January 01, 1970
We define a mass invariant adapted to the spectral scalar curvature for asymptotically hyperbolic 3-manifolds with toroidal infinity and show its positivity under a lower bound on the spectral scalar curvature. In addition, we show a rigidity theorem and some band width estimates under similar assumptions.
A Riemannian manifold \(M\) is called asymptotically flat if there is a compact set \(K\subset M\), such that \(M\setminus K\) is diffeomorphic to \(\Bbb R^n\setminus B_{1}(0)\) and in the standard coordinates in \(\Bbb R^n\) and the metric is close to the flat metric near the infinity. The positive mass theorem (see [1]–[3]) establishes the non-negativity of a geometric quantity called the ADM mass under the assumption of nonnegative scalar curvature. The case of vanishing mass characterizes the Euclidean space.
One can also study a manifold close to the hyperbolic space near the infinity, and such a manifold is called an asymptotically hyperbolic manifold. See [4]–[6] and also an earlier work [7]. Instead, a negative scalar curvature lower bound given by the model hyperbolic space is assumed. Recently, we have seen great progress in the positive mass theorems, in particular the resolution of the higher dimensional positive mass theorem, see [8], [9], [10].
Another model metric of interest in the field of the positive mass theorem is the \(n\)-dimensional hyperbolic cusp \[\mathrm{d} t^2 + e^{2t} g_{\mathbb{T}^{n-1}} \label{horocyclic32model}\tag{1}\] which by a coordinate change \(r = e^t\) is isometric to \[r^{-2}\mathrm{d} r^2 + r^2 g_{\mathbb{T}^{n-1}}. \label{conformal32form}\tag{2}\] In this paper, we are interested in asymptotically hyperbolic 3-manifolds with infinity given by 1 (i.e., \(t, r \to \infty\)) and we give the following definition using the metric 2 .
Definition 1. A manifold \((M, g)\) is called asymptotically hyperbolic with toroidal infinity if there is a compact set \(K\) whose complement is diffeomorphic to a cylinder with torus cross-section and in the coordinates given by the diffeomorphism \(\psi : (1, \infty)\times \mathbb{T}^2 \to M\backslash K\) the metric satisfies \[\psi^{\ast} g = r^{- 2} \mathrm{d} r^2 + r^2 (g_{\mathbb{T}^2} + r^{- \kappa} m) + Q_g \label{asymptotics32g}\qquad{(1)}\] where \(\kappa >0\), \(r \in (1, \infty)\) is the radial coordinate, \(m\) is a symmetric 2-tensor on \(\mathbb{T}^2\) and \(Q_g\) satisfies the asymptotics \[|Q_g | + r | \nabla Q_g | + r^2 | \nabla^2 Q_g | = o (r^{-\kappa}) .\]
It is worth mentioning that such manifolds belong to a more general class of ALH manifolds (see [11], [12]) and the toroidal infinity can also be called a cuspidal end.
Now we recall the mass for such manifolds.
Definition 2 ([5], [13]). Let \(\kappa=1\). If \(r (R_g + 6) \in L^1 (M)\), then \[E = \tfrac{1}{|\mathbb{T}^2 |} \int_{\mathbb{T}^2} 3\operatorname{tr}_{g_{\mathbb{T}^2}} m \mathrm{d} A_{g_{\mathbb{T}^2}}\] is well-defined and called the total energy, and \(\operatorname{tr}_{g_{\mathbb{T}^2}} m\) is called the mass aspect function.
The positive mass theorem for this mass was shown in [11], see also [14] for the 3-dimensional spacetime setting. For the rigidity statement, see [12].
Our main goal is to pursue a spectral positive mass theorem of an asymptotically hyperbolic \(3\)-manifold with toroidal infinity. Such a positive mass theorem deals with the spectral scalar curvature, the spectral mean curvature and requires the notion of a spectral mass. First, we give the following spectral analog of the scalar curvature and the mean curvature.
Definition 3. Given a Riemannian manifold \((M,g)\) (possibly with boundary), a positive number \(\gamma>0\) and a positive function \(u\in C^2(\bar {M})\), the quantity \[-\gamma u^{-1}\Delta_g u + \tfrac{1}{2} R_g \label{scc32def}\qquad{(2)}\] is called the \((\gamma,u)\)-spectral scalar curvature. For a hypersurface \(\Sigma\) in \(M\), \[\gamma u^{-1} u_{\nu} + H\] is called the \((\gamma,u)\)-spectral mean curvature of \(\Sigma\). We often omit the references to \((\gamma,u)\) for convenience. We adopt the convention that the normal of a boundary component of a manifold point to the outside of the manifold; let \(\Sigma\) be a boundary component of \(M\) and \(\nu\) be its normal, and the mean curvature of \(\Sigma\) is given by \(H=\operatorname{div}_\Sigma\nu\).
There are two variations \[\begin{align} S_{c,\gamma} = & -\gamma u^{-1}\Delta_g u + \tfrac{1}{2} R_g + c \gamma u^{-2} |\nabla u|^2 \label{grad32pert} ,\\ P_{ \alpha, \gamma} = & R_g - 2 \gamma \Delta_g f + \alpha |\nabla f | ^2 \label {p form sc} \end{align}\tag{3}\] of the spectral scalar curvature ?? . For the definition of \(P_{\alpha,\gamma}\), see for instance [15]. Evidently, letting \(f = \log u\) in [p form sc] gives the form 3 with suitable values of \(c\). A notable example of [p form sc] is the Perelman’s weighted scalar curvature \[P=R_g + 2 \Delta_g f -|\nabla f |^2 . \label{perelman32sc}\tag{4}\] The form 3 is actually equivalent to ?? by observing that \[- u^{-1}\Delta_g u + c u^{-2} |\nabla u|^2 = - \tfrac{1}{1-c} u^{-(1-c)} \Delta_g u^{1-c} .\] In particular, setting \(f = - 2 \log u\) gives \(P = 2 (-2u^{-1} \Delta_g u + \tfrac{1}{2} R_g)\).
Now we introduce the following spectral mass.
Definition 4. We fix \(0 < \gamma < 2\) and \(\kappa = \tfrac{6 - 2\gamma}{2 - \gamma}\). Let \((M,g)\) be given in Definition 1 and a function \(u\in C^{2}(\bar{M})\) which satisfies \[u = r^{\tfrac{1}{2 - \gamma}} (1 + r^{- \kappa} \zeta + o (r^{- \kappa})). \label{u32asymptotics}\qquad{(3)}\] If \(u^2 (- \gamma u^{-1}\Delta_gu+\tfrac{1}{2}R_g) \in L^1 (M)\), the quantity \[E:= E (M,g,\gamma,u) = \tfrac{4\kappa}{2-\gamma} \int_{T^2} (\tfrac{1}{2} \operatorname{tr}_{g_{\mathbb{T}^2}}m + \gamma \zeta) \sqrt{g_{\mathbb{T}^2}} dA\] is called the spectral mass (or energy) and the quantity \(\tfrac{1}{2}\operatorname{tr}_{g_{\mathbb{T}^2}}m + \gamma \zeta\) on \(T^2\) is called the spectral mass aspect function.
Our definition is related to the Perelman’s weighted scalar curvatrue 4 and the weighted mass in the asymptotically flat setting (see [16], cf. [10]). We can also define the spectral mass with more general asymptotics than ?? , see Chai-Liu-Sun (work in prepration).
Our main result is given in the following which is a positive mass theorem for \(E\) assuming suitable bounds for the spectral scalar curvature and the spectral mean curvature of the boundary.
