1 Introduction↩︎

In this paper, we study complete \(4\)-dimensional negative Einstein manifolds \((M,h)\) with Poincaré–Einstein and cusp ends. We always normalize the metric so that \(\mathop{\mathrm{\mathrm{Ric}}}_h=-3h\). We first introduce the Poincaré–Einstein asymptotic geometry for negative Einstein metrics.

Although the conformal factor is not unique, the conformal class \([g|_{\{0\}\times N}]\) is well-defined, and this is the conformal infinity of the Poincaré–Einstein end. Any such conformal metric \(g\) is called a conformal compactification for the Poincaré–Einstein end. If the Einstein manifold has only Poincaré–Einstein ends, we shall simply say the Einstein manifold is Poincaré–Einstein. Poincaré–Einstein manifolds are asymptotically hyperbolic in the sense that the sectional curvatures approach \(-1\) at infinity.

In the literature, the above Poincaré–Einstein asymptotic is also known as conformally compact Einstein (CCE) or asymptotically hyperbolic Einstein (AHE). The study of local existence, perturbative existence, uniqueness, compactness, and the construction of examples of Poincaré–Einstein metrics has undergone extensive development [1]–[19]. Our recent work gives a non-perturbative construction of Poincaré–Einstein metrics that are conformally Kähler over complex line bundles over Riemann surfaces [20]. For nondegenerate PE \(4\)-manifolds, in the sense that the linearized Einstein operator in Bianchi gauge admits no non-trivial \(L^2\) kernel, there is a well-developed deformation theory [8], [21], [22].

On the other hand, negative Einstein \(4\)-manifolds may exhibit many other types of asymptotic geometry, and at present very little is known about Einstein metrics with such asymptotics. These metrics are of particular interest because they may arise as limits of negative Einstein manifolds under degeneration. The main focus of this paper is on the following asymptotic models.

Here \(\mathcal{H}^3\) denotes the \(3\)-dimensional Heisenberg group \[\mathcal{H}^3= \left\{ \begin{pmatrix} 1 & x & t\\ 0 & 1 & y\\ 0 & 0 & 1 \end{pmatrix} : x,y,t\in\mathbb{R} \right\},\] and \(\widetilde{SL}(2,\mathbb{R})\) is the universal cover of \(SL(2,\mathbb{R})\). The model metrics 1 –3 are Einstein, whereas 4 is only asymptotically Einstein, with \(\mathop{\mathrm{\mathrm{Ric}}}(h_0)+3h_0=O(e^{-2r}).\) The \(r\)-level cross-sections of 1 –4 carry, respectively, \(\mathop{\mathrm{\mathbb{R}}}^3\), \(\mathop{\mathrm{\mathrm{Nil}}}\), \(\mathop{\mathrm{\mathbb{H}}}^2\times \mathop{\mathrm{\mathbb{R}}}\), and \(\widetilde{SL}(2,\mathbb{R})\) geometry.

For brevity, we shall refer to these ends simply as AH cusps, ACH cusps, and \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\)-cusps, respectively. We shall set \(\mathop{\mathrm{\mathscr{N}}}_\ell\) as the nilmanifold that is the \(\mathop{\mathrm{\mathbb{S}}}^1\)-bundle over \(T^2\) of degree \(\ell\). Note that the sign of \(\ell\) depends on the choice of orientation. In particular, \(\mathop{\mathrm{\mathscr{N}}}_0=T^3\). The following three results realize each of these cusp geometries as one end of a complete negative Einstein \(4\)-manifold whose other end is Poincaré–Einstein.

Here, on an oriented 4-manifold, an Einstein metric is called anti-self-dual (ASD) if its self-dual Weyl curvature satisfies \(W^+=0\). There is a previous construction of PE manifolds with intermediate-rank hyperbolic cusps based on perturbing hyperbolic examples [23]. The Einstein metrics we constructed here are actually conformally Kähler with an isometric \(\mathop{\mathrm{\mathbb{S}}}^1\)-action. There is a Kähler metric \(g\) on \(M\) that is conformal to \(h\) and also invariant under the same \(\mathop{\mathrm{\mathbb{S}}}^1\)-action. The Kähler metric \(g\) extends to \(\overline{M}=[0,\infty)\times\mathscr{L}\), which in particular is a conformal compactification for the end \(E_0\). The \(g^\flat\) that appears in the above Theorems 1–3 is precisely the restriction of \(g\) to \(\{0\}\times\mathscr{L}\), while \((\Sigma,g^\sharp)\) is the quotient of \((\mathscr{L},g^\flat)\) by the isometric \(\mathop{\mathrm{\mathbb{S}}}^1\)-action.

It is worth noting that, in Theorems 1–3, the conformal infinity is of negative Yamabe type, except in the hyperbolic cusp case of Theorem 1, where \((T^2,g^\sharp)\) and \((\mathop{\mathrm{\mathscr{N}}}_0,g^\flat)\) are flat. This follows from the results of Witten–Yau [24] and Cai–Galloway [25], or from a direct argument on the conformal infinity.

Next we discuss applications of our construction. Given an Einstein manifold with an AH cusp, there is a well-known generalized Dehn filling procedure that was first proposed by Anderson [26], later refined by Biquard and Bamler [27], [28], which allows us to close up the AH cusp by a so-called toral black hole metric. Note that the black hole metric is also conformally Kähler, and it turns out to be included in our previous construction of conformally Kähler PE manifolds [20]. Previously, generalized Dehn filling has been performed on hyperbolic cusps, while in our situation AH cusps are only asymptotic to hyperbolic cusps. Note that there is also a continuous complex hyperbolic cusp closing procedure [29] based on complex geometry, where a complex hyperbolic cusp is closed by gluing a Tian-Yau gravitational instanton and a neck region constructed via the Calabi ansatz. Performing generalized Dehn filling on our PE manifolds with an AH cusp from Theorem 1, we obtain

The following picture roughly illustrates the convergence of \((M_j,h_j)\) to \((M,h)\). There is a core torus \(T^2\) in the black hole metric, so after filling approximately there is also a core torus in \(M_j\). Taking the pointed Gromov–Hausdorff limit of \((M_j,h_j,p_j)\) we obtain the PE manifold with an AH cusp \((M,h,p)\). If we instead take pointed limits near the core torus, then the sequence of Einstein metrics is collapsing with bounded curvature, while locally after passing to the universal cover and taking a pointed limit we get the universal cover of the black hole metric.

The nontriviality of the above result lies in that the limit PE manifold with an AH cusp is non-hyperbolic and during the degeneration, the conformal infinity is fixed and not locally conformally flat (Lemma 15). Note that previously, generalized Dehn filling produced such degeneration processes but with a hyperbolic metric as the limit. The following question was asked and studied by Anderson [30]. Our Theorem 4 provides a negative answer.

Now we briefly explain our approach. Suppose the negative Einstein manifold \((M,h)\) is non-trivially conformally Kähler, in the sense that there is a complex structure on \(M\) and a Kähler metric \(g=e^{2f}h\) that is conformal to \(h\) with the conformal factor being non-constant. Then associated to the conformally Kähler structure, there is always a Killing field \(\mathcal{K}\) that preserves both \(h\) and \(g\), as observed by [32]–[34]. Assume that \(\mathcal{K}\) induces an isometric free \(\mathop{\mathrm{\mathbb{S}}}^1\)-action on \(M\), then by an ansatz essentially due to LeBrun and Tod [34]–[36], the Einstein equation reduces to the following (twisted-)Toda equation \[\label{eq:twisted-toda-with-k} (e^{v})_{\xi\xi}+\Delta_{\Sigma}{v}-2K_{\Sigma}=-\xi e^{v}\frac{12-6\xi \partial_{\xi}{v}}{12k^3+\xi^3},\tag{5}\] \[\label{eq:W-with-k} W=\frac{12-6\xi \partial_{\xi}{v}}{12+\xi^3/k^3},\tag{6}\] where \(k\in\{1,\infty\}\), \(\Sigma\) is the Kähler reduction, and \(\Delta_{\Sigma}\) is the Laplace operator of the constant curvature metric \(g_\Sigma\) on \(\Sigma\). The constant \(K_\Sigma\in\{-1,0,1\}\) is the curvature of \(g_\Sigma\). Here, the constant \(k\) has geometric meaning that we shall see later, and the Einstein metric \(h\) is anti-self-dual precisely when \(k=\infty\). Once we have a solution to 5 , there is a conformally Kähler negative Einstein metric whenever \(W>0\) \[\label{eq:h-Einstein-introduction} h=\xi^{-2}(Wd\xi^2+W^{-1}\eta^2+We^vg_{\Sigma})\tag{7}\] with \(\eta\) being a connection 1-form for the \(\mathop{\mathrm{\mathbb{S}}}^1\)-bundle, which is determined by \(v\). This approach eventually reduces the construction to boundary value problems for a (twisted-)Toda equation on suitable domains: \[\label{eq:boundary-value-problem-introduction} \left\{\begin{align} &(e^{v})_{\xi\xi}+\Delta_{\Sigma}{v}-2K_\Sigma=-\xi e^{v}\frac{12-6\xi \partial_{\xi}{v}}{12k^3+\xi^3}, \\ &v|_{\xi=0}=\varphi. \end{align}\right.\tag{8}\] Solving this problem on different domains, with different prescribed asymptotic behaviors of the solutions, produces Einstein metrics with corresponding asymptotic behaviors. Here, the boundary value \(\varphi\) corresponds to the metric \(g^\sharp\) that appears in Theorems 2–3 via \(g^\sharp=e^{\varphi}g_{\Sigma}\). For more details on this reduction, see Section 2. This conformally Kähler perspective has been studied by [37]–[39], and more recently by [40]–[42]

Finally, we outline the structure of the paper. In Section 2, we introduce the reduction of the Einstein equations. In Section 3, we study the asymptotics of solutions to the (twisted-)Toda equation on cusp ends and formulate the corresponding boundary value problems. In Section 4, we consider the simplest solutions to the (twisted-)Toda equation. We then solve these boundary value problems in Section 5. Finally, in Section 6, we investigate various degenerations of our Einstein metrics.

