A positive energy theorem for fourth-order gravity

Abstract

In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric analysis. For instance, it underlies positive mass theorems associated to the Paneitz operator, playing a similar role in the positive Q-curvature conformal prescription problem as the Schoen–Yau positive energy theorem does for the Yamabe problem. Several other links to Q-curvature analysis and rigidity phenomena are established.

Appendices

Appendix: Some analytic results concerning AE manifolds

In this appendix we will collect some results concerning AE manifolds which are used in the core of the paper. Most of these results are well-known for experts. We include them for the sake of completeness and to deliver a self-contained presentation. For detailed proofs and discussions on these topics, we refer the reader to Bartnik [9], Lee-Parker [11].

Let us start with the following fundamental theorem regarding the properties of the Laplacian on AE manifolds.

Theorem A.1

Let (M, g) an asymptotically Euclidean manifold with a structure at infinity $\phi :M\setminus K \rightarrow {\mathbb {R}}^n\setminus B_1$ with decay rate $\tau$. If $\delta \not \in ({\mathbb {Z}} \setminus \{-1,\cdots ,3-n\})$ and $q>1$ then

$$\begin{aligned} \Delta _g: W_\delta ^{2,q}(\phi ) \rightarrow L^q_{\delta -2} \end{aligned}$$

is Fredholm. Moreover

$$\begin{aligned} \left\{ \begin{array}{l} \hbox { if } \delta >2-n \hbox { then } \Delta _g \hbox { is surjective,}\\ \hbox { if } 2-n<\delta<0 \hbox { then } \Delta _g \hbox { is bijective,}\\ \hbox { if } \delta <0 \hbox { then } \Delta _g \hbox { is injective.} \end{array} \right. \end{aligned}$$

(A.1)

Remark A.1

The decay rates in the set ${\mathbb {Z}} \setminus \{-1,\cdots ,3-n\}$ are called exceptional and we say that $\delta$ is non-exceptional if $\delta \not \in {\mathbb {Z}} \setminus \{-1,\cdots ,3-n\}$.

Let us now analyse the relation between different potential structures of infinity. In particular, the following theorem concerns the existence of harmonic coordinates.

Theorem A.2

Let (M, g) be an AE manifold, with $g\in W^{k,q}_{loc}$, $k\ge 1$, $q>n$, and $(\Phi ,x):M\backslash K\mapsto E_R\doteq {\mathbb {R}}^n\backslash \overline{B_R(0)}$ where $K\subset \subset M$, $R\ge 1$ is a structure of infinity of order $\tau > 0$ with $1-\tau$ non-exceptional, and fix $1<\eta <2$. There are functions $y^{i}\in W^{k+1,q}_{\eta }$, $i=1,\cdots ,n$, such that $\Delta _gy^i=0$ and $(x^i-y^i)\in W^{k+1,q}_{1-\tau ^{*}}(E_R)$ for $\tau ^{*}\doteq \min \{\tau ,n-2\}$, which implies

$$\begin{aligned} \begin{aligned} |x^i-y^i|&=o_k(r^{1-\tau ^{*}}),\\ |g(\partial _{x^i},\partial _{x^j})-g(\partial _{y^i},\partial _{y^j})|&=o_k(r^{-\tau ^{*}}). \end{aligned} \end{aligned}$$

(A.2)

Furthermore, the set of functions $\{1,y^i\}$ is a basis for $H_1=\{u\in W^{k,q}_{\eta }: \Delta _gu=0\}$.

Proof

Let us first extend the functions $x^i$ smoothly to all of M. Then, near infinity, $\Delta _gx^i=g^{kl}\Gamma ^i_{kl}\doteq \Gamma ^i\in W^{k-1,q}_{-1-\tau }$. Notice that if $1-\tau >2-n$, then Theorem A.1 implies the existence of $v^i\in W^{k+1,q}_{1-\tau }$ solving $\Delta _gv^i=\Gamma ^i$. Now, if $1-\tau < 2-n$ (the equality case is an exceptional case), we cannot, a priori, improve the decay given by $r^{2-n}$. This actually follows from Theorem 1.17 in [9]. Therefore, in any case, we know that there are solutions $v^i\in W^{k+1,q}_{1-\tau ^{*}}$ to $\Delta _gv^i=\Gamma ^i$, where $\tau ^{*}=\min \{\tau ,n-2\}$. Therefore, there is some $v^i\in W^{k+1,q}_{1-\tau ^{*}}$ satisfying $\Delta _g(x^i-v^i)=0$. Let us then define $y^i\doteq x^i-v^i$, which implies the first estimate in (A.2), and furthermore, since $\frac{\partial y^i}{\partial x^j}=\delta ^i_j + o(r^{-\tau ^{*}})$, we see that near infinity $\{y^i\}$ are coordinates which are asymptotically Cartesian. This, in turn, implies the second estimate in (A.2). The final claim concerning $H_1$ follows since $\{1,y^i\}$ spans an $(n+1)$-dimensional subspace of $H_1$, but from Proposition 2.2 in [9] $\mathrm {dim}(H_1)=n+1$. $\square$