Theorem 1. Let \((M,g)\) be asymptotically hyperbolic with toroidal infinity as in Definition 1 with \(K\) being empty, and \(u\) be given by Definition 4. If the spectral scalar curvature satisfies \[-\gamma u^{-1}\Delta_g u + \tfrac{1}{2} R_{g} \geq - \tfrac{(3-\gamma)(4-\gamma)}{(2-\gamma)^{2}}\] and the spectral mean curvature of \(\partial M\) satisfies \[\gamma u^{-1}u_{\nu} + H_{\partial M} \geq -\tfrac{4-\gamma}{2-\gamma},\] then the spectral mass \(E(M, g,\gamma, u)\) is non-negative. The spectral mass vanishes if and only if \((M, g)\) is isometric to \[([r_0, \infty) \times \mathbb{T}^2, r^{- 2} \mathrm{d} r^2 + r^2 g_{\mathbb{T}^2})\] for some \(r_0>0\) and \(u\) is equal to \(r^{\tfrac{1}{2 - \gamma}}\).
Remark 2. We can also allow additional boundaries as in [14]. It is easy to see this is a generalization of the non-spectral positive mass theorem (i.e., the case \(\gamma =0\)).
Our approach to Theorem 1 is based on the spacetime harmonic functions introduced in [17] which are tied to three dimensions. It is an interesting question to extend Theorem 1 to higher dimensions.
The application of spacetime harmonic functions also yields some other geometric results under similar assumptions on the spectral scalar curvature. Using similar techniques, we prove a spectral scalar curvature rigidity theorem for toroidal bands or cusps (Theorem 3) and some band width estimates (Theorem 5), which were obtained earlier by Chai-Sun [18] using the warped \(\mu\)-bubble method. For the band width estimates in the case of a positive spectral scalar curvature bound, see [19]. For related scalar curvature rigidity, see [7], [20], [6].
The spectral scalar curvature rigidity is given as follows.
Theorem 3 (cf. [18]). Let \(M = [- 1, 1] \times \mathbb{T}^2\) with a metric \(g\) and \(u\) be a positive function such that \[\label{scc1} - \gamma u^{- 1} \Delta_g u + \tfrac{1}{2} R_g \geq \Lambda\qquad{(4)}\] where \(0 \leq \gamma < 3\) and \(\Lambda \leq 0\). Let \(\nu_{\pm}\) be the unit normal of the boundary \(\partial_{\pm} M =\{\pm 1\} \times \mathbb{T}^2\) pointing to the direction pointing to the outside of \(M\). Suppose \[H_{\partial_- M} + \gamma u^{- 1} u_{\nu_-} \geq -\sqrt{- \Lambda \tfrac{4 - \gamma}{3 - \gamma}} \text{ and } H_{\partial_+ M} + \gamma u^{- 1} u_{\nu_+} \geq \sqrt{- \Lambda \tfrac{4 - \gamma}{3 - \gamma}} .\] Then \((M, g)\) is isometric to \[([- 1, 1] \times \mathbb{T}^2, g = dt^2 + e^{2 \alpha t} g_{\mathbb{T}^2}), \text{ } \alpha = \tfrac{(2 - \gamma) \sqrt{- \Lambda}}{\sqrt{(3 - \gamma) (4 - \gamma)}},\] and \(u\) is a constant multiple of \(e^{\beta t}\) where \(\beta = \sqrt{\tfrac{- \Lambda}{(3 - \gamma) (4 - \gamma)}}\).
Remark 4. It should be observed that when \(0< \gamma <2\), the metrics in Theorem 3 and Theorem 1 are in fact the same, the 3-dimensional hyperbolic cusp \(\mathrm{d}t^2 + e^{2t} g_{\mathbb{T}^2}\) and \(u\) is \(e^{t / (2-\gamma)}\). The difference only come from by a coordinate change and a scaling, see 1 and 2 . We also call Theorem 3 a spectral cuspidal rigidity.
Now we turn to the band width estimates, which is a research direction pioneered by Gromov [21]. He established that if a toroidal band with positive scalar curvature has bounded width which is defined to be the distance between two boundaries, see also [22], [23], [24].
To facilitate the description of the band width estimates, we introduce some notations. Let \(\Gamma\) and \(\Lambda\) be two constants and we are concerned with ODE \[\Gamma \eta^2 + \eta' + \Lambda = 0 \label{eq32general32ode}\tag{5}\] such that the solution \(\eta\) satisfies \(\eta' < 0\). To ensure \(\eta' < 0\), at least one of \(\Gamma\) and \(\Lambda\) should be positive. Indeed the solution to 5 is given by the following \[\eta (t) := \eta_{\Lambda, \Gamma} (t) := \left\{ \begin{array}{lc} \sqrt{- \Lambda / \Gamma} \coth \left( \sqrt{- \Lambda / \Gamma} t \right), & \Gamma > 0, \Lambda < 0;\\ \frac{1}{\Gamma t}, & \Gamma > 0, \Lambda = 0;\\ \sqrt{\Lambda / \Gamma} \cot \left( \sqrt{\Lambda / \Gamma} t \right), & \Gamma > 0, \Lambda > 0. \end{array} \right.\] Evidently, \(\eta_{\Lambda, \Gamma}\) is only well defined on the interval \(I_{\Lambda, \Gamma}\) given by \[I_{\Lambda, \Gamma} = \left\{ \begin{array}{cc} (0, \infty), & \Gamma > 0, \Lambda \leq 0;\\ \left( 0, \pi / \sqrt{\Lambda \Gamma} \right), & \Gamma > 0, \Lambda > 0. \end{array} \right.\]
Theorem 5 (cf. [18]). Let \(\Lambda\) be a constant, \(0 \leq \gamma < 3\) and \(\Gamma = \frac{3 - \gamma}{4 - \gamma}\). Assume that at least one of \(\Gamma\) and \(\Lambda\) is positive. Let \(M=[-1,1]\times\mathbb{T}^2\), and \(t_- < t_+\) be two numbers such that
(a) there exists a positive function \(u\) with \(- \gamma u^{- 1} \Delta_g u + \frac{1}{2} R_g \geq \Lambda\);
(b) \(H_{\partial_+ M} + \gamma u^{- 1} u_{\nu_+} \geq \eta (t_+)\) on \(\partial_+ M\), \(H_{\partial_- M} + \gamma u^{- 1} u_{\nu_-} \geq - \eta (t_-)\) on \(\partial_- M\),
then \[\operatorname{width} (M, g) \leq t_+ - t_-.\] The equality occurs if and only if \((M, g)\) is isometric to the model \[([t_-, t_+] \times T^2, d t^2 + \phi (t)^2 g_{\mathbb{T}^2})\] where \(\phi (t) = \exp \left( \frac{2 - \gamma}{4 - \gamma} \int^t \eta \right)\) and \(g_{\mathbb{T}^{n - 1}}\) is some flat metric on \(\mathbb{T}^{n - 1}\) and \(u\) is a constant multiple of \(\exp \left( \frac{1}{4 - \gamma} \int^t \eta \right)\).
The article is organized as follows:
In Section 2, we review spacetime harmonic functions and establish a spectral analogue of a fundamental integral inequality. In Section 3, we apply this integral inequality to prove Theorems 3 and 5. Finally, in Section 4, we prove Theorem 1.
Yimin Chen was supported by a start-up grant from Hainan University (XJ2600000222). Juncheol Pyo was supported by the National Research Foundation of Korea (RS-2025-23524266).