Acknowledgment↩︎

The first author would like to thank Claude LeBrun for helpful discussions. The second author thanks Sun-Yung Alice Chang and Paul Yang for many discussions on Poincaré–Einstein metrics. The authors thank Ruobing Zhang for the invitation to a workshop at the Mathematisches Forschungsinstitut Oberwolfach, where this work was initiated. The authors thank Tristan Ozuch for raising the question of whether the cusp asymptotics impose restrictions on the conformal infinity.

We are both grateful to Claude LeBrun for his work on Einstein 4-manifolds, especially on the interplay between Einstein metrics, complex surfaces, and conformally Kähler geometry, which has been an important source of motivation for this work.

Conventions↩︎

• Given a compact Riemann surface \(\Sigma\), we set \(g_{\Sigma}\) as the metric with constant curvature \(K_\Sigma\in\{-1,0,1\}\) on \(\Sigma\). When \(\Sigma=T^2\), we also normalize \(g_{T^2}\) to have volume one.

• We denote a compact Riemann surface of genus \(\mathop{\mathrm{\mathtt{g}}}\geq2\) as \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\).

• We write the real and complex hyperbolic spaces as \((\mathop{\mathrm{\mathbb{RH}}}^4,h_{\mathop{\mathrm{\mathbb{RH}}}^4})\) and \((\mathop{\mathrm{\mathbb{CH}}}^2,h_{\mathop{\mathrm{\mathbb{CH}}}^2})\), normalized to have Einstein constant \(-3\).

• We denote the complex hyperbolic orientation of \(\mathop{\mathrm{\mathbb{CH}}}^2\) as the positive orientation of \(\mathop{\mathrm{\mathbb{CH}}}^2\), and the opposite orientation as the negative orientation.

• We denote by \(\mathop{\mathrm{\mathscr{N}}}_\ell\) the \(\mathop{\mathrm{\mathbb{S}}}^1\)-bundle with degree \(\ell\) over \(T^2\).

2 Background↩︎

In this section, we introduce the reduction of conformally Kähler Einstein metrics to the twisted-Toda equation. For more details on the reduction, see [20].

Let \((M,h)\) be an oriented 4-manifold. With our choice of the orientation, one can decompose the bundle of 2-forms \(\Lambda^2\) into the direct sum \(\Lambda^2=\Lambda^+\oplus\Lambda^-\) of the \((\pm1)\)-eigenspaces of the Hodge star operator. The sections of \(\Lambda^+\) are called self-dual 2-forms and the sections of \(\Lambda^-\) are called anti-self-dual 2-forms. The curvature tensor \(\mathop{\mathrm{\mathrm{Rm}}}_h\) then can be identified as a self-adjoint linear map \(\mathop{\mathrm{\mathrm{Rm}}}_h:\Lambda^+\oplus\Lambda^-\to\Lambda^+\oplus\Lambda^-\). It can also be decomposed into irreducible pieces \[\left(\begin{matrix} W_h^++\frac{s_h}{12}I & \mathop{\mathrm{\mathrm{Ric}}}_h^0\\ \mathop{\mathrm{\mathrm{Ric}}}_h^0 & W_h^-+\frac{s_h}{12}I \end{matrix}\right)\] where \(s_h\) is the scalar curvature, \(\mathop{\mathrm{\mathrm{Ric}}}_h^0=\mathop{\mathrm{\mathrm{Ric}}}_h-\frac{s_h}{4}h\) is the trace-free part of the Ricci curvature, and \(W_h^\pm\) are respectively the self-dual Weyl curvature and anti-self-dual Weyl curvature.

Now we suppose \((M,h)\) is a 4-dimensional Einstein manifold that is non-trivially conformal to a Kähler metric \(g:=\xi^2h\), equipped with the orientation induced by the complex structure. Denote the dual vector fields of a 1-form \(\alpha\) with respect to \(h\) and \(g\) by \(\alpha^{\sharp_h}\) and \(\alpha^{\sharp_g}\), respectively. Then by [33], [36], \((M,h)\) has to be in one of the following two situations.

• Type I: \(W_h^+\equiv0\). That is, the Einstein metric \(h\) is anti-self-dual (ASD). In this situation the Kähler metric \(g\) is Kähler scalar-flat.

• Type II: \(W_h^+\) is nowhere vanishing and has a repeated eigenvalue. In this situation, the Kähler metric \(g\) is Kähler extremal in the sense of Calabi, which means the vector field \((\bar\partial s_{g})^{\sharp_g}\) is a holomorphic vector field.

In both situations, a non-trivial Killing field \(\mathcal{K}\) that preserves \(h\) and \(g\) naturally arises. The Killing field \(\mathcal{K}\) moreover preserves the complex structure. We explain this separately.

• In the Type I case, set the Kähler form of \(g\) as \(\omega\), then the Killing field \(\mathcal{K}\) is \((\delta_h(\xi^{-3}\omega))^{\sharp_h}\), where \(\delta_h\) denotes the divergence of symmetric 2-forms under \(h\) [43].

• In the Type II case, the Killing field \(\mathcal{K}=J_g\nabla^g s_g\), equivalently the imaginary part of \(-2(\bar\partial s_g)^{\sharp_g}\).

The conformal factor \(\xi\) can also be determined explicitly. In the Type I case, the conformal factor \(\xi\) up to scaling is \(|(d (\mathcal{K}^{\sharp_h}))^+|_h^{-1}\), where \(\mathcal{K}^{\sharp_h}\) denotes the 1-form that is dual to \(\mathcal{K}\) under \(h\). In the Type II case, the conformal factor \(\xi\) up to scaling is the scalar curvature \(s_g\). Since \(\mathcal{K}\) is real holomorphic and Killing under \(g\), it is also Hamiltonian; that is, there is a moment map \(\sigma:M\to\mathop{\mathrm{\mathbb{R}}}\) such that \(-d\sigma=\omega(\mathcal{K},\cdot)\). It turns out that up to scaling the conformal factor \(\xi\) is also a moment map for \(\mathcal{K}\). Hence by scaling \(g\), we may suppose the conformal factor \(\xi\) is precisely the moment map, and in the Type II case \(\xi=s_g\), hence \(g=\xi^2h=s_g^2h\). In the Type II case we moreover have \(|s_g|=(2\sqrt{6}|W^+_h|_h)^{1/3}>0\).

For the Kähler metric \((M,g)\) with the Hamiltonian Killing field \(\mathcal{K}\), away from zeros of \(\mathcal{K}\), one can perform Kähler reduction. If we further suppose the Killing field \(\mathcal{K}\) induces a free \(\mathop{\mathrm{\mathbb{S}}}^1\)-action with the Kähler reduction being holomorphically a compact Riemann surface \(\Sigma\), then the Kähler metric \(g\) and the Einstein metric \(h\) can be written as \[\begin{align} g&=Wd\xi^2+W^{-1}\eta^2+We^vg_{\Sigma},\tag{9}\\ h&=\xi^{-2}(Wd\xi^2+W^{-1}\eta^2+We^vg_{\Sigma}),\tag{10} \end{align}\] where \(g_{\Sigma}\) is the constant curvature metric with \(K_\Sigma\in\{-1,0,1\}\) (with volume one if \(\Sigma=T^2\)), and \(\eta\) is the connection 1-form determined by the metric \(g\) on the \(\mathop{\mathrm{\mathbb{S}}}^1\)-bundle associated to the free \(\mathop{\mathrm{\mathbb{S}}}^1\)-action. Diffeomorphically, the manifold \(M\) is \(I\times\mathscr{L}\), where \(I\) is an open interval in the range of \(\xi\) and the link \(\mathscr{L}\) is an \(\mathop{\mathrm{\mathbb{S}}}^1\)-bundle over \(\Sigma\). Following the computation of LeBrun [35], we have that in the Type I and Type II cases, \(h\) being Einstein is equivalent to \(v,W\) satisfying the following (twisted-)Toda equation systems respectively.

Type I. \[\label{eq:case-I} \Delta_{\Sigma} v+(e^{v})_{\xi\xi}-2K_\Sigma=0, \qquad W=1-\frac{1}{2}\xi v_\xi.\tag{11}\] Type II. \[\label{eq:case-II} \Delta_{\Sigma} v+(e^{v})_{\xi\xi}-2K_{\Sigma}=-\xi\,W e^{v}, \qquad W=\frac{12-6\xi v_\xi}{12+\xi^{3}}.\tag{12}\] Here, the (twisted-)Toda equations follow from LeBrun’s calculation [35] of the scalar curvature of 9 , where in the Type I case \(s_g=0\) and in the Type II case \(s_g=\xi\). The equations for \(W\) follow from the conformal change of Ricci curvature for \(g=\xi^2h\). Finally, if we pick a holomorphic coordinate \(x+iy\) on \(\Sigma\), with \(g_{\Sigma}=e^{f(x,y)}\,(dx^2+dy^2)\), then the connection 1-form \(\eta\) is determined via \[\label{eq:deta} d\eta=(We^v)_{\xi}\,e^f\,dx\wedge dy+W_x\,dy\wedge d\xi+W_y\,d\xi\wedge dx.\tag{13}\] By integrating the (twisted-)Toda equation over \(\Sigma\), in both cases there are constants \(a,b\) such that \(\fint_{\Sigma}We^vd\mathrm{vol}_{\Sigma}=b+\frac{a}{2}\xi\). Indeed, since the degree of the \(\mathop{\mathrm{\mathbb{S}}}^1\)-bundle is \(\ell\), if the period of \(\mathcal{K}\) is \(\mathop{\mathrm{\mathfrak{p}}}\), we have \(\frac{1}{\mathop{\mathrm{\mathfrak{p}}}}\int_{\Sigma} d\eta=\ell\), which is the same as \[\label{eq:period-constraint} \frac{1}{\mathop{\mathrm{\mathfrak{p}}}}\int_{\Sigma}(We^v)_{\xi}d\mathrm{vol}_{\Sigma}=\ell\tag{14}\] where \(d\mathrm{vol}_{\Sigma}\) is the volume form of \((\Sigma,g_{\Sigma})\).

