Scattering on singular Yamabe spaces

Scattering theory on asymptotically hyperbolic manifolds has been studied with great profit in the case that the metric is Einstein. In this paper we study the scattering problem in the case that the metric has constant scalar curvature. For every compact Riemannian manifold $(X^{n+1},\bar{g})$ with boundary $M$, it is known that there is precisely one defining function $u$ for the boundary so that the singular Yamabe metric $g=u^{-2}\bar{g}$ has constant scalar curvature, so this study can be seen as bringing scattering theory to bear as a tool in the study of the conformal geometry of compact manifolds with boundary. We apply the methods of [GZ03, FG02, CQY08] and others to the setting of [GW15, Gra17], which used the singular Yamabe problem to study conformal hypersurfaces.

Given an asymptotically hyperbolic (AH) manifold $(X^{n+1},g)$ with boundary $M^{n}=\partial X$, the scattering problem is defined as follows. Let $\bar{g}=r^{2}g$ be a geodesic compactification of $g$, i.e., suppose that $|dr|_{\bar{g}}=1$ on a neighborhood of $M$. Let $f\in C^{\infty}(M)$. Consider the equation

where we assume that $s>\frac{n}{2}$ and that $s(s-n)\notin\sigma_{pp}(\Delta_{g})$ (the set of $L^{2}$ eigenvalues of the Laplacian); in our convention, $\Delta_{g}$ is a negative operator. It is shown in [MM87] that a solution $v$ to this equation has asymptotics

where $F,G\in C^{\infty}(X)$, at least so long as $2s-n\notin\mathbb{Z}$. In [GZ03], it is shown that we can always solve (1.1) with $F|_{M}=f$. The scattering operator $S(s):C^{\infty}(M)\to C^{\infty}(M)$ is then defined by $S(s)f=G|_{M}$, and this operator extends to be meromorphic on $\operatorname{Re}s\geq\frac{n}{2}$, with poles only at $s$ such that $s(n-s)$ is an eigenvalue of $\Delta_{g}$ or at $s=\frac{n+q}{2}$, $q\in\mathbb{N}$.

The case of $g$ Einstein has been thoroughly studied. In that case, the poles for $s=\frac{n+q}{2}$ with $q$ odd actually do not exist – $S(s)$ is regular at these points – while at $s=\frac{n}{2}+j$, the operator $S(s)$ has a simple pole whose residue is the so-called GJMS operator $P_{2j}$ (with $j\leq n/2$ if the boundary dimension $n$ is even). If $n$ is even, $S(s)1$ is holomorphic across $s=n$, and in fact $S(n)1=c_{n}Q_{n}$, where $Q_{n}$ is the $n$th-order $Q$-curvature and $c_{n}$ is a universal constant. On the other hand, if $n$ is odd, then $S(n)1=0$, which reflects the fact that there is no (locally-defined) $Q$-curvature of odd order. However, a nonlocal $Q$-like term was defined in [FG02] by taking $Q_{n}=k_{n}\frac{d}{ds}|_{s=n}S(s)1$ in case $n$ is odd. In addition to these results, there have been numerous interesting results linking the scattering operator with the renormalized volume of an AH Einstein manifold.

We recall the definition of the renormalized volume in that setting. Let $r$ be a geodesic defining function for $M$, and consider the expansion

The quantity $V_{+}$ is called the renormalized volume. While a priori it depends on the choice of geodesic defining function $r$, it is shown in [HS98, Gra00] that, if $n$ is odd, then $\mathcal{E}=0$ and $V_{+}$ is in fact well-defined independent of $r$. If the boundary dimension $n$ is even, on the other hand, then $\mathcal{E}$ is well-defined independent of $r$. Using scattering theory, it was shown in [GZ03, FG02] that in the even case, $\mathcal{E}=\int_{M}Q_{n}dv_{k}$ (where $k=\bar{g}|_{TM}$). In [CQY08], a formula was given relating the renormalized volume in the even-$n$ case to an integral over $M$ of a non-local quantity defined in terms of the scattering operator.

It was shown already in [Gra17] that the renormalized volume can be defined as well in the singular Yamabe case, where $r$ now is the $\bar{g}$-distance to the boundary and $g$ is the corresponding singular Yamabe metric; but in this case, $\mathcal{E}$ is generically nonvanishing in all dimensions and is a conformal invariant, while $V_{+}$ is no longer invariant. In fact, in the case $n=2$, $\mathcal{E}$ is simply a linear combination of the Willmore energy and the Euler characteristic. Thus, in case $n>2$, $\mathcal{E}$ can be seen as a generalization of the Willmore energy.

We generalize the scattering picture to the singular Yamabe setting, and obtain extrinsic analogues to all of the above results. Unlike the Einstein setting, where the existence and uniqueness of Einstein metrics remain extremely challenging and one is often constrained in practice to use a non-unique formal expansion, the analytic picture in the singular Yamabe case is well understood and perhaps optimal: given any compact Riemannian manifold $(X^{n+1},\bar{g})$ with boundary, there is a unique defining function $u$ for the boundary so that the AH metric $g=u^{-2}\bar{g}$ has constant scalar curvature $-n(n+1)$. Any global quantities defined in terms of this singular Yamabe metric are thus well defined for $(X,\bar{g})$. This comes with a price, however: the power series expansion of $u$ at the boundary $M$ is not smooth beyond order $n+1$, after which $\log$ terms appear. Throughout, therefore, we must keep careful track of the expansion of $u$ and its regularity in a way that has not had to be done in the Einstein case, where smooth formally Einstein metrics have largely been considered.