The primary tool employed in this paper is the spacetime harmonic function. A triple \((M^3, g, k)\) consisting of a 3-Riemannian manifolds \((M^3,g)\) and a symmetric 2-tensor \(k\) is called an initial data set. Let us denote \(\Delta\), \(\nabla\) and \(\nabla^2\) the Laplacian, Levi-Civita connection and Hessian on \(M\), respectively.
Definition 5. A positive smooth function \(u\) on \((M^3, g, k)\) is called a spacetime harmonic function with \(k=f g\) for a smooth function \(f\) on \(M\) if it satisfies \[\Delta u + 3 f | \nabla u | = 0. \label{shf}\qquad{(5)}\]
We denote by \(\overline{\nabla}^2\) the modified Hessian given by \[\overline{\nabla}^2 v=\nabla^2 v+f|\nabla v|g\] for any \(v\in C^2(M)\). Then the equation ?? can be simply written as \[\overline{\Delta} v=\operatorname{tr}\overline{\nabla}^2 v=0.\] In [17], an application of the Bochner formula and the Gauss-Bonnet theorem yields the following integral inequality.
Proposition 6 ([17]). Let \((M, \partial_{\pm}M, g)\) be a 3-dimensional Riemannian band, and let \(f\in C^{\infty}(M)\). Let \(v\) be a spacetime harmonic function with \(v = \pm c\) on \(\partial_{\pm} M\), respectively. Then, we have the following integral inequality \[\begin{align} & \int_{\partial_- M} 2 | \nabla v | (2 f - H_{\partial_- M}) d A - \int_{\partial_+ M} 2 | \nabla v | (2 f + H_{\partial_+ M}) d A\\ \geq & \int_{M^3} \left( \frac{| \bar{\nabla}^2 v |^2}{| \nabla v |} + (R_g + 6 f^2) | \nabla v | - 4 \langle \nabla f, \nabla v \rangle \right) - \int_{c_-}^{c_+} 4 \pi \chi (\Sigma_t) d t \end{align}\] where \(\bar{\nabla}^2 v = \nabla^2 v + f | \nabla v | g\).
In order to deal with the condition on spectral curvature condition, we need a spectral analog of Proposition 6.
Proposition 7. For the function \(v\) which satisfies \[\Delta_g v + 3 f | \nabla v| = 0\] with \(v = c_{\pm}\) on \(\partial_{\pm} M\) the following inequality holds \[\begin{align} & \int_{\partial_- M} | \nabla v| \left( \tfrac{3 (4 - \gamma)}{3 - \gamma} f - 2 (H_{\partial_- M} + \gamma u^{- 1} u_{\nu_-}) \right) \nonumber\\ & \qquad - \int_{\partial_+ M} | \nabla v| \left( \tfrac{3 (4 - \gamma)}{3 - \gamma} f + 2 (H_{\partial_+ M} + \gamma u^{- 1} u_{\nu_+}) \right)\nonumber \\ \geq & (6 - 2 \gamma) \int_M \left| \nabla | \nabla v|^{\tfrac{1}{2}} + \tfrac{3}{6 - 2 \gamma} f| \nabla v|^{- \tfrac{1}{2}} \nabla v \right|^2 - \int_{c_-}^{c_+} 4 \pi \chi (\Sigma_t) \mathrm{d} t \nonumber\\ & \quad + \int_M (2 (- \gamma u^{- 1} \Delta_g u + \tfrac{1}{2} R_g) + \tfrac{9 (4 - \gamma)}{2 (3 - \gamma)} f^2 - \tfrac{3 (4 - \gamma)}{3 - \gamma} \langle \nabla f, \tfrac{\nabla v}{| \nabla v|} \rangle) | \nabla v| . \end{align}\]
Proof. We give a proof for \(v\) with its gradient non-vanishing everywhere. For the general case, we can apply similar arguments for \(\varphi = \sqrt{| \nabla v|^2 + \varepsilon^2}\), \(\varepsilon > 0\) and then take limits, see [17].
First, we note that the divergence theorem and that \(v\) is spacetime harmonic yields \[\begin{align} & - \tfrac{\gamma}{3 - \gamma} \int_{\partial M} f \nabla_{\nu} v \\ = & - \tfrac{\gamma}{3 - \gamma} \int_M \operatorname{div}_g (f \nabla v) \\ = & - \tfrac{\gamma}{3 - \gamma} \int_M (\langle \nabla f, \nabla v \rangle + f \Delta_g v) \\ = & - \tfrac{\gamma}{3 - \gamma} \int_M (\langle \nabla f, \nabla v \rangle - 3 f^2 | \nabla v|) . \end{align}\] Since \(v = c_{\pm}\) on \(\partial_{\pm} M\) and \(c_+ > c_-\), \(\nabla_{\nu} v = \pm | \nabla v|\) on \(\partial_{\pm} M\). Hence, \[- \tfrac{\gamma}{3 - \gamma} \left( \int_{\partial_+ M} f| \nabla v| - \int_{\partial_- M} f| \nabla v| \right) = - \tfrac{\gamma}{3 - \gamma} \int_M (\langle \nabla f, \nabla v \rangle - 3 f^2 | \nabla v|) . \label{exra32div32term}\tag{6}\] We add 6 to the integral inequality in Proposition 6, we obtain that \[\begin{align} & \int_{\partial_- M} | \nabla v| \left( \tfrac{3 (4 - \gamma)}{3 - \gamma} f - 2 H_{\partial_- M} \right) - \int_{\partial_+ M} | \nabla v| \left( \tfrac{3 (4 - \gamma)}{3 - \gamma} f + 2 H_{\partial_+ M} \right) \\ \geq & \int_M \left( \frac{| \bar{\nabla}^2 v|^2}{| \nabla v|} + (R_g + \tfrac{3 (6 - \gamma)}{3 - \gamma} f^2) | \nabla v| - \tfrac{3 (4 - \gamma)}{3 - \gamma} \langle \nabla f, \nabla v \rangle \right)\\ &- \int_{c_-}^{c_+} 4 \pi \chi (\Sigma_t) \mathrm{d} t. \label{before32kato} \end{align}\tag{7}\] We make use of a Kato type inequality (see [24] Remark 4.4) on \(\frac{| \bar{\nabla}^2 v|^2}{| \nabla v|}\) as \[\begin{align} & \frac{| \bar{\nabla}^2 v|^2}{| \nabla v|} \\ \geq & \frac{3 | \nabla | \nabla v| + f \nabla v|^2}{2 | \nabla v|} \\ = & 6 | \nabla | \nabla v|^{\tfrac{1}{2}} |^2 + 6 \langle \nabla | \nabla v|^{\tfrac{1}{2}}, f | \nabla v|^{- \tfrac{1}{2}} \nabla v \rangle + \tfrac{3}{2} f^2 | \nabla v| \\ = & (6 - 2 \gamma) \left| \nabla | \nabla