3 Asymptotics and the boundary value problems↩︎

In this section, we derive the asymptotic behavior of the function \(v\) arising from the reductions 11 and 12 for conformally Kähler Poincaré–Einstein four-manifolds with a cusp end. We first establish the asymptotics of the conformal factor \(\xi\) along the cusp end, and then use them to determine the corresponding behavior of \(v\). These asymptotic regimes lead to four natural boundary value problems, according to the type of the reduction and the asymptotic model of the cusp. In Section 5, we will analyze these boundary value problems and obtain sharper asymptotics for \(v\), which in turn yield sharper asymptotics for the Einstein metric \(h\) and the Kähler metric \(g\).

Suppose there is a complete Einstein metric \(h\) that is conformal to the Kähler metric \(g=\xi^2h\), with \(\mathcal{K}\) inducing a free \(\mathop{\mathrm{\mathbb{S}}}^1\)-action and compact Kähler reduction \(\Sigma\). The manifold \(M\) is diffeomorphic to \((0,\infty)\times \mathscr{L}\), with \(\mathscr{L}\) being an \(\mathop{\mathrm{\mathbb{S}}}^1\)-bundle over \(\Sigma\). For such \((M,h)\), there are two ends, which we denote by \(E_0\) and \(E_\infty\). Furthermore, we assume that \(E_0\) is PE and \(E_\infty\) is an AH cusp, an ACH cusp, or a \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp in the sense of Definition 2. We first have

Next, we consider the Type I case and the Type II case separately and establish the behavior of \(v\) towards the cusp end \(E_\infty\).

3.1 The Type I case↩︎

We study the reduction over the cusp end \(E_\infty\). Note that a \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp cannot occur in the Type I case since the model metric is not ASD (under both of the orientations). We first consider the asymptotic behavior of the conformal factor \(\xi\). Recall that we have arranged that \(g=\xi^2h\) and \(\xi\) is a moment map for \(\mathcal{K}\). Moreover, \(\xi\) is proportional to \(|d(\mathcal{K}^{\sharp_h})^+|_h^{-1}\). We may suppose \(\xi>0\). In the Type I case, since \(g\) is Kähler scalar-flat, by calculating the conformal change of scalar curvature for \(g=\xi^2h\) we have \[\label{eq:conformal-relation-for-scalar-ASD} \Delta_h\xi+2\xi=0.\tag{15}\] We have the following Harnack inequality for \(\xi\).

Now for any sequence of points \(p_j\in M\) that diverges to the cusp end, we consider the balls \(B_j:=B_1(p_j)\subset(M,h)\). The sequence of balls \((B_j,h)\) is collapsing in the sense that the injectivity radius converges to zero, but the universal cover \((\widetilde{B}_j,\widetilde{h}_j)\) is non-collapsed. Denote the fundamental group of \(B_j\) as \(G_j\), which acts on \(\widetilde{B}_j\) via deck transformations. Write the hyperbolic metric and complex hyperbolic metric as \[\begin{align} h_{\mathop{\mathrm{\mathbb{RH}}}^4}&:=dr^2+e^{-2r}(dx^2+dy^2+dt^2),\\ h_{\mathop{\mathrm{\mathbb{CH}}}^2}&:=\frac{1}{2}\left(dr^2+e^{-r}(dx^2+dy^2)+e^{-2r}(dt-xdy)^2\right). \end{align}\] Since the end is an AH cusp or an ACH cusp, we know \((\widetilde{B}_j,\widetilde{h}_j,G_j)\) converges smoothly in the equivariant Cheeger-Gromov sense to an open domain \((\widetilde{B}_\infty,\widetilde{h}_\infty, G_\infty)\) that embeds into \(\mathop{\mathrm{\mathbb{RH}}}^4\) or \(\mathop{\mathrm{\mathbb{CH}}}^2\). Here we can isometrically identify the limit as \(\widetilde{B}_\infty=\{1<r<3\}\), with the action \(G_\infty\) being the standard \(\mathop{\mathrm{\mathbb{R}}}^3\)-action in the \(\mathop{\mathrm{\mathbb{RH}}}^4\) case, and being the standard \(\mathcal{H}^3\)-action in the \(\mathop{\mathrm{\mathbb{CH}}}^2\) case. We may normalize \(\xi\) on \(\widetilde{B}_1(p_j)\) to \(\xi_j:=\xi/\xi(p_j)\). Then because of the Harnack inequality and the elliptic equation 15 , one has

Since \(\xi_\infty\) is \(G_\infty\)-invariant, it is easy to see that

• in the \(\mathop{\mathrm{\mathbb{RH}}}^4\) case, there are constants \(a,b\) such that \[\xi_\infty=a e^r+be^{2r};\]

• in the \(\mathop{\mathrm{\mathbb{CH}}}^2\) case, there are constants \(a,b\) such that \[\xi_\infty=(a+br)e^r.\]

However, since the conformal metric \(\widetilde{g}_j:=\xi_j^2\widetilde{h}_j\) is Kähler on \(\widetilde{B}_j\), the limit metric \(\widetilde{g}_\infty:=\xi_\infty^2\widetilde{h}_\infty\) is Kähler as well.

As a direct consequence of the above, we have

Since the cusp end is exponentially asymptotic to the corresponding model cusp metric, we may apply Fredholm theory to 15 .

We will show \(b=0\) for the second case in the proof of Proposition 6.

Since \(\xi\) is a defining function for the PE end \(E_0\), its range on \(M\) is \((0,\infty)\). Hence the reduction is defined globally: \[\begin{align} g&=Wd\xi^2+W^{-1}\eta^2+We^vg_{T^2},\\ h&=\xi^{-2}(Wd\xi^2+W^{-1}\eta^2+We^vg_{T^2}), \end{align}\] where \(v\) and \(W\) satisfy \[\label{eq:toda95equation95asd95uniqueness95proof} (e^v)_{\xi\xi}+\Delta_{T^2}v=0 \qquad W=1-\frac{1}{2}\xi v_\xi.\tag{18}\]

3.2 The Type II case↩︎

Now we consider the Type II case. Recall that in the Type II case we arrange \(g=s_g^2h\) with \(\xi=s_g\) being the scalar curvature of the Kähler extremal metric \(g\), which is either strictly positive or strictly negative in \(M\). Moreover, \[\label{eq:xi-and-Weyl} |\xi|=(2\sqrt{6}|W^+_h|_h)^{\frac{1}{3}}.\tag{21}\] It is impossible for \(M\) to have both a PE end and an AH cusp end; otherwise, by 21 , \(\xi \to 0\) at both ends, so \(\xi\) has an interior critical point, contradicting the freeness of the \(\mathop{\mathrm{\mathbb{S}}}^1\)-action.

We first consider the case that \(E_\infty\) is an ACH cusp. Then the orientation induced by the complex structure agrees with the positive orientation of \(\mathop{\mathrm{\mathbb{CH}}}^2\); otherwise, 21 again gives \(\xi \to 0\) on both ends, yielding the same contradiction.

Since \(E_\infty\) is an ACH cusp with the positive orientation of \(\mathop{\mathrm{\mathbb{CH}}}^2\), \(W_h^+\) approaches the self-dual Weyl tensor of positively oriented \(\mathop{\mathrm{\mathbb{CH}}}^2\), which, in an orthogonal frame of \(\Lambda^+\), is \(\mathrm{diag}(-2,1,1)\). Hence \(|\xi|=(2\sqrt{6}|W_h^+|)^{1/3}\to \sqrt[3]{12}\) as we approach infinity of \(E_\infty\). Since the conformal factor tends to a constant, the Kähler metric \(g\) is also asymptotic, up to scaling, to the standard complex hyperbolic cusp metric. As \(\mathop{\mathrm{\mathbb{CH}}}^2\) has scalar curvature \(-12\), it follows that \(\xi<0\) on \(E_\infty\), thus \(\xi<0\) on \(M\). For simplicity we shall set \[\xi_\ast:=-\sqrt[3]{12}.\]

The conformal change of scalar curvature associated to \(g=\xi^2h\) is \[\label{eq:conformal-change-of-scalar-Type-II} 6\Delta_h\xi+\xi(12+\xi^3)=0,\tag{22}\] which can be written as \[\label{eq:modified-conformal-change-of-scalar-Type-II} 6\Delta_h(\xi- \xi_\ast)+\xi(\xi- \xi_\ast)(\xi^2+ \xi_\ast\xi+ \xi_\ast^2)=0.\tag{23}\]

Now we identify the end \(E_\infty\) as \((R_0,\infty)\times\mathop{\mathrm{\mathscr{N}}}_{\ell}\) via the diffeomorphism in Definition 2.

Therefore \(\xi=s_g\) takes values in \((\xi_\ast,0)\) on \(M\).