Also unlike AH Einstein spaces, singular Yamabe metrics have no particular parity properties, and this has numerous consequences: in general, this theory tends to behave more like the even-dimensional-boundary Einstein theory in all dimensions than like the odd-dimensional Einstein theory, whose special behaviors are entirely a product of parity considerations. In some cases, though, even features of the even-dimensional Einstein case are not reproduced. A first consequence, for example, is that $S(s)$ generically has poles at $s=\frac{n+q}{2}$ for all $q$ (not only even $q$). Therefore, we get residue operators $P_{q}^{SY}=\operatorname{res}_{s=n/2+q/2}S(s)$ for all $q\leq n$. Of course, since Einstein metrics are the singular Yamabe metrics in their conformal class, our results must in all case reduce to the usual ones in that special case. Similarly, for all $n$ – not only even – we can define $Q^{SY}_{n}=c_{n}^{-1}S(n)1$. Here, $Q^{SY}_{n}$ is a locally determined curvature quantity along $M$, which however depends on $\bar{g}$ and not merely $\bar{g}|_{TM}$. Similarly, for each $q$, $P_{q}^{SY}$ is a conformally covariant differential operator along $M$ whose coefficients depend on extrinsic data.

We now state our main results.

Several remarks are in order regarding this statement.

First, note that in the Einstein case considered by [GZ03], the integer $q$ is assumed to be even, since otherwise $P_{q}^{SY}$ vanishes identically in that setting. Moreover, the indexing of our operators $P_{q}^{SY}$ differs from the indexing in that paper by a factor of two – what we call $P_{2}$ was there $P_{1}$, etc. This is because in the other case, what are here the odd-order operators vanish. Already in the literature, there is some difference in the numbering of these operators. The operators defined in [GZ03], of course (unlike those here) are the GJMS operators. The operators defined here differ from the GJMS operators by terms depending on the extrinsic geometry of $M$ in $X$.

Next, since in the Einstein case, the odd-order operators vanish, the notation $P_{q}$ for $q$ odd has there developed another meaning that, unfortunately, is not analogous to that used here: see [FG02]. Given the strong analogy between the even-order operators and those in the Einstein case, and the equivalence of definition here of the even- and odd-order operators, it seems there is no very good way around having confusing notation. Observe also that the odd-order operators defined in those settings do not exist in this setting, since their definition depends on the vanishing of the residue at $s=\frac{n+q}{2}$. A similar remark applies to the definition, below, of $Q_{n}^{SY}$ for odd $n$; again, see [FG02].

As discussed more fully in section 4, the restriction on $s$ in the statement of the meromorphicity of the scattering operator is due to the non-regularity at order $n+2$ of the singular Yamabe solution $u$. In fact, stronger statements could be made, but the nonregularity makes it somewhat delicate to discuss and define just what this means. Since we are not interested behavior at higher $s$ in any event, we give the weaker statement.

Finally, in work such as [CdMG11], it is common to use the notation $s=\frac{n}{2}+\gamma$. The relationship between $q$ and $\gamma$, then, is $q=2\gamma$.

This follows from their definition in terms of the scattering operator.

We now define the $Q_{n}^{SY}$-curvature, as in [GZ03], by $Q_{n}^{SY}=c_{n}^{-1}S(n)1$. This is well-defined since $\mathbb{R}\subseteq\ker P_{n}^{SY}$. This extrinsic $Q$-curvature quantity follows a conformal transformation law like its intrinsic cousin (which however exists only for even order) and, like it, gives the conformally invariant term in the volume expansion:

A corollary then follows immediately from Theorem 3.1 of [Gra17]. To state it, we must say slightly more about the singular Yamabe function. Given $\bar{g}$, there is a unique defining function $u$ for $M$ such that $g=u^{-2}\bar{g}$ has scalar curvature $-n(n+1)$. As discussed above and in [Gra17], $u$ has the expansion (in terms of the distance function $r$ from the boundary $M$ in $X$)

where $\mathcal{L}$ is a locally and extrinsically defined function on $M$, conformally invariant of weight $-(n+1)$. With this notation fixed, we have the following:

The results so far stated about $P_{q}^{SY}$ and $Q_{n}^{SY}$ have actually appeared previously in the recent literature. They were derived in a somewhat different framework by Gover and Waldron ([GW14, GW15, GW17]), where they are defined in terms of the tractor calculus and so-called Laplace-Robin operators. When the current paper was quite advanced in preparation, the paper [JO21] of Juhl and Orsted was brought to our attention. Among other things, it reinterprets the results of Gover and Waldron in the setting of scattering theory shared by this paper. Our contribution with respect to this part of our material is thus a different treatment, more similar in spirit to [GZ03, FG02, CQY08]. On the other hand, our perspective here is also focused more than those papers on the unique global scattering operator associated to every compact Riemmanian manifold with boundary. In particular, rather than doing only asymptotic analysis at the boundary and thus neglecting the impact of the non-regularity of the singular Yamabe solution, we take account of the logarithmic term. The following results, more global in spirit, are new to this paper.

