# The Higher-Dimensional Chern–Gauss–Bonnet Formula for Singular Conformally Flat Manifolds

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## Abstract

In a previous article, we generalised the classical four-dimensional Chern–Gauss–Bonnet formula to a class of manifolds with finitely many conformally flat ends and singular points, in particular obtaining the first such formula in a dimension higher than two which allows the underlying manifold to have isolated conical singularities. In the present article, we extend this result to all even dimensions \(n\ge 4\) in the case of a class of conformally flat manifolds.

## Keywords

Chern–Gauss–Bonnet Conical singularities Conformal metrics Integral estimates*Q*-curvature

## Mathematics Subject Classification

Primary 53A30 35J30 Secondary 58J05## 1 Introduction

*Q*-curvature introduced by Branson and Branson-Ørsted [2, 3, 4]. Here, \(\mathrm {R}_g\) denotes the scalar and \(\mathrm {Rc}_g\) the Ricci curvature of \((M^4,g)\). As in the two-dimensional case, also the four-dimensional formula (1.1) requires correction terms if the smoothness or compactness assumptions are dropped. The most basic situation where this can be observed is for a conformal metric on \(\mathbb {R}^4\setminus \{0\}\) with one end (at infinity) and one singular point (at the origin). For such metrics, we proved the following result in [5].

### Theorem 1.1

*g*has finite total

*Q*-curvature, \(\int _{\mathbb {R}^4}|Q_{g,4}|\, \mathrm{{d}}V_g<\infty \), and non-negative scalar curvature at infinity and at the origin, then we have

*N*,

*g*) is a compact manifold with boundary \(\partial N=\big (\bigcup _{i=1}^k \partial E_i\big )\cup \big (\bigcup _{j=1}^{\ell } \partial S_j\big )\), each \(E_i\) is a conformally flat complete simple end satisfying

*B*denotes the unit ball in \(\mathbb {R}^4\). Localising Theorem 1.1 to such ends and singular regions (obtaining Chern–Gauss–Bonnet formulas with boundary terms), and gluing all the pieces together, we obtained the following more general theorem.

### Theorem 1.2

*g*has finite total

*Q*-curvature, \(\int _{M}|Q_{g,4}|\, \mathrm{{d}}V_g<\infty \), and non-negative scalar curvature at every singular point and at infinity at each end. Then we have

The Chern–Gauss–Bonnet formulas in the above two theorems, generalising in particular the formulas of Chang et al. [6, 7] for smooth but non-compact four-manifolds, are the first such formulas in a dimension higher than two which allow the underlying manifold to have isolated branch points or conical singularities. It is natural to ask whether Theorems 1.1 and 1.2 can be generalised to higher even dimensions \(n = 2m \ge 4\) using the *n*-dimensional *Q*-curvature. In the present article, we give an affirmative answer in the case of Theorem 1.1 and prove an analogue of Theorem 1.2 for a class of conformally flat manifolds.

*Q*-curvature are not unique for general manifolds but for a conformally flat metric \(g=e^{2w}|\text {d}x|^2\), the

*n*-dimensional

*Q*-curvature is uniquely determined. We define the

*n*-dimensional

*Q*-curvature by the formula

*n*-th order partial differential equation (while it would yield a pseudo-differential equation involving a fractional Laplacian in odd dimensions). Let us remark here that an explicit formula for \(Q_{g,n}\) in terms of the Riemann curvature tensor and its covariant derivatives, similar to (1.2) in the four-dimensional case, is difficult to obtain in higher dimensions and is currently unknown for dimensions higher than 8. On the other hand, a second advantage of restricting to conformally flat manifolds is that in this case the

*Q*-curvature is a multiple of the Pfaffian modulo a divergence term. Thus the Chern–Gauss–Bonnet theorem can be written as

*n*-manifold \((M^n,g)\) with

### Theorem 1.3

*g*has finite total

*Q*-curvature, \(\int _{\mathbb {R}^n}|Q_{g,n}|\, \mathrm{{d}}V_g<\infty \), and non-negative scalar curvature at infinity and at the origin, then we have

### Remark 1.4

Using a partition of unity argument, we then obtain the following more general theorem.