So we can apply the reduction over the entire \(M\) to write \[\begin{align} g&=Wd\xi^2+W^{-1}\eta^2+We^vg_{T^2},\\ h&=\xi^{-2}(Wd\xi^2+W^{-1}\eta^2+We^vg_{T^2}), \end{align}\] with \(\xi\in(\xi_\ast,0)\), and \(v,W\) satisfy the twisted-Toda system \[\begin{align} \label{eq:twisted-toda-system} (e^v)_{\xi\xi}+\Delta_{T^2}v=-\xi We^v,\qquad W=\frac{12-6\xi v_\xi}{12+\xi^3}. \end{align}\tag{26}\]

With \(W^{-1}=\xi^{-2}|d\xi|^2_h\), this shows that \[|\nabla_{h_0}^k(W^{-1}-2 \xi_\ast^{-2}a^2e^{-2r})|_{h_0}=O(e^{-(2+\delta_2)r})\] where \(\delta_2=\min\{1,\delta_1\}\), which implies \[|W-\frac{1}{2} \xi_\ast^2(\xi- \xi_\ast)^{-2}|=O((\xi- \xi_\ast)^{-2+\delta_2}).\] Together with \(W=\frac{12-6\xi v_\xi}{12+\xi^3}\), we have \(v_\xi=3(\xi- \xi_\ast)^{-1}+\frac{2}{\xi_\ast}+O((\xi- \xi_\ast)^{-1+\delta_2})\). By integration, it follows that

Next we consider the situation that \(E_\infty\) is a \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp. For the same reason as in the ACH cusp case, we know that the range of \(\xi=s_g\) on \(M\) is \((\xi_\ast,0)\). We have Lemma 7 as well: note that for the model \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp metric \(h_0\) (both trivial and non-trivial case), instead of 25 , we have \[(\Delta_{h_{0}}-6)e^{-\alpha r} =3(\alpha-1)(\alpha+2)e^{-\alpha r}.\]

We also have \[|\nabla_{h_0}^k(\xi-\xi_\ast-ae^{-r})|_{h_0}=O(e^{-(1+\delta_1)r})\] as in Lemma 8, which gives that \[|\nabla_{h_0}^k(W^{-1}-3\xi_\ast^{-2}a^2e^{-2r})|_{h_0}=O(e^{-(2+\delta_2)r}),\] where \(\delta_2=\min\{1,\delta_1\}\). This implies \[|W-\frac{1}{3} \xi_\ast^2(\xi- \xi_\ast)^{-2}|=O((\xi- \xi_\ast)^{-2+\delta_2}).\] Recall \(W=\frac{12-6\xi v_\xi}{12+\xi^3}\). By integration, it follows that

3.3 The boundary value problems↩︎

Finally, we formulate our boundary value problems. Based on our analysis of the asymptotic behavior of solutions \(v\) to the (twisted-)Toda equation in Sections 3.1 and 3.2, we are led to the following four natural boundary value problems with smooth Dirichlet boundary data \(\varphi\).

\[\begin{align} &\tag{27}\left\{ \begin{aligned} &(e^v)_{\xi\xi}+\Delta_{T^2}v = 0 \qquad \text{in } (0,\infty)\times T^2,\\ &v|_{\{0\}\times T^2} = \varphi,\\ &v \in L^\infty\bigl((0,\infty)\times T^2\bigr). \end{aligned} \right.\\ \notag\\ &\tag{28}\left\{ \begin{align} &(e^v)_{\xi\xi}+\Delta_{T^2}v = 0 \qquad \text{in } (0,\infty)\times T^2,\\ &v|_{\{0\}\times T^2} = \varphi,\\ &v-\log(\xi+1) \in L^\infty\bigl((0,\infty)\times T^2\bigr). \end{align} \right.\\ \notag\\ &\tag{29}\left\{ \begin{align} &(e^v)_{\xi\xi}+\Delta_{T^2}v = -\xi\frac{12-6\xi v_\xi}{12+\xi^3}e^v \qquad \text{in } (\xi_\ast,0)\times T^2,\\ &v|_{\{0\}\times T^2} = \varphi,\\ &v-3\log(\xi- \xi_\ast) \in L^\infty\bigl((\xi_\ast,0)\times T^2\bigr). \end{align} \right.\\ \notag\\ &\tag{30}\left\{ \begin{aligned} &(e^v)_{\xi\xi}+\Delta_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}v =-2 -\xi\frac{12-6\xi v_\xi}{12+\xi^3}e^v \qquad \text{in } (\xi_\ast,0)\times \Sigma_{\mathop{\mathrm{\mathtt{g}}}},\\ &v|_{\{0\}\times \Sigma_{\mathop{\mathrm{\mathtt{g}}}}} = \varphi,\\ &v-2\log(\xi- \xi_\ast) \in L^\infty\bigl((\xi_\ast,0)\times \Sigma_{\mathop{\mathrm{\mathtt{g}}}}\bigr). \end{aligned} \right. \end{align}\]

Suppose that there exists a smooth solution \(v\), and suppose moreover that the corresponding functions \(W\), determined by \(v\) through the reductions 11 –12 , are positive. As explained in Section 2, \(\int_{\Sigma}(We^v)_\xi d\mathrm{vol}_{\Sigma}\) is a constant. For given \(\ell\), by choosing the period \(\mathop{\mathrm{\mathfrak{p}}}\) correctly we can arrange 14 . Then geometrically these solutions give rise to Einstein metrics \(h\) on \(M=(0,\infty)\times\mathscr{L}\) with the following asymptotic behavior:

• BVP1: \(h\) is Type I, \(E_0\) is a PE end, and \(E_\infty\) is an AH cusp end;

• BVP2: \(h\) is Type I, \(E_0\) is a PE end, and \(E_\infty\) is an ACH cusp end;

• BVP3: \(h\) is Type II, \(E_0\) is a PE end, and \(E_\infty\) is an ACH cusp end;

• BVP4: \(h\) is Type II, \(E_0\) is a PE end, and \(E_\infty\) is a \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp end.

Moreover, there is the Kähler metric \(g\) that extends to a smooth complete Kähler metric with boundary \(\bar g\) on \(\overline{M}=[0,\infty)\times\mathscr{L}\). The quotient of \(\bar g|_{\{0\}\times\mathscr{L}}\) by \(\mathop{\mathrm{\mathbb{S}}}^1\) is \((\Sigma,e^{\varphi}g_{\Sigma})\), and hence is determined by the boundary data \(\varphi\). Conversely, by Section 2 and Sections 3.1–3.2, we know any conformally Kähler PE metric with a cusp on \((0,\infty)\times\mathscr{L}\), whose \(\mathcal{K}\) induces an \(\mathop{\mathrm{\mathbb{S}}}^1\)-action, arises from solutions to these boundary value problems.

In Sections 5.1–5.4, we shall prove the following results.

For \(\Sigma=T^2\) or \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\), set \[\bar\varphi=\log(\fint_{\Sigma} e^\varphi \, d\mathrm{vol}_{\Sigma}).\] We will also prove that all solutions are asymptotic to model solutions:

Here, for 28 , each \(a>0\) and boundary value \(\varphi\) determine a unique solution \(v_{a,\varphi}\) to 28 with the above asymptotic. Given the solution \(v_{1,\varphi}\), the function \(v_{1,\varphi}(a\xi,\cdot)-2\log a\) is also a solution, and is exactly \(v_{a^{-1},\varphi-2\log a}\). These solutions give rise to the same Einstein metric up to isometry.

The decay of \(v\) shows that the cusp ends arising from 27 –29 are asymptotic to their model cusp metrics at a super-exponential rate, while the \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp end arising from 30 has exponential decay. Note that the \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp has degree zero if and only if \(a=0\), i.e., \(\fint_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}} e^\varphi \,d\mathrm{vol}_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}=\frac{1}{3}\xi_\ast^2 .\)

Summarizing this section, we have

In particular, the weak assumption in Definition 2 that \(h\) is asymptotic to a model cusp metric \(h_0\) implies the much stronger asymptotic behavior described above.

4 Model solutions and their geometries↩︎

Before solving the boundary value problems 27 –30 , we first study the simplest solutions of the (twisted-)Toda equation, namely those depending only on \(\xi\). Part of our solutions here, namely the subclass for which the connection one-form \(\eta\) is flat, in the sense that \(d\eta=0\), can be identified with the so-called Euclidean topological black holes [45].

4.1 The case of Type I↩︎

Solutions to 11 that only depend on \(\xi\) take the form of \[We^v=b+\frac{a}{2}\xi, \qquad e^v=b+a\xi.\] The corresponding metric is \[\label{eq:typeI-metric} h_{\mathrm{mod}} = \frac{1}{\xi^{2}}\left( \frac{b+\frac{a}{2}\xi}{b+a\xi}\,d\xi^{2} + \frac{b+a\xi}{b+\frac{a}{2}\xi}\,\eta^{2} + \left(b+\frac{a}{2}\xi\right)g_{T^{2}} \right), \qquad d\eta=\frac{a}{2}\,d\mathrm{vol}_{T^2}.\tag{31}\] Here \(h_{\mathrm{mod}}\) is Riemannian precisely when \[\label{eq:typeI-positivity} b+\frac{a}{2}\xi>0, \qquad b+a\xi>0.\tag{32}\] Let \(I\subset (0,\infty)\) be the maximal interval on which 32 holds. We now determine \(I\) and describe the geometry at the endpoints of \(I\). The maximal interval \(I\subset(-\infty,0)\) such that 32 can be determined similarly. For an endpoint \(\hat{\xi}\) of \(I\), we say the \(\hat{\xi}\)-end is

• conical if the Einstein metric \(h\) at \(\hat{\xi}\) completes to a metric with cone angle along \(T^2\);

• \(\frac{2}{3}\)-horn-type if reparametrizing \(\xi\) as a suitable \(\zeta=\zeta(\xi)>0\) with \(\zeta(\hat{\xi})=0\), \[h\sim d\zeta^2+c_1\zeta^{-2/3}\eta^2+c_2\zeta^{2/3}g_{T^2},\] and \(\frac{4}{3}\)-horn-type if \[h\sim d\zeta^2+c_1\zeta^{-2/3}\eta^2+c_2\zeta^{4/3}g_{T^2},\] for some constants \(c_1,c_2\).

We summarize the maximal Riemannian intervals and the corresponding endpoint geometries in the following table.

5pt

4.2 The case of Type II with base \(T^2\)↩︎

We now turn to the twisted-Toda equation with \(\Sigma=T^2\). Solutions to 12 that only depend on \(\xi\) are \[We^v=b+\frac{a}{2}\xi, \qquad e^v=b+a\xi-\frac{b}{6}\xi^3-\frac{a}{24}\xi^4.\] Set \[P(\xi):=b+a\xi-\frac{b}{6}\xi^3-\frac{a}{24}\xi^4.\] Then \[\label{eq:typeII-metric} h_{\mathrm{mod}} = \frac{1}{\xi^2}\left( \frac{b+\frac{a}{2}\xi}{P(\xi)}\,d\xi^2 + \frac{P(\xi)}{b+\frac{a}{2}\xi}\,\eta^2 + \left(b+\frac{a}{2}\xi\right)g_{T^2} \right), \qquad d\eta=\frac{a}{2}\,d\mathrm{vol}_{T^2}.\tag{33}\] Hence \(h_{\mathrm{mod}}\) is Riemannian precisely when \[\label{eq:typeII-positivity} b+\frac{a}{2}\xi>0, \qquad P(\xi)>0.\tag{34}\] Set \(\underline{\xi}:=-\frac{2b}{a}\). We make the following convention.