v|^{\tfrac{1}{2}} + \tfrac{3}{6 - 2 \gamma} f| \nabla v|^{- \tfrac{1}{2}} \nabla v \right|^2 + 2 \gamma | \nabla | \nabla v|^{\tfrac{1}{2}} |^2 + (\tfrac{3}{2} - \tfrac{9}{6 - 2 \gamma}) f^2 | \nabla v| . \end{align}\] Hence with the above in 7 , we see \[\begin{align} & \int_{\partial_- M} | \nabla v| \left( \tfrac{3 (4 - \gamma)}{3 - \gamma} f - 2 H_{\partial_- M} \right) - \int_{\partial_+ M} | \nabla v| \left( \tfrac{3 (4 - \gamma)}{3 - \gamma} f + 2 H_{\partial_+ M} \right) \\ & \geq\;(6 - 2 \gamma) \int_M \left| \nabla | \nabla v|^{\tfrac{1}{2}} + \tfrac{3}{6 - 2 \gamma} f| \nabla v|^{- \tfrac{1}{2}} \nabla v \right|^2 + \int_M 2 \gamma | \nabla | \nabla v|^{\tfrac{1}{2}} |^2 \tag{8} \\ & \quad + \int_M \left( (R_g + \tfrac{9 (4 - \gamma)}{2 (3 - \gamma)} f^2) | \nabla v| - \tfrac{3 (4 - \gamma)}{3 - \gamma} \langle \nabla f, \nabla v \rangle \right) - \int_{c_-}^{c_+} 4 \pi \chi (\Sigma_t) \mathrm{d} t.\tag{9} \end{align}\] Now we give an estimate of \(\int_M | \nabla | \nabla v|^{\tfrac{1}{2}} |^2\) in terms of \(u\): by integration by parts, \[\begin{align} &- \int_M \varphi^2 u^{- 1} \Delta_g u + \int_{\partial M} \varphi^2 u^{- 1} u_{\nu} \\ = & - \int_M \varphi^2 u^{- 1} \Delta_g u + \int_M div_g (\varphi^2 u^{- 1} \nabla u) \\ = & \int_M \langle \nabla (\varphi^2 u^{- 1}), \nabla u \rangle \\ = & - \int_M | \nabla \varphi - \tfrac{\varphi \nabla u}{u} |^2 + \int_M | \nabla \varphi |^2 \leq \int_M | \nabla \varphi |^2 . \end{align}\] Letting \(\varphi = | \nabla v|^{\tfrac{1}{2}}\) gives \[- \int_M u^{- 1} \Delta_g u | \nabla v| + \int_{\partial M} u^{- 1} u_{\nu} | \nabla v| \leq \int_M | \nabla | \nabla v|^{\tfrac{1}{2}} |^2 .\] The inequality above and 8 then finish the proof. ◻
In this section, we prove the spectral scalar rigidity for cusps (Theorem 3) and the band width estimate (Theorem 5). The idea is to select a suitable \(f\) in Proposition 7 and then perform the rigidity analysis.
Now we give the proof for Theorem 3.
Proof of Theorem 3. Let \(v\) be the function which solves \[\Delta_g v - (6 - 2 \gamma) \sqrt{\tfrac{- \Lambda}{(3 - \gamma) (4 - \gamma)}} | \nabla v| = 0\] with \(v = c_{\pm}\) on \(\partial_{\pm} M\), that is, we set \(f = - \tfrac{6 - 2 \gamma}{3} \sqrt{\tfrac{- \Lambda}{(3 - \gamma) (4 - \gamma)}}\) in Proposition 7. This leads to an integral inequality for \(v\) as follows \[\begin{align} & 2 \int_{\partial_- M} | \nabla v| \left( - \sqrt{- \Lambda \tfrac{4 - \gamma}{3 - \gamma}} - (H_{\partial_- M} + \gamma u^{- 1} u_{\nu_-}) \right) \\ & \qquad - 2 \int_{\partial_+ M} | \nabla v| \left( - \sqrt{- \Lambda \tfrac{4 - \gamma}{3 - \gamma}} + (H_{\partial_+ M} + \gamma u^{- 1} u_{\nu_+}) \right) \\ \geq & (6 - 2 \gamma) \int_M \left| \nabla | \nabla v|^{\tfrac{1}{2}} - \sqrt{\tfrac{- \Lambda}{(3 - \gamma) (4 - \gamma)}} | \nabla v|^{- \tfrac{1}{2}} \nabla v \right|^2 - \int_{c_-}^{c_+} 4 \pi \chi (\Sigma_t) \mathrm{d} t \\ & \quad + \int_M 2 (- \gamma u^{- 1} \Delta_g u + \tfrac{1}{2} R_g - \Lambda) | \nabla v| . \end{align}\] It follows that \(\chi (\Sigma_t) \leq 0\) from the toroidal structure of \(M\); then we apply the spectral scalar curvature and spectral mean curvature bound, we obtain that \[\nabla | \nabla v|^{\frac{1}{2}} - \sqrt{\tfrac{- \Lambda}{(3 - \gamma) (4 - \gamma)}} | \nabla v|^{- \frac{1}{2}} \nabla v = 0 \label{gradeq}\tag{10}\] almost everywhere. This implies that \[\nabla | \nabla v| - 2 \sqrt{\tfrac{- \Lambda}{(3 - \gamma) (4 - \gamma)}} | \nabla v| \nabla v = 0.\] Since \(| \nabla v| \neq 0\) almost everywhere, the metric splits, that is, \(g = \frac{dv^2}{| \nabla v|^2} + g_v\), where \(g_v\) is a family of metrics on \(\mathbb{T}^2\).
Firstly from \[d (| \nabla v|^2) = 2| \nabla v|d (| \nabla v|) = \tfrac{4 \sqrt{- \Lambda}}{\sqrt{(3 - \gamma) (4 - \gamma)}} | \nabla v|^2 dv,\] we have \[d (| \nabla v|^2) \wedge dv = 0, \label{integrable}\tag{11}\] and therefore \(d \left( \frac{\nabla v}{| \nabla v|} \right) = 0\) for \(| \nabla v| \neq 0\), which implies that there exist \(t\), such that \(dt = dv / | \nabla v|\), and the metric can be written by \[g = dt^2 + \phi^2 (t) g_0 .\] From the argument in [19], \[\begin{align} 2 \frac{\phi'}{\phi} = H (t) = \frac{\Delta v - \nabla_{tt} v}{| \nabla v|} & = & \frac{2 \alpha (\gamma - 3)}{\gamma - 2} + \frac{2 \alpha}{\gamma - 2} = 2 \alpha, \end{align}\] therefore we have \(\phi = e^{\alpha t}\). Therefore, the metric is given by \[g = dt^2 + e^{2 \alpha t} g_{\mathbb{T}^2} .\] Tracing back all the equalities in Proposition 7, we see from 9 that \[\frac{\nabla|\nabla v|^{\frac{1}{2}}}{|\nabla v|^{\frac{1}{2}}}=\frac{\nabla u}{u}\] \(u\) must be a constant multiple of \(| \nabla v|^{\tfrac{1}{2}}\), and by a direct calculation, it is a constant multiple of \(e^{\beta t}\). ◻
Now we prove the band width estimate Theorem 5.
