that the case $\left|F_{A}^{+}\right|=\sqrt{3}$ and $\left|F_{A}^{-}\right|=\sqrt{3}$ is not possible. We see that either (i) $\left|F_{A}^{+}\right|=\sqrt{3}$ and $F_{A}^{-}=0$ or (ii) $F_{A}^{+}=0$ and $\left|F_{A}^{-}\right|=\sqrt{3}$. Without loss of generality we suppose that (ii) is true. Using the fact that $vol\left(S^{4}\right)=\frac{8\pi^{2}}{3}$ and

[TABLE]

we get Claim 1.

Suppose that $A$ is not flat. Using Claim 1 we have $charge\left(E\right)=1$. Using a stereographic projection of $S^{4}\setminus\left\{p\right\}$ we can view $A$ as an instanton in $R^{4}$. Using the Claim 1 and the fact that $g_{S^{4}}=\frac{4}{\left(1+\left|x\right|^{2}\right)^{2}}g_{R^{4}}$ we have

[TABLE]

Let $\left\{A_{x_{0},a}\right\}$ be the five-parameter family of instantons in $R^{4}$ with group $SU\left(2\right)$ and charge $1$ where $x_{0}$ is a point in $R^{4}$ (the center) and $a>0$ (the scale). This family is explicit and we have

[TABLE]

See for example [3], [15], [20] and [33].

Using the above formulas we have $x_{0}=0$ and $a=1$. We conclude that $A$ is the basic instanton (the ADHM [2] - BPST [4] instanton in $R^{4}$ with group $SU\left(2\right)$, charge $1$, center [math] and scale $1$). ∎

We remark that the $L^{2}$ version of Theorem 4 is not true. Indeed for any $A_{x_{0},a}$ in the five-parameter family of instantons in $S^{4}$ with group $SU\left(2\right)$ and charge $1$ we have $\int_{S^{4}}\left|F\left(A_{x_{0},a}\right)\right|^{2}=\left(\frac{4}{\gamma}\right)^{2}vol\left(S^{4}\right)$.