• When \(a>0,b\neq0\), the largest root of \(P\) is positive, denoted as \(\xi_+\).

• When \(a<0,b>0\), \(P\) has a unique root in \((0,\underline{\xi})\) denoted as \(\xi_+\).

• When \(3a^3>2b^3>0\), \(P\) has a unique root in \((\underline{\xi},0)\) denoted as \(\xi_-\).

• When \(2b^3<3a^3<0\), \(P\) has a unique root in \((-\infty,\underline{\xi})\) denoted as \(\xi_-\).

We summarize the maximal Riemannian intervals and the corresponding endpoint geometries in the following Table 2.

5pt

4.3 The case of Type II with base \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\)↩︎

We now consider the case where the base \((\Sigma_{\mathop{\mathrm{\mathtt{g}}}},g_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}})\) is a hyperbolic surface with constant Gauss curvature \(-1\). For solutions of 12 depending only on \(\xi\), we have \[We^v=b+\frac{a}{2}\xi, \qquad e^v=b+a\xi-\frac{b}{6}\xi^3-\frac{a}{24}\xi^4-\xi^2.\] Set \[Q(\xi):=b+a\xi-\frac{b}{6}\xi^3-\frac{a}{24}\xi^4-\xi^2.\] Then \[\label{eq:model95metric95cusp95cross95hyperbolic95surface} h_{\mathrm{mod}} = \frac{1}{\xi^2}\left( \frac{b+\frac{a}{2}\xi}{Q(\xi)} d\xi^2 + \frac{Q(\xi)}{b+\frac{a}{2}\xi} \eta^2 + \left(b+\frac{a}{2}\xi\right)g_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}} \right), \qquad d\eta=\frac{a}{2} d\mathrm{vol}_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}.\tag{35}\]

Note that \(Q\) cannot have a triple root. For \(h_{\mathrm{mod}}\) to define a complete Riemannian metric with two ends, it is necessary that \(Q\) have a double root at some \(\xi_\ast\in\mathbb{R}\) satisfying \(b+\frac{a}{2}\xi_\ast\neq 0\). Since \(Q(\xi_\ast)=Q'(\xi_\ast)=0\), the identity \[Q(\xi)-\frac{\xi}{2}Q'(\xi)=\left(b+\frac{a}{2}\xi\right)\left(1+\frac{\xi^3}{12}\right)\] implies \(\xi_\ast=-\sqrt[3]{12}\). Substituting this into \(Q(\xi_\ast)=0\) yields \(b+\frac{a}{2}\xi_\ast=\frac{\xi_\ast^2}{3}.\)

Conversely, if \[\xi_\ast=-\sqrt[3]{12}, \qquad b>0, \qquad b+\frac{a}{2}\xi_\ast=\frac{\xi_\ast^2}{3},\] then \(h_{\mathrm{mod}}\) defines a complete Riemannian metric on \((\xi_\ast,0)\) with two ends. Indeed,

\[\label{eq:Q95expression95sigma95g95cusp} Q(\xi)=(\xi-\xi_\ast)^2\left(\frac{1}{3}-\frac{\xi_\ast^2}{18}\xi+\frac{a}{24}(\xi_\ast^2-\xi^2)\right)\tag{36}\] as \(\xi\to\xi_\ast^+\), \[h_{\mathrm{mod}} \sim \frac{1}{3}\left( \frac{d\xi^2}{(\xi-\xi_\ast)^2} + \frac{9(\xi-\xi_\ast)^2}{\xi_\ast^4}\eta^2 + g_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}\right).\] Therefore, the metric defined by 35 on \((\xi_\ast,0)\) with \(Q(\xi)\) defined by 36 defines a complete Riemannian metric with two ends: one Poincaré–Einstein end and one \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\)-cusp end.

5 Solving the boundary value problems↩︎

In this section we establish existence, uniqueness, and decay estimates for the boundary value problems 27 –30 . We consider the following classes of PDEs. We shall eventually make a change of variables in the (twisted-)Toda equation, which puts it into this form. Let \((\Sigma,g_{\Sigma})\) be a closed surface. Consider \[\label{eq:u-PDE-t} \Delta_{\Sigma}u+\Psi(t)\,(e^{u})_{tt}+B(t)(e^{u})_{t}+2K(t)\,(e^u-1)=0 \qquad\text{on }[0,\infty)\times \Sigma.\tag{37}\] We suppose \[\label{eq:coeff-assumptions} \begin{align} &0<c^{-1}\le \Psi(t)\le c,\qquad K(t)\le 0,\qquad \Psi,B,K\in C^\infty([0,\infty)),\\ &\sup_{t\ge 0}\bigl(|\partial_t^j\Psi(t)|+|\partial_t^jB(t)|+|\partial_t^jK(t)|\bigr)<\infty, \qquad \forall j\ge 0. \end{align}\tag{38}\] By integrating 37 over \(\Sigma\), \(\fint_{\Sigma}e^{u(t,\cdot)} d\mathrm{vol}_{\Sigma}\) satisfies the ODE \[\Psi(t)\left(\fint_{\Sigma}e^{u(t,\cdot)}d\mathrm{vol}_{\Sigma}\right)_{tt}+B(t)\left(\fint_{\Sigma}e^{u(t,\cdot)}d\mathrm{vol}_{\Sigma}\right)_{t}+2K(t)\left(\fint_{\Sigma}e^{u(t,\cdot)}d\mathrm{vol}_{\Sigma}-1\right)=0.\] We study solutions such that \(\fint_{\Sigma}e^{u(t,\cdot)} d\mathrm{vol}_{\Sigma}=1\). The main conclusion of this section is

We will actually prove the following exponential stability result that is stronger than the uniqueness part and decay part of the above.

Taking \(u_2=0\) in Theorem 13, we get exponential decay of the solution, and taking \(\varphi_1=\varphi_2\) we get the uniqueness part in Theorem 12.

We prove the existence first.

Now we prove Theorem 13 through a slice-wise \(H^{-1}\) energy differential inequality. Proposition 14 and Theorem 13 together imply Theorem 12. Let \(I\subset\mathbb{R}\) be an interval and suppose \(w:I\times \Sigma\to\mathbb{R}\) is a function where

• \(w(t,\cdot)\in L^{2}(\Sigma)\) for each \(t\in I\);

• \(\int_{\Sigma}w(t,x) d\mathrm{vol}_{\Sigma}=0\).

For each \(t\), let \(\phi(t,\cdot)\in H^{2}(\Sigma)\) be the unique solution of \[-\Delta_{\Sigma}\phi(t,\cdot)=w(t,\cdot),\qquad \int_{\Sigma}\phi(t,x) d\mathrm{vol}_{\Sigma}=0,\] and define \[E_{w}(t):=\frac{1}{2}\|w(t,\cdot)\|_{H^{-1}(\Sigma)}^{2} :=\frac{1}{2}\int_{\Sigma}|\nabla\phi(t,x)|^{2} d\mathrm{vol}_{\Sigma}=\frac{1}{2}\int_{\Sigma}w(t,x)\phi(t,x)d\mathrm{vol}_{\Sigma}.\] Let \(u_i\in C^{2}([0,\infty)\times \Sigma)\cap L^\infty([0,\infty)\times \Sigma)\) \((i=1,2)\) be solutions of 37 with boundary values \(u_i(0,\cdot)=\varphi_i\), \(\|\varphi_i\|_{L^\infty(\Sigma)}\le A\), and \(A':=\max_i\|u_i\|_{L^\infty([0,\infty)\times \Sigma)}<\infty\). Suppose \(\fint_{ \Sigma} e^{u_i(t,\cdot)} d\mathrm{vol}_{\Sigma}=1\) for \(t\geq0\). Define \[w(t,x):=e^{u_1(t,x)}-e^{u_2(t,x)}.\] We first prove a differential inequality.

Next we conclude the decay of the \(H^{-1}\) energy.

Finally we prove the exponential stability.

Finally we apply our study to the boundary value problems 27 –30 and prove Theorem 9 and 10.

5.1 The boundary value problem 27↩︎

We consider 27 . Integrating the Toda equation over \(T^{2}\) at fixed \(\xi\) gives \[\label{eq:slice-affine-hyperbolic-cusp} \fint_{T^{2}}e^{v(\xi,\cdot)} d\mathrm{vol}_{T^2}=a\xi+b \quad\text{for some }a,b\in\mathbb{R}.\tag{48}\] Since the left-hand side is positive on \([0,\infty)\) and for 27 we only consider bounded solutions, we have \(a=0\) and \(b>0\). Plugging in \(v|_{\xi=0}=\varphi\), we find \(b=\fint_{T^2}e^{\varphi}\mathop{\mathrm{d\mathrm{vol}}}_{T^2}\). Setting \(u=v-\log b\), we have \(\fint_{T^2}e^{u(\xi,\cdot)}\mathop{\mathrm{d\mathrm{vol}}}_{T^2}\equiv 1\). By Theorems 12 and 13, we obtain existence, uniqueness, and exponential decay of the solution to 27 . It remains to show \(W=1-\frac{1}{2}\xi v_{\xi}>0\) so that the solutions to 27 correspond to Einstein metrics.

5.2 The boundary value problem 28↩︎

We now consider 28 . Since we consider solutions with logarithmic growth, by integrating the Toda equation over \(T^2\) we have \(\fint_{T^2}e^vd\mathrm{vol}_{T^2}=b+a\xi\) with \(a>0\). Define \[u:=v-\log(b+a\xi).\] Then the Toda equation is equivalent to \[\label{eq:u-PDE-asd-complex-cusp} \Delta_{T^2}u+\bigl((b+a\xi)e^u\bigr)_{\xi\xi}=0,\tag{50}\] and we have \(\fint_{T^2}e^{u(\xi,\cdot)}d\mathrm{vol}_{T^2}=1\). We consider bounded solutions of 50 .