### Theorem 1.5

*Q*-curvature and non-negative scalar curvature and each \(q _ j\) is a finite area singular point of finite total

*Q*-curvature and non-negative scalar curvature. Then

*R*is chosen small enough such that the balls \(\{B_R(p)\}_{p\in \Lambda }\) are pairwise disjoint.

*smooth*locally conformally flat

*n*-manifolds with finitely many ends where the scalar curvature is non-negative. Our Theorems 1.3 and 1.5 do not only generalise this result to manifolds with singularities, but we also get an explicit formula for the error terms in this inequality. Note that for singular manifolds the inequality (1.14) might no longer hold if the conical angle (and hence the isoperimetric ratio) is larger than the Euclidean one. Later, Ndiaye and Xiao [16] obtained a result similar to Theorem 1.3, but again only for

*smooth*metrics. They did not consider the case of manifolds with several ends but only studied conformal metrics on \(\mathbb {R}^n\). Let us remark that their work also contains some small errors which we correct here, in particular in their main theorem they only assume (1.13) and not (1.12), which in view of the counterexample from the above remark is not a sufficiently strong assumption. Nevertheless, we clearly profited from their results and, in fact, some of the ideas in the present article are in parts inspired by [11, 16]—combined with the approach we developed in [5] in order to deal with isolated singularities. Let us also mention here that various normalisations of the

*Q*-curvature exist in the literature and that, in particular, [11, 16] define \(Q_{g,n}\) without the factor 2 in (1.8). We prefer to put this factor to make the results consistent with formula (1.2) and with our earlier work [5] in dimension four.

In the *singular* case, the problem has previously only been studied for manifolds with edge-cone singularities and *V*-manifolds, see e.g. [1, 15, 18], but these results are technically very different and none of them allows for isolated singular points.

Let us now describe how this article is organised and how the arguments differ from our previous four-dimensional results in [5]. First, in Sect. 2, we collect some integral estimates which are needed for the arguments in the following sections. Then, in Sects. 3–5, we prove Theorems 1.3 in three steps as follows. In Sect. 3, we prove the result in the special case where \(w=w(r)\) is a radial function on \(\mathbb {R}^n\setminus \{0\}\) and (1.8) reduces to an ODE. To make it easier to deal with all dimensions at once, we do not solve explicitly for the non-linearity in this ODE as we did in [5] but rather prove asymptotic estimates for an abstract kernel (see Lemma 3.1), employing the integral estimates from Sect. 2. In Sect. 4, we introduce an *n*-dimensional version of our notion of generalised normal metrics from [5] and prove Theorem 1.3 for this class of metrics. Then, in Sect. 5, we show that every metric \(g=e^{2w}|\text {d}x|^2\) on \(\mathbb {R}^n\setminus \{0\}\) satisfying the assumptions of Theorem 1.3 is actually such a generalised normal metric. The necessary singularity removal argument in this section is much more involved than our short argument in dimension \(n=4\) from [5], as we cannot directly apply Bôcher’s Theorem in higher dimensions. Finally, in Sect. 6, we prove Theorem 1.5. As currently no good notion of boundary *T*-curvature associated to \(Q_{g,n}\) is known in dimensions \(n>4\), we cannot localise Theorem 1.3 as we did in [5]. However, in the special case of domains \(\Omega \) as in Theorem 1.5, we can deduce the desired result from an easy partition of unity argument.