Proof. We prove it by contradiction. Suppose that \(\operatorname{width} (M, g) \geq t_+ - t_-\). Let \(\zeta (x) = \min \{ t_+, t_- +\operatorname{dist}_g (x, \partial_- M) \}\). Then it is easy to see from the assumption that \[\zeta ({\partial_\pm M}) = t_\pm, \text{ and } | \nabla \zeta (x) | \leq 1. \label{zeta32property}\tag{12}\] Let \(v\) be the spacetime harmonic function given in Proposition 7 with \(f = - \tfrac{2 (3 - \gamma)}{3 (4 - \gamma)} \eta \circ \zeta\). A direct calculation with this choice of \(f\) in Proposition 7 leads to \[\begin{align} & 2 \int_{\partial_- M} | \nabla v| (-\eta \circ \zeta - (H_{\partial_- M} + \gamma u^{- 1} u_{\nu_-})) \tag{13} \\ & \qquad - 2 \int_{\partial_+ M} | \nabla v| (-\eta \circ \zeta + (H_{\partial_+ M} + \gamma u^{- 1} u_{\nu_+})) \tag{14} \\ \geq & (6 - 2 \gamma) \int_M \left| \nabla | \nabla v|^{\tfrac{1}{2}} - \tfrac{1}{4 - \gamma} \eta \circ \zeta | \nabla v|^{- \tfrac{1}{2}} \nabla v \right|^2 - 4 \pi \int_{c_-}^{c_+} \chi (\Sigma_t) \mathrm{d} t \\ & \quad + \int_M 2 \left( \tfrac{3 - \gamma}{4 - \gamma} (\eta \circ \zeta)^2 + \langle \nabla (\eta \circ \zeta), \tfrac{\nabla v}{| \nabla v|} \rangle + (- \gamma u^{- 1} \Delta_g u + \tfrac{1}{2} R_g) \right) | \nabla v| . \tag{15} \end{align}\] We estimate the lines 13 , 14 and 15 . First, we estimate the integrands in the lines [neg boundary line] and 14 . Using the assumptions and 12 , we see that \[\gamma u^{- 1} u_{\nu_+} + H_{\partial_+ M} \geq \eta \circ \zeta \text{ on } \partial_+ M \text{ and } \gamma u^{- 1} u_{\nu_-} + H_{\partial_- M} \geq -\eta \circ \zeta \text{ on } \partial_- M. \label{mc32by32model}\tag{16}\] Next, by the chain rule and that \(| \nabla \zeta | \leq 1\), \[\langle \nabla (\eta \circ \zeta), \tfrac{\nabla v}{| \nabla v|} \rangle = \eta' \circ \zeta \langle \nabla \zeta, \tfrac{\nabla v}{| \nabla v|} \rangle \geq \eta' \circ \zeta ;\] and \(- \gamma u^{- 1} \Delta_g u + \tfrac{1}{2} R_g \geq \Lambda\), so the integrand of the line 15 is greater than \(2 ((\eta \circ \zeta)^2 + \eta' \circ \zeta + \Lambda) |\nabla v|\) which vanishes by [eq general ode]. Lastly by the toroidal structure, \(\chi (\Sigma_t) \leq 0\). Therefore, we can conclude that \[\nabla | \nabla v|^{\tfrac{1}{2}} - \tfrac{1}{4 - \gamma} \eta \circ \zeta | \nabla v|^{- \tfrac{1}{2}} \nabla v = 0\] almost everywhere, equivalently, \[\nabla | \nabla v| - \tfrac{2}{4 - \gamma} \eta \circ \zeta | \nabla v| \nabla v = 0.\] In the case that equality holds such that the width is \(t_+ - t_-\), the metric can be split into \(g = d t^2 + \phi (t)^2 g_{\mathbb{T}^2}\), where \(d t = | \nabla v |^{- 1} d v\) and \(\phi\) satisfies that \[2 \frac{\phi'}{\phi} = \frac{\Delta v - \nabla_{t t} v}{| \nabla v |} = \frac{4 - 2 \gamma}{4 - \gamma} \eta,\] therefore we have \(\phi (t) = \exp \left( \frac{2 - \gamma}{4 - \gamma} \int^t \eta \right)\).
Tracing back all the equalities in Proposition 7, we see from 9 that \(u\) must be a constant multiple of \(| \nabla v|^{\tfrac{1}{2}}\), and by a direct calculation, it is a constant multiple of \(\exp \left( \frac{1}{4 - \gamma} \int^t \eta \right)\). ◻
In this section, we prove our main result the spectral positive mass theorem (Theorem 1). The proof of the spectral positive mass theorem deals with the same spacetime harmonic function as in the proof of Theorem 3, see Remark 4.
Let \(T_\rho\) be the constant radial coordinate torus in the asymptotic end, and set \(M_\rho\) to be the bounded component of \(M\backslash T_\rho\).
We start with a solution of the spacetime harmonic function \(w\) which solves \[\label{w} \Delta_g w + 3f |\nabla w| =0 \text{ in } M_\rho\text{, }w =0 \text{ on } \partial_-M_\rho, \; w =1 \text{ on } \partial_+ M_\rho,\tag{17}\] where we set \(f = - \tfrac{2(3-\gamma)}{3(2-\gamma)}.\) Let \(v_\rho = \rho^{\frac{2}{2-\gamma}} w\).
Lemma 1. There exists constants \(C > 0\) and \(r_{\ast} > 1\), such that for all \(\rho > r_{\ast}\), \[|v_{\rho} - r^{\tfrac{2}{2 - \gamma}} | \leq C \text{ on } M_{\rho} \backslash M_{r_{\ast}} .\]
Proof. Let \(M_\rho = [r_0, \rho] \times \mathbb{T}^2\) for any \(\rho>r_0\), \(\partial_+ M_\rho = \{\rho\} \times \mathbb{T}^2\) and \(\partial_- M_\rho = \{r_0 \} \times \mathbb{T}^2\). For any \(\rho > r_0\), we denote \(w_{\rho }\) the spacetime harmonic function with \(w_\rho = 0\) on \(\partial_- M_\rho\) and \(w_\rho = 1\) on \(\partial_+ M_\rho\).
Let \(r_1\in(r_0,+\infty)\) be determined later. By the Hopf lemma, \[\tfrac{\partial w_{r_1}}{\partial \nu_{r_1}} > 0 \text{ along } \partial_+ M_{r_1} . \label{normal32derivative32wr1}\tag{18}\] Define \[z^+ = \left\{\begin{array}{ll} c_1 w_{r_1} & \text{ on } M_{r_1},\\ r^{\tfrac{2}{2 - \gamma}} + (c_1 - r_1^{\frac{2}{2-\gamma}} - \lambda r_1^{- 2}) + \lambda r^{- 2} & \text{ on } M\backslash M_{r_1} . \end{array}\right. \label{z32plus}\tag{19}\] We show that \(z^+\) is a super solution on \(M\backslash M_{r_1}\) if \(\lambda\) and \(r_1\) are chosen appropriately.
By the asymptotics ?? of \(g\), \[\det g = r^2 (1 + r^{- \kappa} \operatorname{tr}_g m + o (r^{- \kappa})), \label{full32det}\tag{20}\] and \[g^{r r} = r^2 (1 + o (r^{- \kappa})) . \label{inverse32metric32rr}\tag{21}\] So for all sufficiently large \(r > r_1\), \[\begin{align} \Delta z^+ = & \tfrac{1}{\sqrt{\det g}} \partial_r (g^{r r} \sqrt{\det g} \partial_r z^+) \\ = & \tfrac{2}{2 - \gamma} r^{\tfrac{2}{2 - \gamma}} (\kappa - \tfrac{\kappa}{2} r^{- \kappa} tr_g m + o (r^{- \kappa})) \label{laplace32z} \end{align}\tag{22}\] by a tedious calculation. Moreover, \[| \nabla z^+ |^2 = g^{r r} (\partial_r z^+)^2 = r^{\tfrac{4}{2 - \gamma}} (\tfrac{4}{(2 - \gamma)^2} - 2 \lambda r^{- \kappa} + o (r^{- \kappa})),\] and hence \[| \nabla z^+ | = r^{\tfrac{2}{2 - \gamma}} (\tfrac{2}{2 - \gamma} - \lambda \tfrac{2 - \gamma}{2} r^{- \kappa} + o (r^{- \kappa})) . \label{grad32z32length}\tag{23}\] It follows from 22 and 23 that \[\Delta z^+ - \kappa | \nabla z^+ | = \kappa \tfrac{2 - \gamma}{2} r^{- 2} (\lambda - \tfrac{2}{(2 - \gamma)^2} tr_g m + o (1))< 0 \label{ss32prop32outside}\tag{24}\] on \(M\backslash M_{r_1}\) if we choose \(\lambda < \inf_{\mathbb{T}^2} \tfrac{2}{(2 - \gamma)^2} tr_g m\) and \(r_1\) sufficiently large.