Remark A.2

Let us highlight that the statement of the above theorem is slightly different than the well-known Theorem 3.1 in [9]. The difference relies in the fact that, in [9], it does not seem to be explicitly stated that, a priori, the decay rate $r^{2-n}$ cannot be improved, regardless of how large $\tau$ may be. Nevertheless, as can be seen in the above proof, the above mild correction comes about from results contained in the same reference, as is also clear from a careful examination of the proof of Theorem 3.1 in [9]. Similar comments apply to the following theorem, which can be found in [9] as Corollary 3.2, and presents the relation between two different structures of infinity.

Theorem A.3

Let (M, g) an asymptotically Euclidean manifold, $g\in W^{k,q}_{loc}$, $k\ge 1$ and $q>n$, with two structures at infinity $\phi ,\psi :M\setminus K \rightarrow {\mathbb {R}}^n\setminus \overline{B_1(0)}$ with decay rates $\tau _\phi$ and $\tau _\psi$, where of each of these weights satisfies that $1-\tau$ is non-exceptional. There exists $(O,a)\in O(n)\times {\mathbb {R}}^n$ such that

$$\begin{aligned} x^i-(O^i_jz^j +a^i) \in W^{k,q}_{1-\tau }({\mathbb {R}}^n) \end{aligned}$$

which implies

$$\begin{aligned} \vert x^i-(O^i_jz^j +a^i)\vert =o_{k}(r^{1-\tau }), \end{aligned}$$

(A.3)

where $\tau \doteq \min \{\tau _\phi ,\tau _\psi ,n-2\}$, $x=\phi ^{-1}$ and $z=\psi ^{-1}$.

Proof

Let $y^i$ and $w^i$ be the harmonic coordinates constructed in the previous theorem associated to $\phi$ and $\psi$ respectively. Then, since $H_1$ is intrinsic to M and $\{1,y^i\}$ and $\{1,w^i\}$ are bases for this space, we get that

$$\begin{aligned} w^i=A^i_jy^j + a^i, \end{aligned}$$

where $A\in \mathrm {GL}(n,{\mathbb {R}})$ a priori, but actually, from the construction of the previous theorem, we know that $z^i=A^i_jx^j + a^i - A^i_jv^j + {\bar{v}}^i$, with $v^i\in W^{k,q}_{1-\tau ^{*}_{\phi }}$ and ${\bar{v}}^i\in W^{k,q}_{1-\tau ^{*}_{\psi }}$, where $\tau ^{*}$ was defined in the previous theorem. Since these last two systems are Cartesian, then $A\in O(n)$ and we explicitly see that $z^i-A^i_jx^j-a^i\in W^{k,q}_{1-\tau }$. $\square$

The above estimate is the best possible and it is determined by the Ricci curvature as shown by the next theorem which corresponds to Proposition 3.3 in [9], see also [44].

Theorem A.4

Let (M, g) an asymptotically Euclidean manifold with a structure at infinity $\phi :M\setminus K \rightarrow {\mathbb {R}}^n\setminus B(0,1)$ with decay rate $\tau$ such that $(\phi _{*} g-\delta )\in W^{2,q}_{-\tau }({\mathbb {R}}^n\setminus B(0,1))$ for $q>n$ and such that

$$\begin{aligned} Ric_g \in L^q_{-2-\eta }(M) \hbox { for some } \eta >\tau \hbox { and } \eta \not \in ({\mathbb {Z}} \setminus \{-1,\cdots ,3-n\}) . \end{aligned}$$

Then there exists a structure at infinity $\Theta :M\setminus K' \rightarrow {\mathbb {R}}^n\setminus B_1$ such that $(\Theta _* g-\delta )\in W^{2,q}_{-\eta }({\mathbb {R}}^n\setminus B_1)$.

Finally, the following statement concerning the conformal Laplacian will be important in our analysis. It is a variant of theorem 9.2 of [11]

Corollary A.1

Let (M, g) an asymptotically Euclidean manifold of order $\tau >0$ and assume $R_g\ge 0$. If $2-n<\delta <0$ and $q>1$, then

$$\begin{aligned} L_g =\Delta _g -c_n R_g : W_\delta ^{2,q} \rightarrow L^q_{\delta -2} \end{aligned}$$

is an isomorphism.