## 2 Some Integral Estimates

*u*is a radial function solving \((-\triangle )^k u=0\) on \(\mathbb {R}^n\setminus \{0\}\), where

*k*is an integer satisfying \(1\le k\le \frac{n}{2}\), then

*u*is given by

*r*in Euclidean \(\mathbb {R}^n\) and consider the following four integrals:

### Proposition 2.1

- (i)For any \(r,s > 0\) and even integer \(n\ge 4\), \(I_n(r,s)\) evaluates to$$\begin{aligned} I_n(r,s) = {\left\{ \begin{array}{ll} \frac{1}{r^{n-2}}, &{} \text { if } s\le r,\\ \frac{1}{s^{n-2}}, &{} \text { if } s> r. \end{array}\right. } \end{aligned}$$(2.9)
- (ii)There exists \(C>0\) such that for any \(r,s > 0\) and even integer \(n\ge 4\), we haveIn particular, in both cases, we obtain$$\begin{aligned} \begin{aligned} |r^2 J_n(r,s)-1|&\le C \frac{s^2}{r^2}, \quad \text { if } s \le r,\\ J_n(r,s)&\le C\frac{1}{s^2}, \quad \text { if } s>r. \end{aligned} \end{aligned}$$(2.10)$$\begin{aligned} r^2 J_n(r,s) \le C. \end{aligned}$$(2.11)
- (iii)There exists \(C>0\) such that for any \(r,s > 0\) and even integer \(n\ge 4\), we have$$\begin{aligned} K_n(r,s)\le C. \end{aligned}$$(2.12)
- (iv)There exists \(C>0\) such that for any \(r,s > 0\) satisfying \(\frac{1}{2}r\le s\le \frac{3}{2}r\) and any even integer \(n\ge 4\), we have$$\begin{aligned} |L_n(r,s)| \le C. \end{aligned}$$(2.13)

### Proof

*r*, we see that there is a constants \(C_1(s)\) such thatFixing

*y*(and hence \(|y| = s\)) and letting \(r\rightarrow \infty \), we see that \(C_1(s) = 0\). This proves (2.9) in the first case. If instead we have \(s>r\), (2.14) impliesso thatAs \(s > r\), we can fix \(s=|y|\) and let \(r \rightarrow 0\) this time to obtainThis proves (2.9) in the second case.

*n*is an even integer satisfying \(n\ge 6\). In the case \(s\le r\), we computeWe therefore obtain

*p*is a polynomial of degree \(\frac{n}{2}-2\) with no constant term. Obviously, this is bounded when \(s \le r\), proving the first case of (2.10). If \(s > r\), we simply apply Hölder’s inequality with \(p=\frac{n}{2}-1\), yieldingThis proves the second case of (2.10).

## 3 The Rotationally Symmetric Case

In this section, we prove Theorem 1.3 for a conformal metric \(g = e^{2w}|\text {d}x|^2\) on \(\mathbb {R}^n\setminus \{0\}\) when \(w=w(r)\) is a *radial* function. Here and in the following, we always use the notation \(r=|x|\). In this situation, (1.8) becomes an ODE, but instead of solving for the non-linearity in this ODE as we did in our previous four-dimensional work [5] by an explicit integration, we rather apply the integral estimates derived in the previous section. We first prove the following lemma.

### Lemma 3.1

*Q*-curvature

### Proof

*w*and \(\log |x|\) are both radially symmetric. As in Sect. 3 of [5], we have

*w*, we obtainNow, we get thatwhere \(K_n(r,s)\) is defined in (2.7) and we used Proposition 2.1(iii) as well as the assumption of finite total

*Q*-curvature (3.1) in the last steps. We can therefore apply the dominated convergence theorem, implying

*y*we have

*w*and hence of \(Q_{g,n}\). Using \(J_n(r,s)\) defined in (2.6) and estimated in (2.11), we then easily getTherefore \(|x|^2|\triangle f_\alpha (x)|\) is uniformly bounded, thus finishing the proof of the lemma. \(\square \)

Next, we prove that *w* agrees with some \(f_\alpha \) up to a constant.