Now we show that \(z^+\) is a weak super solution on \(M\) if \(c_1\) is chosen appropriately. Indeed, first observe that by the choice of \(r_1\) above, \(z^+\) is a super solution on \(M\backslash M_{r_1}\); on \(M_{r_1}\), \(z^+ = w_{r_1}\) is spacetime harmonic.
With 18 , we choose \(c_1\) such that \[\begin{align} c_1 \tfrac{\partial w_{r_1}}{\partial \nu_{r_1}} \geq & \tfrac{\partial}{\partial \nu_{r_1}} (r^{\tfrac{2}{2 - \gamma}} + (c_1 - r_1 - \lambda r_1^{- 2}) + \lambda r^{- 2})\\ = &r^{\tfrac{2}{2 - \gamma}} (\tfrac{2}{2 - \gamma} - 2 \lambda r^{- \kappa} + o (r^{- \kappa}))\label{bdry32deri32comparison} \end{align}\tag{25}\] holds along \(T_{r_1}\). The following steps the lines are almost identical to [14]. For the sake of completeness, we present it here with our notation.
It is easy to see that the spacetime harmonic function \(v_\rho\) satisfies the following boundary condition \[\left\{\begin{array}{ll} v_\rho=\rho^{\frac{2}{2-\gamma}} & on\;\partial_+ M_\rho \\ v_\rho=0 & on\;\partial_-M_\rho \end{array}\right.\]
From the definition of \(z^+\) and 24 , the function \(z^+-v_\rho\) is a super solution for a linear elliptic equation with bounded coefficients as follows \[\mathcal{L}(z^+- v_\rho):=\Delta(z^+- v_\rho)+\kappa\frac{\nabla(z^++ v_\rho)}{|\nabla z^+|+|\nabla v_\rho|}\nabla(z^+- v_\rho)\leq 0\] holds both on \(M_{r_1}\) and \(M_\rho/M_{r_1}\). We denote \(\overrightarrow{\mathcal{G}}=\kappa\frac{\nabla(z^++v_\rho)}{|\nabla z^+|+|\nabla v_\rho|}\) Then for any non-negative test function \(\phi\in C_c^{\infty}(M_{r_1})\), we have \[\begin{align} 0\leq &-\int_{M_{r_1}}\phi\mathcal{L}(z^+-v_{r_1})dV\\ = &\int_{M_{r_1}}(\nabla \phi\cdot\nabla(z^+-v_\rho)-\phi \overrightarrow{\mathcal{G}}\cdot\nabla(z^+-v_\rho))dV\\ &-\int_{T_{r_1}}\phi\frac{\partial}{\partial\nu_{r_1}}(c_1\omega_{r_1}- v_\rho)dA\label{LHS32week32sub32harmonic} \end{align}\tag{26}\] and also \[\begin{align} 0\leq &-\int_{M_\rho\setminus M_{r_1}}\phi\mathcal{L}(z^+-v_{r_1})dV\\ = &\int_{M_\rho\setminus M_{r_1}}(\nabla \phi\cdot\nabla(z^+-v_\rho)-\phi \overrightarrow{\mathcal{G}}\cdot\nabla(z^+-v_\rho))dV\\ &+\int_{T_{r_1}}\phi\frac{\partial}{\partial\nu_{r_1}}(r^{\frac{2}{2-\gamma}}+\lambda r^{-2}-v_\rho)dA\label{RHS32week32subharmonic} \end{align}\tag{27}\] By summing both sides of inequalities 26 and 27 , in the spirit of 25 , we obtain \[\begin{align} &\int_{M_\rho}(\nabla \phi\cdot\nabla(z^+-v_\rho)-\phi \overrightarrow{\mathcal{G}}\cdot\nabla(z^+- v_\rho))dV\\\geq &\int_{T_{r_1}}\phi\frac{\partial}{\partial\nu_{r_1}}(c_1w_{r_1}-r^{\frac{2}{2-\gamma}}-\lambda r^{-2})dA\\ \geq & 0\label{weak32sup32harmonic} \end{align}\tag{28}\]
We derive from the weak maximum principle that \[\inf_{M_{r_1}} (z^+ - v_{r_1}) \geq \inf_{\partial M_{r_1}} (z^+ - v_{r_1}) \geq 0.\] On the other hand, let \(r_2>1\). The construction of a lower barrier \(z^-\) is analogous, which is given by \[z^- = \left\{\begin{array}{ll} c_2 \tilde{w}_{r_2} & \text{ on } M_{r_2},\\ r^{\tfrac{2}{2 - \gamma}} + (c_2 - r_2^{\frac{2}{2-\gamma}} - \chi r_2^{- 2}) + \chi r^{- 2} & \text{ on } M\backslash M_{r_2} . \end{array}\right.\] where \(\tilde{\omega}_{r_2}\) is a spacetime harmonic function satisfying that \(\tilde{\omega}_{r_2}=-1\) on \(\partial_+M_{r_2}\) and \(\tilde{\omega}_{r_2}=0\) on \(\partial_-M_{r_2}\), and similarly \(\chi\) and \(r_2\) are chosen such that \(z^-\) is a sub solution for the spacetime harmonic equation. The same comparison argument proves that \(v_\rho \geq z^-\) on \(M_\rho\) for any \(\rho>r_2\). Therefore, by choosing \(r_*=\max\{r_0,r_1\}\), we have \(z^-\leq v_\rho\leq z^+\) on \(M\rho\) for any \(\rho>r_*\). Hence, the lemma is proved. ◻
Lemma 2. There exists constants \(C > 0\) and \(r_{\ast} > 1\), such that for all \(\rho > r_{\ast}\), \[| \nabla v_{\rho} - \nabla r^{\tfrac{2}{2 - \gamma}} | \leq C \text{ on } M_{\rho} \backslash M_{r_{\ast}} .\]
Proof. The proof is basically the same with [14] with appropriate adjustments similar to those of Lemma 1.