Proof

Since $L_g$ is a compact perturbation of the Laplacian, it is also Fredholm, hence thanks to its self-adjointness it suffices to proof injectivity to get the result. Consider $u\in W_\delta ^{2,q}$ such that $\Delta _g u=c_n R_g u$, and first notice that by elliptic regularity we can assume that $u\in W^{2,q}_{\delta }$ with $q>n$. Also, notice that $R_g u \in W^{2,q}_{\delta '-2}$ for any $\delta '>\delta -\tau$. Since $\tau >0$, we can pick $\delta '$ satisfying $\max \{\delta -\tau ,2-n\}<\delta '<\delta$ and appeal to Theorem A.1 to obtain $u\in W^{2,q}_{\delta '}$. Since we can repeat the argument as much as necessary, we can assume that $u\in W^{2,q}_{\delta '}$ with $\delta '> 2-n$ taken arbitrary close to $2-n$. Then, we can multiply $L_g(u)=0$ by u and integrate by parts to obtain

$$\begin{aligned} \int _{M\setminus ({\mathbb {R}}^n \setminus B_R(0))} \left( \vert \nabla u \vert _g^2 + c_n R_g u^2\right) dv_g = \int _{\partial B_R(0)} u\partial _\nu u \; d\sigma , \end{aligned}$$

Since $u=O_2(r^{\delta '})$ near infinity, we can estimate $\vert \nabla u \vert _g^2=O(r^{2\delta '-2})$ and $R_gu^2=O(r^{2\delta '-\tau -2})$ which implies these quantities are in $L^1(M)$ as long as chose $2-n<\delta '<\frac{2-n}{2}$. Furthermore, under these condition one finds $u\partial _\nu u=O(r^{2\delta '-1})$ and therefore

$$\begin{aligned} \int _{M\setminus ({\mathbb {R}}^n \setminus B_R(0))} \left( \vert \nabla u \vert _g^2 + c_n R_g u^2\right) dv_g = O(R^{n-2\delta '-2}) \end{aligned}$$

so that we can pass to the limit as $R\rightarrow +\infty$, which gives that $u\equiv 0$ since $R_g{\ge }0$ and $u\rightarrow 0$ as $|x|\rightarrow \infty$, which concludes the proof. $\square$

Appendix: Conventions on Q-curvature

Let us adopt the following general definition of Q-curvature for an arbitrary Riemannian manifold $(M^n,g)$ with $n\ge 3$:

$$\begin{aligned} Q_g=-\Delta _g\sigma _1(S_g) + 4\sigma _2(S_g) + \frac{n-4}{2}\left( \sigma _1(S_g)\right) ^2, \end{aligned}$$

(B.1)

where $S_g\doteq \frac{1}{n-2}\left( \mathrm {Ric}_g - \frac{1}{2(n-1)}R_gg \right)$ stands for the Schouten tensor and $\sigma _k(S_g)$ stands for the k-th elementary symmetric function of the eigenvalues of $S_g$. In this context, the Paneitz operator is defined by

$$\begin{aligned} P_gu\doteq \Delta ^2_gu + \mathrm {div}_{g}\left( \left( 4S_g - (n-2)\sigma _1(S_g)g \right) (\nabla u,\cdot )\right) + \frac{n-4}{2}Q_gu, \end{aligned}$$

(B.2)

for all $u\in C^{\infty }(M)$. In this context, the following relations hold:

$$\begin{aligned} \begin{aligned} \sigma _1(S_g)&=\frac{R_g}{2(n-1)},\\ \sigma _2(S_g)&=\frac{1}{2}\left( \frac{n^2-n}{4(n-1)^2(n-2)^2}R^2_g - \frac{|\mathrm {Ric}_g|^2_{g}}{(n-2)^2} \right) , \end{aligned} \end{aligned}$$

(B.3)

which implies that

$$\begin{aligned} \begin{aligned} Q_g&=-\frac{1}{2(n-1)}\Delta _gR_g - \frac{2}{(n-2)^2}|\mathrm {Ric}_g|^2_g + \frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}R_g^2,\\ P_gu&=\Delta ^2u + \mathrm {div}_g\left( \left( \frac{4}{n-2}\mathrm {Ric}_g - \frac{n^2-4n+8}{2(n-1)(n-2)}R_g\, g \right) (\nabla u,\cdot ) \right) + \frac{n-4}{2}Q_gu. \end{aligned} \end{aligned}$$

(B.4)

In particular, for $n\ne 4$, if ${\bar{g}}=u^{\frac{4}{n-4}}g$, then

$$\begin{aligned} Q_{{\bar{g}}}=\frac{2}{n-4}u^{-\frac{n+4}{n-4}}P_gu. \end{aligned}$$