Introduce \[t:=\sqrt{\frac{b+a\xi}{b}}\in [1,\infty).\] Then \[\partial_\xi=\frac{a}{2bt}\partial_t.\] Therefore 50 becomes \[\label{eq:PDE-PsiB} \Delta_{T^{2}}u+\Psi(t) (e^{u})_{tt}+B(t) (e^{u})_{t}=0, \qquad \Psi(t)=\frac{a^{2}}{4b},\;\;B(t)=\frac{3a^{2}}{4bt}.\tag{51}\] The coefficients satisfy 38 . Hence Theorems 12 and 13 apply, giving existence, uniqueness, and exponential decay. The positivity \(W:=1-\frac{1}{2}\xi v_\xi>0\) follows by the same argument as in Proposition 15. For any solution \(v=\log(b+a\xi)+u\), \[W=1-\tfrac12 \xi v_\xi =1-\frac{a\xi}{2(b+a\xi)}-\frac{1}{2}\xi u_{\xi}.\] Since \[W(0,x)\equiv 1, \qquad \lim_{\xi\to\infty} W(\xi,x)=\frac{1}{2} \;\;\text{uniformly in }x\in T^{2},\] we can argue as in Proposition 15 to conclude \(W>0\). This shows that, for each fixed \(a>0\) and boundary value \(\varphi\), 28 has a unique solution.

5.3 The boundary value problem 29↩︎

Consider the twisted Toda equation on \((\xi_\ast,0)\times T^{2}\), \[\label{eq:v-PDE} \Delta_{T^{2}} v + (e^{v})_{\xi\xi} +\xi \frac{12e^v-6\xi (e^v)_{\xi}}{12+\xi^3}=0.\tag{52}\] Recall that we have the model solution \(v_{\mathrm{mod}}\) from Section 4.2 that corresponds to a Type II PE metric with an ACH cusp \[e^{v_{\mathrm{mod}}} = -\frac{a}{24}(\xi-\xi_\ast)^3(\xi+\xi_\ast), \qquad a>0, \quad \xi_\ast=-\sqrt[3]{12}.\] This is the model solution from Case 2 in Table 2 of Section 4.2 with \(2b^3=3a^3\). Set \[\label{eq:u-def} u(\xi,x):=v(\xi,x)-v_{\mathrm{mod}}(\xi).\tag{53}\] By picking the correct \(a\), we have \(\fint_{T^{2}} e^{u(\xi,\cdot)} d\mathrm{vol}_{T^2}=1\). Then 52 is equivalent to \[\label{eq:u-PDE-xi} \Delta_{T^{2}}u +e^{v_{\mathrm{mod}}}(e^{u})_{\xi\xi} +\left(2(e^{v_{\mathrm{mod}}})_{\xi} -\frac{6\xi^{2}}{12+\xi^{3}} e^{v_{\mathrm{mod}}}\right)(e^{u})_{\xi}=0.\tag{54}\]

Introduce \[t:=\Bigl(1-\frac{\xi}{\xi_\ast}\Bigr)^{-1/2}\in[1,\infty), \qquad \xi=\xi_\ast(1-t^{-2}).\] Then \[e^{v_{\mathrm{mod}}}(t) = -\frac{a\xi_\ast}{2} \frac{2t^{2}-1}{t^{8}}, \qquad W_{\mathrm{mod}} = \frac{t^{6}}{2t^{2}-1},\] and 54 becomes \[\label{eq:u-PDE-t-typeII} \Delta_{T^{2}}u+\Psi(t)(e^{u})_{tt}+B(t)(e^{u})_{t}=0,\tag{55}\] where \[\Psi(t)=-\frac{a}{8\xi_\ast}\left(2-\frac{1}{t^{2}}\right), \qquad B(t)= \frac{a}{8\xi_\ast} \frac{30t^{6}-33t^{4}+9t^{2}-1}{t^{3}(3t^{4}-3t^{2}+1)}.\] Since \(a>0\) and \(\xi_\ast<0\), we have \[0<-\frac{a}{8\xi_\ast}\le \Psi(t)\le -\frac{a}{4\xi_\ast},\] and \(B\) is bounded on \([1,\infty)\). Therefore the coefficients satisfy 38 , so Theorems 12 and 13 apply directly. This shows 29 has a unique solution.

Moreover, for \(t\in [1,\infty),\) \[W = W_{\mathrm{mod}} -\frac{\frac{1}{2}\xi u_\xi}{1+\frac{\xi^3}{12}} = \frac{t^{6}}{2t^{2}-1} -\frac{t^{7}(t^{2}-1)}{4(3t^{4}-3t^{2}+1)} u_t\ge c_1t^4-c_2t^5e^{-\kappa_2 t}\] for some \(c_1,c_2,\kappa_2>0\). In particular, \(\lim_{t\to\infty}W(t)=+\infty.\) The argument in Proposition 15 then implies \(W>0\) on \([1,\infty)\times T^{2}\).

5.4 The boundary value problem 30↩︎

Consider the twisted Toda equation on \((\xi_\ast,0)\times \Sigma_{\mathop{\mathrm{\mathtt{g}}}}\), \[\label{eq:v-PDE-sigma95g} \Delta_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}} v + (e^{v})_{\xi\xi} +\xi \frac{12e^v-6\xi (e^v)_{\xi}}{12+\xi^3}=-2.\tag{56}\]

Recall that we have model solutions \(v_{\mathrm{mod}}\) from Section 4.3 that correspond to a Type II PE metric with \(\Sigma_{\mathop{\mathrm{\mathtt{g}}}}\) cusp \[e^{v_{\mathrm{mod}}} =(\xi-\xi_\ast)^2\left(\frac{1}{3}-\frac{\xi_\ast^2}{18}\xi+\frac{a}{24}(\xi_\ast^2-\xi^2)\right).\] Here \(a\) is determined by \(\fint_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}} e^\varphi \mathop{\mathrm{d\mathrm{vol}}}_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}+\frac{a}{2}\xi_\ast=\frac{1}{3}\xi_\ast^2.\)

Define \[\label{eq:u-def-sigma95g} u(\xi,x):=v(\xi,x)-v_{\mathrm{mod}}(\xi).\tag{57}\] It follows that \(\fint_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}} e^{u(\xi,\cdot)} d\mathrm{vol}_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}=1\) and \(u\in L^\infty((\xi_\ast,0)\times \Sigma_{\mathop{\mathrm{\mathtt{g}}}})\). Then 56 is equivalent to \[\label{eq:u-PDE-xi-sigma95g} \Delta_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}u +e^{v_{\mathrm{mod}}}(e^{u})_{\xi\xi} +\left(2(e^{v_{\mathrm{mod}}})_{\xi} -\frac{6\xi^{2}}{12+\xi^{3}} e^{v_{\mathrm{mod}}}\right)(e^{u})_{\xi}-2(e^u-1)=0.\tag{58}\]

Introduce \[t:=-\log\Bigl(1-\frac{\xi}{\xi_\ast}\Bigr)\in[0,\infty), \qquad \xi=\xi_\ast(1-e^{-t}).\]

Substituting into 58 , we obtain \[\label{eq:u-PDE-t-typeII-hyp} \Delta_{\Sigma_{\mathop{\mathrm{\mathtt{g}}}}}u+\Psi(t)(e^{u})_{tt}+B(t)(e^{u})_{t}-2(e^{u}-1)=0,\tag{59}\] where \[\Psi(t) = 1+\left(\frac{a\xi_\ast^2}{12}-\frac{2}{3}\right)e^{-t} -\frac{a\xi_\ast^2}{24}e^{-2t},\] and \[B(t) = 2\Psi_t(t)-3\Psi(t) +\frac{6(1-e^{-t})^2}{3-3e^{-t}+e^{-2t}}\Psi(t).\]

Since \(e^{v_{\mathrm{mod}}}>0\) on \((\xi_\ast,0)\), the function \(\Psi\) is positive on \([0,\infty)\); moreover \(\Psi(t)\to1\) as \(t\to\infty\), so there exist constants \(0<c\le C\) such that \[c\le \Psi(t)\le C \qquad\text{for all }t\in[0,\infty).\] Also, \(B\) is bounded on \([0,\infty)\). Therefore 59 is uniformly elliptic with bounded coefficients, hence Theorems 12 and 13 are applicable.

Moreover, \[W_{\mathrm{mod}} = \frac{\frac{\xi_\ast^2}{3}+\frac{a}{2}(\xi-\xi_\ast)}{e^{v_{\mathrm{mod}}}} = \frac{e^{2t}\left(\frac{1}{3}-\frac{a}{2\xi_\ast}e^{-t}\right)}{\Psi(t)}.\] Hence \[W = W_{\mathrm{mod}} -\frac{\frac{1}{2}\xi u_\xi}{1+\frac{\xi^3}{12}} = \frac{e^{2t}\left(\frac{1}{3}-\frac{a}{2\xi_\ast}e^{-t}\right)}{\Psi(t)} -\frac{e^{t}(e^{t}-1)}{2(3-3e^{-t}+e^{-2t})}\,u_t =\frac{1}{3} e^{2t}+O(e^{(2-\kappa_2)t})\] for some \(\kappa_2>0\). In particular, \(\lim_{t\to\infty}W(t)=+\infty.\) The argument in Proposition 15 then implies \(W>0\) on \([0,\infty)\times \Sigma_{\mathop{\mathrm{\mathtt{g}}}}\).

6 Degenerations↩︎

In this section we investigate various degenerations of our PE manifolds (with a cusp).

6.1 Limiting geometry of a family of Type I PE metrics↩︎

Recall our Type I model solutions in Section 4.1, where the model solution is \(e^{v_{\mathrm{mod}}}=b+a\xi\) and the Einstein metric is \[h_{\mathrm{mod}} = \frac{1}{\xi^{2}}\left( \frac{b+\frac{a}{2}\xi}{b+a\xi}\,d\xi^{2} + \frac{b+a\xi}{b+\frac{a}{2}\xi}\,\eta^{2} + \left(b+\frac{a}{2}\xi\right)g_{T^{2}} \right), \qquad d\eta=\frac{a}{2}\,d\mathrm{vol}_{T^2}.\] Suppose \(a>0,b>0\). After choosing a suitable period \(\mathop{\mathrm{\mathfrak{p}}}\), the model solution induces a PE metric with an ACH cusp on \((0,\infty)\times \mathop{\mathrm{\mathscr{N}}}_{\ell}\). We then let \(b\to0^+\). By 14 , in this process the period \(\mathop{\mathrm{\mathfrak{p}}}=\frac{a}{2\ell}\) does not change. In this limit, the solution \(v_{\mathrm{mod}}\) converges to \(\log(a\xi)\). Geometrically, if we take a pointed Cheeger-Gromov limit at a suitable base point, for instance at \(\xi=1\), then the PE metrics with an ACH cusp converge to a complex hyperbolic metric, namely the metric associated to \(\log(a\xi)\).