Given \(\rho>r_*\), let \(h_\rho=v_\rho-r^{\frac{2}{2-\gamma}}\). Note that \(h_\rho\) satisfies the following equation \[\Delta h_\rho+\kappa\frac{\nabla(v_\rho+r)}{|\nabla v_\rho|+|\nabla r^{\frac{2}{2-\gamma}}|}\nabla{h_\rho}=-\Delta r^\frac{2}{2-\gamma}-\kappa|\nabla r^{\frac{2}{2-\gamma}}|:=G.\label{equation32for32gradient32estimate}\tag{29}\] It is straightforward to verify that the coefficients of the first-order terms in 29 are uniformly bounded. Fix \(p_0\in\partial_+M_\rho\), and denote by \(B_\epsilon\) (resp. \(B_{\epsilon/2}\)) the geodesic ball centered at \(p_0\) of radius \(\epsilon\) (resp. \(\epsilon/2\)). We fix \(\epsilon>0\) so that it is smaller than the injectivity radius at every point \(x\in M\setminus M_{r_*}\). For \(1<p<\infty\), the boundary \(L^p\)-estimates, together with the condition \(h_\rho=0\) on \(\partial_1^+M_\rho\), then yield \[\|h_\rho\|_{W^{2,p}(B_{\epsilon/2}\cap M_\rho)}\leq C_0(\|G\|_{L^p(B_{\epsilon}\cap M_\rho)}+\|h_\rho\|_{L^p(B_\epsilon\cap M_\rho)})\] Since the metric is asymptotically locally hyperbolic, the constant \(C_0\) is uniform over all \(x_0\in\partial_1^+M_\rho\) and all \(\rho>r_*\). On the other hand, a direct calculation gives \[\Delta r^{\frac{2}{2-\gamma}}+\kappa|\nabla r^{\frac{2}{2-\gamma}}|=- \tfrac{\kappa}{(2 - \gamma)} tr_g m r^{-2} + o (1).\] Therefore, \(G\) is uniformly bounded. Together with the last lemma, this implies that \(h_\rho\) is also uniformly bounded on \(M_\rho\setminus M_1\), independently of \(\rho\). By choosing \(p>3\) and applying the Sobolev embedding theorem, there exists a uniform constant \(C\) such that \[\|h_\rho\|_{C^{1,1-\frac{3}{p}}(B_{\epsilon/2}\cap M_\rho)}\leq C_1\|v_\rho\|_{W^{2,p}(B_{\epsilon/2}\cap M_\rho)}\leq C.\] Interior \(L^p\)-estimates can be used to obtain the same conclusion for balls away from the boundary. The desired result follows. ◻
We derive the asymptotics of the spectral mean curvature in the following lemma.
Lemma 3. The spectral mean curvature of the coordinate torus \(T_r\) satisfies the following \[H + \gamma u^{- 1} u_{\nu} = (2 + \tfrac{\gamma}{2 - \gamma}) - \kappa (\tfrac{1}{2} \operatorname{tr}_{g_{\mathbb{T}^{2}}} m + \gamma \zeta) r^{- \kappa} + o (r^{- \kappa}) .\]
Proof. The asymptotics ?? is written in terms of components of \(g\) as \(g_{r r} = r^{- 2} (1 + o (r^{- \kappa}))\), \(g_{r i} = o (r^{- \kappa})\) and \(g_{i j} = r^2 (\delta_{i j} + r^{- \kappa} m_{i j} + o (r^{- \kappa}))\). Then \(g^{r r} = r^2 (1 + o (r^{- \kappa}))\), \(g^{r i} = o (r^{- \kappa})\). The unit normal \(\nu\) of \(r\)-level set \(T_r\) is given by \[\nu = (g^{r r})^{- \tfrac{1}{2}} g^{r \alpha} \partial_{\alpha} \label{normal}\tag{30}\] where \(\alpha\) ranges from \(1\) to \(3\). The mean curvature \(H\) of \(T_r\) is \[H = \sigma^{i j} \langle \nu, \nabla_{\partial_i} \partial_j \rangle = - \sigma^{i j} (g^{r r})^{- \tfrac{1}{2}} \Gamma_{i j}^r, \label{mean103232curvature}\tag{31}\] where \(\sigma_{i j} = g_{i j}\) is the the induced metric on \(T_r\) and \(\sigma^{i j}\) is its inverse.
We compute componentwise and write the quantities in the form of an asymptotic expansion.
First of all,
\[(g^{r r})^{- \tfrac{1}{2}} = r^{- 1} (1 + o (r^{- \kappa})).\label{grr}\tag{32}\]
Secondly, in terms of \(\sigma\), we have \[\det \sigma = r^4 (1 + r^{- \kappa} tr_{g_{\mathbb{T}^{2}}}m + o (r^{- \kappa})),\label{det}\tag{33}\] and
\[\sigma^{- 1} = (\det \sigma)^{- 1} r^2 \left(\begin{array}{cc} 1 + r^{- \kappa} m_{22} + o (r^{- \kappa}) & - r^{- \kappa} m_{12} + o (r^{- \kappa})\\ - r^{- \kappa} m_{12} + o (r^{- \kappa}) & 1 + r^{- \kappa} m_{11} + o (r^{- \kappa}) \end{array}\right).\label{sigmainverse}\tag{34}\]
Set \(\hat{m} = \left(\begin{array}{cc} m_{22} & - m_{1 2}\\ - m_{1 2} & m_{11} \end{array}\right)\), we have \[\begin{align} \sigma^{i j} = & (\det \sigma)^{- 1} r^2 (\delta_{i j} + r^{- \kappa} \hat{m}_{i j} + o (r^{- \kappa})) \\ = & r^{- 2} (1 - r^{- \kappa} tr_{g_{\mathbb{T}^{2}}} m + o (r^{- \kappa})) (\delta_{i j} + r^{- \kappa} \hat{m}_{i j} + o (r^{- \kappa} )) .\label{sigmainverse2} \end{align}\tag{35}\] Then we compute the Christoffel symbols of \(g\) \[\begin{align} \Gamma_{i j}^r = & \tfrac{1}{2} g^{r r} (g_{i r, j} + g_{j r, i} - g_{i j, r}) + \tfrac{1}{2} g^{r k} (g_{i k, j} + g_{j k, i} - g_{i j, k}) \\ = & - \tfrac{1}{2} g^{r r} g_{i j, r} + o (r^{2 - 2 \kappa}) \\ = & - \tfrac{1}{2} r^2 (1 + o (r^{- \kappa})) \tfrac{\partial}{\partial r} (r^2 (\delta_{i j} + r^{- \kappa} m_{i j} + o (r^{- \kappa}))) + o (r^{2 - 2 \kappa}) \\ = & - r^3 \delta_{i j} - \tfrac{1}{2} (2 - \kappa) r^{- \kappa + 3} m_{i j} + o (r^{- \kappa + 3}) .\label{chsymbol} \end{align}\tag{36}\] By Combining 33 , 34 , 35 , 36 and [mean curvature], we obtain \[\begin{align} H = & - \sigma^{i j} (g^{r r})^{- \tfrac{1}{2}} \Gamma_{i j}^r \\ = & (1 - r^{- \kappa} tr_{g_{\mathbb{T}^{2}}} m+ o (r^{- \kappa})) (\delta_{i j} + r^{- \kappa} \hat{m}_{i j} {+ o (r^{- \kappa}} )) \\ & \quad (1 + o (r^{- \kappa})) (- \delta_{i j} - \tfrac{1}{2} (2 - \kappa) r^{- \kappa} m_{i j} + o (r^{- \kappa})) \\ = & 2 - \tfrac{1}{2} \kappa r^{- \kappa} tr_{g_{\mathbb{T}^{2}}} m + o (r^{- \kappa}).\label{mean32curvature2} \end{align}\tag{37}\] Next we evaluate the term \(u^{- 1} u_{\nu}\). By 30 , we have \[\nu = r (1 + o (r^{- \kappa})) \partial_r + \sum_i o (r^{- \kappa - 1}) \partial_i .\] Therefore, \[\begin{align} u^{- 1} u_{\nu} & = r^{- \tfrac{1}{2 - \gamma}} (1 + r^{- \kappa} \zeta + o (r^{- \kappa}))^{- 1} \\ & \quad (r (1 + o (r^{- \kappa})) \partial_r + \sum_i o (r^{- \kappa - 1}) \partial_i) (r^{\tfrac{1}{2 - \gamma}} (1 + r^{- \kappa} \zeta + o (r^{- \kappa}))) \\ & = r^{- \tfrac{1}{2 - \gamma}} (1 - r^{- \kappa} \zeta + o (r^{- \kappa})) r \partial_r (r^{\tfrac{1}{2 - \gamma}} (1 + r^{- \kappa} \zeta)) + o (r^{- \kappa}) \\ & = \tfrac{1}{2 - \gamma} - \kappa \zeta r^{- \kappa} + o (r^{- \kappa}) . \end{align}\] Hence the spectral mean curvature \(H + \gamma u^{- 1} u_{\nu}\) is given by \[H + \gamma u^{- 1} u_{\nu} = (2 + \tfrac{\gamma}{2 - \gamma}) - \kappa (\tfrac{1}{2} \operatorname{tr}_{g_{\mathbb{T}^{2}}} m + \gamma \zeta) r^{- \kappa} + o (r^{- \kappa}) .\] This completes the proof of the lemma. ◻
With the PDE estimates and the asymptotics of the spectral mean curvature, we can finally prove our main result Theorem 1.