First, as stated above, $S(s)1$ is smooth across $s=n$. We now define

This is a function dependent on the global geometry of $(X,\bar{g})$. Next, it follows from [GZ03] that for $s\in\mathbb{C}$ near 1, there exists $u_{s}=r^{n-s}F_{s}+r^{s}G_{s}$ satisfying $(\Delta_{g}+s(n-s))u_{s}=0$ and $F_{s}|_{M}=1$. Moreover, $F_{s}$ and $G_{s}$ may be uniquely determined by the requirement that they be holomorphic in $s$. We then write

where each $a_{j}$ is a smooth function on $M$, and $a_{j}$ is locally (and extrinsically) determined for $1\leq j\leq n$. Finally, we recall the definition in [Gra17] (see also [Gra00]) of the renormalized volume coefficients. Since $g=u^{-2}\bar{g}$ for a defining function $u$ for the boundary, we may write

on a collar neighborhood $[0,\varepsilon)_{r}\times M$ near $M$ in $X$. The functions $v^{(j)}\in C^{\infty}(M)$ are the renormalized volume coefficients. With these notations in hand, we may state our next result, which is analogous to a theorem proved in [CQY08] (based on [FG02]) in the Einstein setting.

All the terms on the right-hand side except the first are local.

It was shown in [FG02] that, if $g$ is in fact Einstein and $n$ is odd, then this formula holds with only the first term on the right-hand side present. Although, in such a case, $g$ is also the singular Yamabe metric and thus (1.5) applies, the results are not inconsistent, because in that case, the $v^{(k)}$ with $k$ odd and the $a_{k}$ with $k$ odd are both identically zero, and one of these is a factor in each of the integrands multiplied above by $\frac{1}{n}$ when $n$ is odd.

The following theorem, which was proved in [CQY08] in the the special case of even-dimensional Einstein metrics, states (in all dimensions) that the scattering term in (1.5) is a conformal primitive for total $Q^{SY}$.

Finally, for $X$ of dimension four, we apply Theorem E and the main result of [GG19] to obtain a Gauss-Bonnet theorem in terms of the scattering operator.

The following corollary also follows from Corollary 1.4 of that paper, or from the previous result.

We point out also that having a uniquely defined scattering operator, as stated in Theorem A, allows one to define unique fractional extrinsic GJMS operators $P_{2\gamma}^{SY}=S(\frac{n}{2}+\gamma)$ of order $2\gamma$ as well, as in [CdMG11]. In the intrinsic case, such operators are unique only when a global Einstein metric can be found. As there, these operators are nonlocal.

The paper is organized as follows. In section 2 we review the background necessary for the paper, and also introduce geodesic coordinates for the singular Yamabe setting. These are a useful tool for studying the singular Yamabe metric, and in particular performing computations. In section 3, we develop the existence theory for the class of singular Yamabe GJMS operators of integral order, which entails formal analysis of the scattering operator at the boundary. The analysis is a variation of that carried out in section four of [GZ03]. The result is summarized in Theorem 3.1. In section 4, we turn to the global existence of the scattering operator and the results that follow from it, including Theorems A - F and their corollaries. In section 5, we perform specific computations in low dimensions. This is done both to illuminate and illustrate the theory, and to demonstrate the usefulness of geodesic normal coordinates for carrying out computations with singular Yamabe metrics. We also there prove Theorem G.

Appendix A contains more thorough discussion of some analytic ramifications of the limited regularity of the singular Yamabe metric.

Acknowledgements The authors found an error in their formula for $c_{k}$ in section 3 of an earlier draft thanks to an explicit computation in [JO21], and have corrected it here. Stephen McKeown carried out part of the work while a postdoctoral research associate at Princeton University, whom he thanks for the hospitality and support.

In this section we analyze formal solutions to the equation $(\Delta_{g}+s(n-s))u=0$ for a singular Yamabe metric $g$. Let $\bar{g}$ be a smooth metric on $X^{n+1}$, and $u$ the solution to the singular Yamabe equation (2.1), so that $g=u^{-2}\bar{g}$ has constant scalar curvature $-n(n+1)$. We again let $M=\partial X$. Near $M$, we write $\bar{g}=dr^{2}+h_{r}$, where $h_{r}$ is a one-parameter family of metrics on $M$ and $r$ is the $\bar{g}$-distance to $M$. We write $k=h_{0}=\bar{g}|_{TM}$.

The following result, which is the primary result of this section, is directly analogous to Proposition 4.2 of [GZ03] for the Einstein case, and the proof is modified accordingly. One difference is that a log term arises here for every integer, whereas in the Einstein cases it arises only for the even integers. The other significant difference is that the metric itself is non-smooth, and via $u$ has logarithmic terms that must be considered.