We show that this convergence to the complex hyperbolic metric also occurs for the PE manifolds with an ACH cusp arising from 28 . More precisely, we work in the following setting. Recall that, for each boundary value \(\varphi\) and constant \(a>0\), there is a unique solution \(v\) to 28 with boundary value \(\varphi\) satisfying \[\left|v-\log(\fint_{T^2} e^{\varphi}\mathop{\mathrm{d\mathrm{vol}}}_{T^2}+a\xi)\right|=O(e^{-\delta\sqrt{\xi}}).\] As discussed at the end of Section 3.3, solutions corresponding to different values of \(a\) determine isometric Einstein manifolds, after shifting the boundary value by an appropriate constant. We may assume without loss of generality that \(a=1\). Next, fix \(\phi\in C^\infty(T^2)\) and, for \(N\gg1\), let \(v_N\) denote the unique solution to 28 with \(a=1\) and boundary data \(\varphi=-N+\phi\). We prove the following.

Geometrically, for each solution \(v_N\), after fixing a degree \(\ell\) and choosing the correct period \(\mathop{\mathrm{\mathfrak{p}}}\), we obtain a PE manifold with an ACH cusp \((M,h_N)\), where \(M\) is diffeomorphic to \((0,\infty)\times \mathop{\mathrm{\mathscr{N}}}_\ell\). Let \(p_N\) be a base point with \(\xi(p_N)=1\). We then have

It suffices to prove Theorem 16.

An interesting phenomenon is that there is another non-trivial scale. By taking a pointed Cheeger-Gromov limit at suitably chosen base points, we obtain a non-trivial limit that is different from the complex hyperbolic space above.

For each \(N\gg1\), let \(q_N\in M\) be a base point with \(\xi(q_N)=e^{-N}\). We then analyze the pointed Cheeger–Gromov limit of the family \((M,h_N,q_N)\) as \(N\to \infty\). Set \(z:=\log\xi+N\).

First we consider the model case. The metric becomes \[h_N=\frac{1+\frac{1}{2}e^z}{1+e^z}dz^2+\frac{1+e^z}{1+\frac{1}{2}e^z}\frac{1}{e^{-2N}e^{2z}}\eta^2+\frac{1+\frac{1}{2}e^z}{e^{-N}e^{2z}}g_{T^2}.\] Hence, taking the limit we obtain the following smooth limit Einstein metric on \(\mathop{\mathrm{\mathbb{R}}}^4\), which is neither hyperbolic, nor complex hyperbolic. \[\label{eq:blow-up-limit-Einstein-on-R4} h_{\infty}=\frac{1+\frac{1}{2}e^z}{1+e^z}dz^2+\frac{1+e^z}{1+\frac{1}{2}e^z}\frac{1}{e^{2z}}\left(dt+\frac{1}{2}xdy\right)^2+\frac{1+\frac{1}{2}e^z}{e^{2z}}(dx^2+dy^2).\tag{60}\] As \(z\to\infty\), the metric \(h_\infty\) is asymptotic to the complex hyperbolic space, whereas as \(z\to-\infty\), it is asymptotic to the real hyperbolic space. Note that the metric \(h_\infty\) was discovered in [46].

Next we investigate the general case. We identify the Riemann surface \(T^2\) as \(\mathbb{C}/\Lambda\), so the flat metric \(g_{T^2}\) can be identified as \(dx'^2+dy'^2\). The Toda equation then becomes \[(e^v)_{\xi\xi}+v_{x'x'}+v_{y'y'}=0.\] Set \(\zeta=\xi e^N, x=x'e^{N/2},y=y' e^{N/2}\). For the solution \(v_N\), we set \[\label{eq:blow-up-R4-Einstein-metric} u_N(\zeta,x,y):=v_{N}(\xi,x',y')+N=v_N(e^{-N}\zeta,e^{-N/2}x, e^{-N/2}y)+N.\tag{61}\] Then \(u_N\) satisfies \[(e^{u_N})_{\zeta\zeta}+(u_N)_{xx}+(u_N)_{yy}=0,\qquad u_N|_{\{\zeta=0\}}(x,y)=\phi(e^{-N/2}x,e^{-N/2}y).\] In terms of \(u_N\), the Einstein metric is \[h_N=\zeta^{-2}\left(W(d\zeta^2+e^{u_N}(dx^2+dy^2)))+W^{-1}\eta^2\right),\qquad W=1-\frac{1}{2}\zeta (u_N)_{\zeta}.\] Denote \(\bar{u}_N(\zeta)=\log(\fint_{T^2} e^{u_N(\zeta, x,y)} dxdy)=\log(e^{\bar\phi}+\zeta)\). Applying the maximum principle to the PDE for \(w_N:=u_N-\bar{u}_N\), we have \[\sup |w_N|\leq \sup |\phi-\bar\phi|.\] Thus we can take a limit of \(u_N\) to \(u_\infty\), which satisfies the equation \[(e^{u_\infty})_{\zeta\zeta}+(u_\infty)_{xx}+(u_\infty)_{yy}=0,\quad u_\infty|_{\{\zeta=0\}}(x,y)=\phi(0,0),\] with \(u_\infty-\log(e^{\bar\phi}+\zeta)\in L^\infty([0,\infty)\times\mathbb{R}^2).\) We have the following lemma.

Geometrically, if we consider \((M,h_N,q_N)\) with a base point \(q_N\) where \(\xi(q_N)=e^{-N},x'(q_N)=x_0',y'(q_N)=y_0'\), then the limit is again the Einstein metric on \(\mathop{\mathrm{\mathbb{R}}}^4\) defined by 60 (after coordinate change).

By similar blow-up arguments, we provide a complete understanding of our Einstein metric during the process \(N\to\infty\).

Note that in the third case, instead of using 61 , we use \(u_N(\zeta,x,y)=v_N(\lambda_N\zeta,\lambda_N^{1/2}x,\lambda_N^{1/2}y)-\log(\lambda_N)\) with \(\lambda_N=\xi(\tilde{p}_N)\) and we apply Lemma 17.

6.2 Generalized Dehn filling↩︎

In this section we outline how to perform generalized Dehn filling on the PE manifold with an AH cusp constructed above. This gives a proof of Theorem 4. We follow the approach of [27]. Suppose \((M,h)\) is an ASD PE manifold with an AH cusp that arises from our construction. Since the end \(E_\infty\) is an AH cusp, we may choose a diffeomorphism identifying \(E_\infty\) with \((0,\epsilon_0)\times T^3\) such that, for some small \(\delta_0>0\), \[\label{eq:asymptotic-to-cusp} h=\frac{ds^2}{s^2}+s^2g_{T^3}+O(s^{\delta_0})=h_{\mathop{\mathrm{\mathbb{RH}}}^4_{cusp}}+O(s^{\delta_0}).\tag{62}\] Here the cusp is at \(s=0\), and the asymptotics are understood in the sense that \(|\nabla_{h_{\mathop{\mathrm{\mathbb{RH}}}^4_{cusp}}}^k (h-h_{\mathop{\mathrm{\mathbb{RH}}}^4_{cusp}})|_{h_{\mathop{\mathrm{\mathbb{RH}}}^4_{cusp}}} = O(s^{\delta_0})\) for all \(k \ge 0\). There is a toral black hole metric on \(\mathop{\mathrm{\mathbb{R}}}^2\times\mathop{\mathrm{\mathbb{R}}}^2\) \[\label{eq:BH-metric} h_{BH}:=\frac{ds^2}{V(s)}+V(s)d\theta^2+s^2 g_{\mathop{\mathrm{\mathbb{R}}}^2},\quad V(s)=s^2-\frac{a}{s},\tag{63}\] where \(s_+=a^{\frac{1}{3}}\), \(s\in[s_+,\infty)\), and the period for \(\theta\) is \(\beta=\frac{4\pi}{3s_+}\). A different choice of \(a\) gives the same metric up to isometry. The metric \(h_{BH}\) is asymptotic to the hyperbolic metric \(\frac{ds^2}{s^2}+s^2(d\theta^2+g_{\mathop{\mathrm{\mathbb{R}}}^2})\) as \(s\to\infty\), with \[\left|h_{BH}-\left(\frac{ds^2}{s^2}+s^2(d\theta^2+g_{\mathop{\mathrm{\mathbb{R}}}^2})\right)\right|<C\frac{a}{s^3}\] for a constant \(C>0\), measured under the hyperbolic metric.

Next we describe the approximation metric. We would like to patch 62 and 63 near \(s=e^{-R}\) for \(R\gg1\). For this, we pick a primitive closed geodesic \(\sigma\) on the flat \(T^3\) with length \(l\gg1\), such that \[\beta^2V(e^{-R})=l^2.\] This is equivalent to \[\frac{16\pi^2(1-e^{3R}a)}{9a^{\frac{2}{3}}e^{2R}}=l^2.\] So once we fix \(l,R\), we can determine \(a\). There is a unique lattice \(\Lambda\subset \mathop{\mathrm{\mathbb{S}}}^1\times\mathbb{R}^2\) such that

• there is an isometry that identifies the quotient of the slice \(s=e^{-R}\) in 63 by \(\Lambda\), which is a flat torus, to the \(T^3\) at \(s=e^{-R}\) of the hyperbolic cusp metric \(h_{\mathop{\mathrm{\mathbb{RH}}}^4_{cusp}}=\frac{ds^2}{s^2}+s^2g_{T^3}\);

• the isometry identifies the \(\theta\)-geodesic in the quotient torus to \(\sigma\).