(B.5)

In the case of $n=4$ we can apply the above definitions to get

$$\begin{aligned} \begin{aligned} Q_g&=-\frac{1}{6}\Delta _gR_g - \frac{1}{2}|\mathrm {Ric}_g|^2_g + \frac{1}{6}R_g^2,\\ P_gu&=\Delta ^2_gu + \mathrm {div}_g\left( \left( 2\mathrm {Ric}_g - \frac{2}{3}R_g\,g \right) (\nabla u,\cdot )\right) , \end{aligned} \end{aligned}$$

(B.6)

and in this case, if ${\bar{g}}=e^{2u}g$, we have that [42]

$$\begin{aligned} Q_{{\bar{g}}}=e^{-4u}\left( P_gu+Q_g \right) . \end{aligned}$$

(B.7)

Now, let us notice that it is also quite standard to redefine the Q-curvature in dimension 4 via (see [29, 41])

$$\begin{aligned} Q^{(4)}_g=\frac{1}{2}Q_g=-\frac{1}{12}\Delta _gR_g - \frac{1}{4}|\mathrm {Ric}_g|^2_g + \frac{1}{12}R_g^2 \end{aligned}$$

(B.8)

and in this case, it follows that

$$\begin{aligned} 2Q^{(4)}_{{\bar{g}}}=e^{-4u}\left( P_gu+2Q^{(4)}_g \right) . \end{aligned}$$

(B.9)

We will not adopt this redefinitions and keep a unified notation via (B.1) along this paper.

Appendix: Conformal normal coordinates

In order to deliver a presentation as self-cointained as possible, this section is meant to summarise some of the results concerning conformal normal coordinates presented in [11], which are used in the core of this paper. The basic construction is given on a smooth Riemannian manifold $(M^n,g)$ where we intend to expand g around a fixed point $p\in M$. In particular, we are interested in finding an element ${\tilde{g}}$ within the conformal class [g] for which, in ${\tilde{g}}$-normal coordinates $\{x^i\}$, the following expansion holds around p:

$$\begin{aligned} \det ({\tilde{g}})=1+O(r^{N}) \end{aligned}$$

(C.1)

for any chosen $N\ge 2$, where $r=|x|$ (see Theorem 5.1 in [11]). This type of expansion is achieved by first noticing that, for any Riemannian metric g, in g-normal coordinates $\{x^i \}_{i=1}^n$ around $p\in M$, the following holds

$$\begin{aligned} g_{ij}(x)&=\delta _{ij}+\frac{1}{3}R_{iklj}x^kx^l + \frac{1}{6}R_{iklj,a}x^kx^lx^a\\&\quad +\left( \frac{1}{20}R_{iklj,ab} + \frac{2}{45}R_{iklc}R_{jabc} \right) x^kx^lx^ax^b + O(r^{5}), \end{aligned}$$

where all the coefficients are evaluated at p. From this expression it is possible to compute $\det (g)$ around p for any such metric, so as to get

$$\begin{aligned} \begin{aligned} \det (g)(x)&=1-\frac{1}{3}R_{ij}x^ix^j - \frac{1}{6}R_{ij,k}x^ix^jx^k \\&\quad - \left( \frac{1}{20}R_{ij,kl} + \frac{1}{90}R_{aijb}R_{aklb} -\frac{1}{18}R_{ij}R_{kl} \right) x^ix^jx^kx^l + O(r^5), \end{aligned} \end{aligned}$$

(C.2)

where again all the coefficients are evaluated at p. Assuming an expanssion of the form $\det (g)=1+O(r^N)$ with $N\ge 2$ (the case $N=2$ is valid for any metric in its own normal coordinates) and appelaing to Theorem 5.2 in [11], the authors can find a homogeneous polynomial $f\in {\mathcal {P}}_N$ so that the expansion of $\det ({\tilde{g}})=1+O(r^{N+1})$ for ${\tilde{g}}=e^{2f}g$, which establishes an inductive proof (see the proof of Theorem 5.1). But notice that this implies that the symmetrization of the coefficients in (C.2) of order up to N must vanish for ${\tilde{g}}$. Thus, in the case we do this construction for $N\ge 4$, we see that this implies

$$\begin{aligned} \begin{aligned}&{\tilde{R}}_{ij}(p)=0,\\&{\tilde{R}}_{ij,k} + {\tilde{R}}_{ki,j} + {\tilde{R}}_{jk,i}(p)=0. \end{aligned} \end{aligned}$$

(C.3)

Putting together the second condition above with the contracted Bianchi identities, we also see that

$$\begin{aligned} {\tilde{R}}_{,k}(p)=0. \end{aligned}$$

(C.4)