### Lemma 3.2

*w*be as in Lemma 3.1 and assume in addition either that

### Proof

*Q*-curvature in (1.8), we obtain \((-\triangle )^{n/2}(w-f) = 0\). As \((w-f)\) is a radial function, we can use (2.3) to conclude that

### Remark 3.3

The proof of Lemma 3.2 shows that the scalar curvature assumption (3.6) can be replaced by the (much less geometric) condition that \(|x|^2 \mathrm {R}_g e^{2w}\) is bounded from below. Note that the metric \(g=e^{2|x|^2}|\text {d}x|^2\) from the counterexample stated in Remark 1.4 satisfies \(\mathrm {R}_g\rightarrow 0\) as \(|x|\rightarrow \infty \), but has \(|x|^2 \mathrm {R}_g e^{2w} \rightarrow -\infty \) as \(|x|\rightarrow \infty \). Hence, (the first part of) condition (3.6) cannot be weakened to \(\liminf _{|x|\rightarrow \infty } \mathrm {R}_g(x) \ge 0\) as claimed in [16]. A similar argument shows that also the second part of condition (3.6) cannot be weakened to \(\liminf _{|x|\rightarrow 0} \mathrm {R}_g(x) \ge 0\).

Combining Lemma 3.2 with Eq. (3.2), we immediately obtain the following consequence, which corresponds to [5, Corollary 2.3] in the four-dimensional case.

### Corollary 3.4

To finish the proof of Theorem 1.3 in the rotationally symmetric case, it remains to express the two limits in Corollary 3.4 as isoperimetric ratios. Let us first recall the mixed volumes \(V_k(\Omega )\) defined by Trudinger [20] for a convex domain \(\Omega \subseteq \mathbb {R}^n\). We can restrict to the special situation where \(\Omega \) is a ball \(B_r(0)\) and \(k=n\) or \(k=n-1\), in which case we obtain the following.

### Definition 3.5

*k*-volumes \(V_k(r)\) for \(k=n\) or \(k=n-1\) by

Theorem 1.3 for rotationally symmetric metrics then follows from Corollary 3.4 combined with the following result.

### Lemma 3.6

*w*be as in Lemma 3.2 and assume in addition that

*g*is complete at infinity and has finite area over the origin. Then we have

### Proof

*w*(

*x*), we can rewrite the

*k*-volumes defined in (3.11) as

*r*by (3.14). In the case where \(\lim _{r\rightarrow \infty } r\frac{\mathrm{d}w}{\mathrm{d}r}(r) + 1> 0\), (3.13)–(3.14) imply that both \(V_n(r)\) and \(V_{n-1}(r)\) tend to infinity and therefore we can again apply L’Hôpital’s rule as above, obtaining

We have checked all possible cases and thus finished the proof of Lemma 3.6 and in view of Corollary 3.4 also proved Theorem 1.3 for rotationally symmetric metrics.\(\square \)

In the remainder of this section, we study the case where the origin is a second complete end rather than a finite area singular point. This result will be used in Sect. 6.

### Lemma 3.7

*Q*-curvature \(\int _{\mathbb {R}^n} |Q_{g,n}| \mathrm{{d}}V_g < \infty \). Then

This lemma in particular has the consequence that no such metric *g* can have positive *Q*-curvature everywhere. The proof is almost identical to the above, and we therefore only give a short sketch.

### Proof

## 4 Generalised Normal Metrics

In this section, we first define generalised normal metrics on \(\mathbb {R}^n\) as an extension of our definition of generalised normal metrics in \(\mathbb {R}^4\) in [5]. This in turn was a generalisation of normal metrics in [6] and [12]. We then prove Theorem 1.3 for this class of metrics.