The quotient of \(h_{BH}\) by \(\Lambda\) yields an Einstein metric on \(\mathop{\mathrm{\mathbb{R}}}^2\times T^2\). Denote \(M\setminus \{0<s<e^{-R}\}\) by \(M_R\) and \(\mathop{\mathrm{\mathbb{R}}}^2\times T^2 \setminus \{s\ge e^{-R}\}\) by \(N_R\). Via the isometry between the boundary tori, we identify the slice \(s=e^{-R}\) of \(h_{\mathop{\mathrm{\mathbb{RH}}}^4_{cusp}}\) with the slice \(s=e^{-R}\) of the quotient toral black hole metric, thereby obtaining the manifold \(M_\sigma = M_R \amalg N_R\). The function \(s\) on the end \((0,\epsilon_0)\times T^3\) patches continuously with the one on \(D^2\times T^2\) at \(s=e^{-R}\), hence we may define the function \(s\) on \(M_\sigma\), via extending it as \(s\equiv\epsilon_0\) outside of the end \((0,\epsilon_0)\times T^3\) in \(M\). We define the approximate metric \(\bar h_{R,l}\) on \(M_\sigma\)

• as \(h\) on the part \(s\geq e^{-R+10}\);

• as the quotient of \(h_{BH}\) on the part \(s\leq e^{-R-10}\);

• we interpolate them smoothly via a cut-off function \(\gamma = \chi(\log(s)+R)\) on \(e^{-R-10} < s < e^{-R+10}\), where \(\chi \in C^\infty(\mathbb{R})\), \(0\leq \chi\leq 1\), \(\chi \equiv 0\) on \([10, \infty)\), and \(\chi \equiv 1\) on \((-\infty, -10]\).

So \(\bar h_{R,l}=(1-\chi) h+\chi h_{BH}\). For the PE end \(E_0\) of \(M\), we can find a geodesic defining function \(r\) and identify the Einstein metric \(h\) with \(\frac{1}{r^2}(dr^2+g_{r})\) on \((0,\epsilon_1/2)_r\times T^3\). The function \(r\) can also be extended to a smooth function on \(M_{\sigma}\) such that \(r\equiv \epsilon_1\) on the region \(\{s<\epsilon_0\}\).

Next we define weighted Hölder norms for symmetric \(2\)-tensors on \(M_\sigma\). Fix weights \[0<\delta<\frac{3}{2},\qquad \frac{3}{2}<\tau<3.\] Set \[\|u\|_{C^{m,\alpha}_{\delta,\tau}(M_\sigma)} := \left\| r^{-\tau}\left(\frac{s}{s_+}\right)^{\delta}u \right\|_{C^{m,\alpha}(M_\sigma)}.\] Here \(m\geq 2, \;0<\alpha<1\), and the \(C^{m,\alpha}\)-norm of a symmetric \(2\)-tensor \(k\) is defined by \[\|k\|_{C^{m,\alpha}(M_\sigma)} := \sup_{p\in M_\sigma} \bigl\|(\exp_p)^*k\bigr\|_{C^{m,\alpha}\left(B_{r_0}(0)\subset T_pM_\sigma\right)}, \qquad r_0:=\frac{1}{2} \inf_{p\in M_\sigma}\operatorname{conj}_{\bar h_{R,l}}(p),\] where \(\exp_p:T_pM_\sigma\to M_\sigma\) is the exponential map with respect to \(\bar h_{R,l}\). Note that the sectional curvatures of \((M_\sigma, \bar{h}_{R,l})\) are uniformly bounded; thus, the conjugacy radius of \((M_\sigma, \bar{h}_{R,l})\) is bounded uniformly from below, independently of \(\sigma\), \(R\), and \(l\).

Recall that \(s\equiv\epsilon_0\) outside a compact region of \(M_\sigma\) while \(r\equiv \epsilon_1\) inside a compact region of \(M_\sigma\). For a background metric \(\bar g\) and a nearby metric \(g\), consider the Einstein operator in the Bianchi gauge \[\Phi_{\bar g}(g)=\mathop{\mathrm{\mathrm{Ric}}}_{g}+3g+\mathop{\mathrm{\mathrm{div}}}_{g}^*\left(\mathop{\mathrm{\mathrm{div}}}_{\bar g}(g)+\frac{1}{2}d\mathrm{tr}_{\bar g}(g)\right),\] whose linearization at \(\bar g\) is \[(d\Phi_{\bar g})_{\bar g}(k)=\frac{1}{2}\Delta_{L}k+3k.\] Here \(\Delta_L\) is the Lichnerowicz Laplace of \(\bar g\), given by \((\Delta_L k)_{ij}=(\nabla^*\nabla k)_{ij}+\mathop{\mathrm{\mathrm{Ric}}}_i{}^p k_{pj}+\mathop{\mathrm{\mathrm{Ric}}}_j{}^p k_{ip} -2R_{ipqj}k^{pq}\). In what follows, we denote \(L_{\bar g}:=(d\Phi_{\bar g})_{\bar g}\) to denote the linearization at \(\bar g\). Note that our ASD PE manifold with an AH cusp \((M,h)\) is non-degenerate [22], in the sense that the kernel of \(L_h:=(d\Phi_h)_h\) is trivial in \(L^2\).

We prove the following uniform estimate of \(L_{\bar h_{R,l}}\) for all sufficiently large \(R\) and \(l\).

It remains to estimate the defect \(\Phi_{\bar h_{R,l}}(\bar h_{R,l})\). Since both \(h\) and \(h_{BH}\) are Einstein, this defect is supported in the transition region \(A_R=\{e^{-R-10}<s<e^{-R+10}\}\). On \(A_R\), relative to \(h_c=s^{-2}ds^2+s^2g_{T^3}\), we have \(h=h_c+O(s^{\delta_0})\) and \(h_{BH}=h_c+O(as^{-3})\), with the same bounds for derivatives. Since \(s\sim e^{-R}\) on \(A_R\), and the cutoff \(\chi(\log s+R)\) has uniformly bounded derivatives, it follows that \[\|\Phi_{\bar h_{R,l}}(\bar h_{R,l})\|_{C^{m-2,\alpha}_{\delta,\tau}} \le C\left(\frac{e^{-R}}{s_+}\right)^\delta \left(e^{-\delta_0R}+ae^{3R}\right).\] Here \(r\equiv \epsilon_1\) on \(A_R\), so the factor \(r^{-\tau}\) is harmless. Using the matching relation \(s_+e^R\sim l^{-1}\), equivalently \(ae^{3R}=(s_+e^R)^3\sim l^{-3}\), we obtain \[\|\Phi_{\bar h_{R,l}}(\bar h_{R,l})\|_{C^{m-2,\alpha}_{\delta,\tau}} \le C\bigl(l^\delta e^{-\delta_0R}+l^{\delta-3}\bigr).\] Thus, for \(0<\delta<\frac{3}{2}\) and for \(R,l\to\infty\) chosen so that \(l^\delta e^{-\delta_0R}\to0\), the defect tends to zero. For \(0<\tau<3\), the operator \(L_{\bar h_{R,l}}:C_{\delta,\tau}^{m,\alpha}\rightarrow C_{\delta,\tau}^{m-2,\alpha}\) is bounded and Fredholm of index zero; by Lemma 14, it is invertible, and \(\|L_{\bar h_{R,l}}^{-1}\|\) is uniformly bounded independently of \(R,l\). Moreover, the nonlinear remainder satisfies the corresponding uniform quadratic estimate. The inverse function theorem therefore gives a unique small perturbation \(u_{R,l}\in C_{\delta,\tau}^{m,\alpha}\) such that \(h_{R,l}:=\bar h_{R,l}+u_{R,l}\) solves \(\Phi_{\bar h_{R,l}}(h_{R,l})=0\) on \(M_\sigma\). Since \(\tau>0\), the Bianchi one-form decays at the PE end; applying the standard Bianchi argument to the gauged equation gives \(\mathop{\mathrm{\mathrm{Ric}}}_{h_{R,l}}+3h_{R,l}=0.\) Thus \(h_{R,l}\) is a Poincaré–Einstein metric on \(M_\sigma\) with the same conformal infinity as \((M,h)\).

6.3 Further Discussion↩︎

The examples constructed in Theorem 1 show that Proposition 5.7 of [30] fails. Their conformal infinities have negative Yamabe constants, except in the flat \(T^3\) case. Recall that for a closed \(n\)-manifold \(N\), \(n\geq 3\), with conformal class \([\gamma]\), the Yamabe constant is \[Y(N,[\gamma]) := \inf_{\tilde{\gamma}\in[\gamma]} \frac{\int_N R_{\tilde{\gamma}}\,d\mathrm{vol}_{\tilde{\gamma}}}{\operatorname{Vol}(N,\tilde{\gamma})^{\frac{n-2}{n}}}.\] We mention that Proposition 5.7 of [30] is true if we assume further that the conformal infinity of a PE end has non-negative Yamabe constant by the result of Witten-Yau [24] and Cai-Galloway [25]. For completeness we recall their argument.

For completeness, we also verify non-local conformal flatness and determine the Yamabe sign intrinsically on the conformal infinities of the metrics we constructed.

7 Liouville theorems↩︎

Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA

E-mail address: mingyang.li@scgp.stonybrook.edu

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA.

E-mail address: hongyil@princeton.edu

References↩︎

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1. We take this opportunity to clarify a point in the literature. Anderson [30] asserted that, under certain mild hypotheses, a limiting PE manifold with an AH cusp must be hyperbolic. However, there appears to be a gap in the proof. The argument involves translating into the cusp region and blowing up the difference between the Einstein metric and the hyperbolic cusp metric. It is claimed that the corresponding blow-up limit exists, but this need not be the case. For instance, examples constructed in the present paper converge to the hyperbolic cusp at a super-exponential rate, in which case the proposed blow-up limit does not arise.

   Craig [31] constructed counterexamples by applying generalized Dehn filling to hyperbolic manifolds with PE ends and AH cusp ends. Nevertheless, that paper still refers to the result of [30], that such a limiting PE manifold with an AH cusp is hyperbolic. The examples in [31] are perturbative in nature and hence remain close to hyperbolic metrics, whereas the examples in the present paper may be far from hyperbolic, since the boundary data can be prescribed.↩︎