### Definition 4.1

*Q*-curvature

*g*a

*generalised normal metric*, if

*w*has the expansion

*averaged metric*\(\bar{g}=e^{2\bar{w}}|\text {d}x|^2\) by

### Proposition 4.2

### Proof

The proof of this statement is merely a modification of our four-dimensional version from [5, Lemma 3.4]. The proof for \(r\rightarrow \infty \) was essentially covered in [6, Lemma 3.2] in the four-dimensional case and later generalised in [16, Prop. 3.1 (ii)] to higher dimensions. Note that in [6, 16] the formula (4.3) is proved for *normal metrics* which differ from our definition of *generalised* normal metrics by our additional term \(\alpha \log |x|\) in (4.1). But this additional term, the fundamental solution of the \(\frac{n}{2}\)-Laplacian, is rotationally symmetric and thus in Eq. (4.3), \(e^{\alpha \log |x|}\) appears on both sides and hence cancels. For this reason, we only need to prove the proposition for \(r\rightarrow 0\).

### Claim 1

### Proof

### Claim 2

### Proof

*Q*-curvature and the dominated convergence theorem, (4.8) follows.

*n*-dimensional disc centred at the point \(\frac{y}{r}\) orthogonal to

*y*, in which case we get

Proposition 4.2 then immediately implies the following two corollaries.

### Corollary 4.3

*g*be a generalised normal metric on \( \mathbb {R}^ n \setminus \{ 0 \} \) with averaged metric \( {\bar{g}}\) and define the mixed volumes \( V_k\) (with respect to

*g*) and \( {\bar{V}} _k \) (with respect to \({\bar{g}}\)) as in Definition 3.5. Then we have

### Corollary 4.4

Suppose that the metric \(g = e^{ 2w }|\mathrm{d}x|^2\) on \( \mathbb {R}^n \setminus \{0\} \) is a generalised normal metric that is complete and has a finite area singularity at \( \{ 0 \} \), then \(\bar{g}=e^{2 \bar{w}} |\mathrm{d}x|^2\) is complete and has a finite area singularity over \( \{ 0 \} \).

In order to conclude that Theorem 1.3 holds for \({\bar{w}}\), we need to either show the geometric property that \({\bar{g}}\) has positive scalar curvature at infinity and the origin, as in (3.6), or alternatively verify the analytical assumption (3.7). These latter bounds are easy to verify for the averaged conformal factor of a generalised normal metric. In fact, we prove a slightly more general result here which we can then also use in the next section.

### Lemma 4.5

*Q*-curvature

*v*(

*x*) by

### Proof

*Q*-curvature. Similarly, using (3.4), we deduce thatHence, Eq. (2.12), the assumption of finite total

*Q*-curvature, and the rotational symmetry of \(\bar{v}(r)\) imply

Obviously, if \(g =e^{2w}|\text {d}x|^2\) is a generalised normal metric on \( \mathbb {R}^n\setminus \{0\}\), then \(v=w\) and hence (4.15) gives the desired bounds for \({\bar{w}}\). This allows us to now prove Theorem 1.3 under the assumption that \(g=e^{2w}|\text {d}x|^2\) is a generalised normal metric.

### Proof of Theorem 1.3 for generalised normal metrics

*g*be a generalised normal metric with average metric \({\bar{g}}\) (see Definition 4.1). Corollary 4.4, Lemma 4.5 and the results from the last section show that Theorem 1.3 holds for the rotationally symmetric metric \({\bar{g}}\). Moreover, Corollary 4.3 implies that

*g*and \({\bar{g}}\), respectively. Thus, in order to obtain Theorem 1.3 for the generalised normal metric

*g*, we need to only show that

## 5 Singularity Removal Theorem

Let \(g = e ^{2w} |\text {d}x|^2\) be a metric on \( \mathbb {R}^n \setminus \{0\} \) satisfying the assumptions of Theorem 1.3. In this section, we show then that *g* is a generalised normal metric. Together with the results from Sect. 4, this completes the proof of Theorem 1.3.

### Proposition 5.1

*Q*-curvature

*w*with respect to \( x _0\) byClearly \( {\bar{w}}_{ x_0 } \) is rotationally symmetric with respect to \(x_0\). If \( x_0 = 0\) then we will often write \( {\bar{w}} _{ 0}( x ) = {\bar{w}} (x)\).

### Lemma 5.2

Let \(g = e^{2w} |\mathrm{{d}}x|^2\) be a complete finite area metric on \(\mathbb {R}^n \setminus \{0\}\) with finite total *Q*-curvature and non-negative scalar curvature at infinity and at the origin. Then for \(x_0\) close enough to the origin, the symmetrised metric \(\bar{g}_{x_0} = e ^{2 {\bar{w}}_{x_0}} |\mathrm{{d}}x|^ 2\) has finite total *Q*-curvature and non-negative scalar curvature at infinity and at the origin.

### Proof

*x*sufficiently close to the origin, the integrand on the right-hand side of (5.1) is non-positive and hence \(\mathrm {R}_{{\bar{g}}_{x_0}}(x) \ge 0\) if \(x_0\) and

*x*are sufficiently close to the origin. Moreover, \(\mathrm {R}_{{\bar{g}}_{x_0}}(x) \ge 0\) also holds whenever

*x*is sufficiently large. Furthermore \(Q_{ {\bar{g}}_{x_0}, n }\) is absolutely integrable with respect to \(\text {d}V _ {{\bar{g}}_{x_0}}\). This follows from

*Q*-curvature. \(\square \)

### Proof of Proposition 5.1

*v*be given by (5.2) and recall from the above that

### Claim 1

\(w ( x) - v ( x ) - \alpha \log |x|\) is harmonic on \(\mathbb {R}^n\setminus \{0\}\).

### Proof

*C*. This shows that \(\psi _{x_0}(x)\equiv c_1\). (Of course, in view of (5.3), we have \(c_1 = C_0\).) Let us point out here that we do not need that \(\bar{g}_{x_0} = e^{2 {\bar{w}}_{x_0}} |\text {d}x|^ 2\) has non-negative scalar curvature at the origin, and hence smallness of \(|x_0|\) is not needed.

To complete the proof of the proposition, we show that \(w(x) - v(x) - \alpha \log |x|\) is in fact a constant. This part of the proof is similar to the four-dimensional case [5, Sect. 4].

*v*(

*x*) we get thatThis is due to the fact thatby (2.11), for any \(y\in \mathbb {R}^n\). Furthermore, using

## 6 Multiple Ends and Cone Points

In this section we extend the Chern–Gauss–Bonnet formula to cover the case of a domain conformal to \( \mathbb {S}^n\) with several ends and cone points as in Theorem 1.5.

### Proof of Theorem 1.5

*N*to infinity. We now identify \(\Lambda = \{p_1, \ldots , p_k, q_1, \ldots , q _\ell \} \) with its images in \(\mathbb {R}^n\) under this stereographic projection and interpret the metric

*g*on \(\Omega \) as a metric on \(\mathbb {R}^n \setminus \{p_1,\ldots ,p_k\}\). Hence, there is a function

*w*that is smooth away from \(\Lambda \subset \mathbb {R}^n\) such that

*R*disjoint by choosing

*R*small enough. The function \(\varphi _0(x)\) is given by the condition

*Q*-curvature and is complete at infinity with zero scalar curvature. Fixing some index

*i*, we can assume without loss of generality that \(p_i\) is the origin. We consider the symmetrisationClearly, the arguments from Sects. 4 and 5 go through, and we can therefore argue as in Lemma 3.7 to conclude that

*g*in a neighbourhood of

*N*and therefore can be smoothly extended to all of \(\mathbb {S}^n\), we obtain by (1.8) and (1.9)

## Notes

### Acknowledgements

Parts of this work were carried out during a visit of RB at The University of Queensland in Brisbane as well as two visits of HN at Queen Mary University of London. We would like to thank the two universities for their hospitality. These visits have been financially supported by RB’s EPSRC Grant number EP/M011224/1 and HN’s AK Head Travelling Scholarship from the Australian Academy of Science.

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