Proof of Theorem 1. We apply Proposition 7 to \(v_\rho\), and we obtain the following integral inequality \[\begin{align} & 2 \int_{\partial_- M_\rho} | \nabla v_\rho| \left( - { \tfrac{4 - \gamma}{2 - \gamma}} - (H_{\partial_- M_{r}} + \gamma u^{- 1} u_{\nu_-}) \right) \tag{38} \\ & \qquad - 2 \int_{\partial_+ M_\rho} | \nabla v_\rho| \left( - { \tfrac{4 - \gamma}{2 - \gamma}} + (H_{\partial_+ M_\rho} + \gamma u^{- 1} u_{\nu_+}) \right) \tag{39} \\ \geq & (6 - 2 \gamma) \int_{M_\rho} \left| \nabla | \nabla v_\rho|^{\tfrac{1}{2}} - {\tfrac{1}{ 2 - \gamma}} | \nabla v_{r}|^{- \tfrac{1}{2}} \nabla v_\rho \right|^2 - \int_{c_-}^{c_+} 4 \pi \chi (\Sigma_t) \mathrm{d} t \tag{40}\\ & \quad + \int_{M_\rho} 2 \left(- \gamma u^{- 1} \Delta_g u + \tfrac{1}{2} R_g +\tfrac{(3-\gamma)(4-\gamma)}{(2-\gamma)^2} \right) | \nabla v_\rho| \tag{41}. \end{align}\] By the assumptions of the theorem, \[\label{bdry32integral} \qquad - 2 \int_{\partial_+ M_\rho} | \nabla v_\rho| \left( - { \tfrac{4 - \gamma}{2 - \gamma}} + (H_{\partial_+ M_\rho} + \gamma u^{- 1} u_{\nu_+}) \right) \geq 0.\tag{42}\]
To show the spectral positive mass theorem, it suffices to show that the above converges to the spectral mass \(E= E(M,g,\gamma,u)\) as \(\rho\to \infty\).
Using Lemma 1, \(v_\rho\) is locally uniformly bounded. Standard elliptic estimates yield locally uniform \(C^{2,\alpha}\) bounds for \(v_\rho\) for any \(\alpha \in (0,1)\). Then we can use the Arzelà-Ascoli lemma to extract a convergent subsequence \(v_{\rho_i}\). We denote the limit by \(v\).
It follows from Lemma 2 that \[|\nabla v_\rho | = |\nabla r ^{\tfrac{2}{2-\gamma}} |\big|_{r=\rho} + O(1) = \tfrac{2}{2-\gamma} \rho^{\tfrac{2}{2-\gamma}} + O(1).\] Note that \(T_\rho = \partial_+ M_\rho\). Hence the boundary integral 42 is \[\begin{align} 0 \leq & - 2 \int_{\partial_+ M_\rho} | \nabla v_\rho | \left( - \tfrac{4 - \gamma}{2 - \gamma} + (H_{\partial_+ M_\rho} + \gamma u^{- 1} u_{\nu_+}) \right) \\ = & 2 \int_{\mathbb{T}^2} (\tfrac{2}{2 - \gamma} \rho^{\tfrac{2}{2 - \gamma}} + O (1)) \kappa (\tfrac{1}{2} \operatorname{tr}_{g_{\mathbb{T}^2}} m + \gamma \zeta) \rho^{- \kappa} (\rho^2 + o (\rho^2)) \sqrt{g_{\mathbb{T}^2}} \\ = & \tfrac{4}{2 - \gamma} \kappa \int_{\mathbb{T}^2} (\tfrac{1}{2} \operatorname{tr}_{g_{\mathbb{T}^2}} m + \gamma \zeta) \sqrt{g_{\mathbb{T}^{2}}} + o (1) = E+o(1) \label{mass32convergence} , \end{align}\tag{43}\] where we have used Lemma 3. Hence, we have shown the spectral positive mass theorem.
It remains to show the case of vanishing spectral mass. To this end, we need to take limits of both sides of the inequality in the lines 38 41 . We only have to deal with the convergence of \[\int_{M_{\rho_i}} \left| \nabla | \nabla v_{\rho_{i}}|^{\tfrac{1}{2}} - {\tfrac{1}{ 2 - \gamma}} | \nabla v_{r_{i}}|^{- \tfrac{1}{2}} \nabla v_{\rho_{i}} \right|^2,\] since the convergence of other terms follows simply from \(C^{2,\alpha}\) convergence. The issue is due to the points where \(\nabla v_{r_{i}}\) vanishes.
To remedy this, we fix a compact subset \(\Omega\) of \(M\), and define \[\Omega_{\epsilon} = \{ x\in \Omega:\text{ } |\nabla v| \geq \epsilon \}\] The convergence \[\left| \nabla | \nabla v_{r_{i}}|^{\tfrac{1}{2}} - {\tfrac{1}{ 2 - \gamma}} | \nabla v_{r_{i}}|^{- \tfrac{1}{2}} \nabla v_{r_{i}} \right|^2 \to \left| \nabla | \nabla v|^{\tfrac{1}{2}} - {\tfrac{1}{ 2 - \gamma}} | \nabla v|^{- \tfrac{1}{2}} \nabla v \right|^2 \text{ on }\Omega_{\epsilon}\] follows again from the \(C^{2,\alpha}\) convergence of \(v_{\rho_i}\) to \(v\). And it follows from Fatou’s lemma that \[\begin{align} & \liminf_{i \to \infty} \int_{\Omega} | \nabla | \nabla v_{\rho_i} |^{\tfrac{1}{2}} - \tfrac{1}{2 - \gamma} | \nabla v_{\rho_i} |^{- \tfrac{1}{2}} \nabla v_{\rho_i} |^2 \\ \geq & \liminf_{i \to \infty} \int_{\Omega_{\varepsilon}} | \nabla | \nabla v_{\rho_i} |^{\tfrac{1}{2}} - \tfrac{1}{2 - \gamma} | \nabla v_{r_i} |^{- \tfrac{1}{2}} \nabla v_{\rho_i} |^2 \\ \geq & \int_{\Omega_{\varepsilon}} | \nabla | \nabla v|^{\tfrac{1}{2}} - \tfrac{1}{2 - \gamma} | \nabla v|^{- \tfrac{1}{2}} \nabla v |^2 . \end{align}\] Hence by letting \(\epsilon\rightarrow 0\), \[\begin{align} & \liminf_{i \to \infty} \int_{\Omega} | \nabla | \nabla v_{\rho_i} |^{\tfrac{1}{2}} - \tfrac{1}{2 - \gamma} | \nabla v_{\rho_i} |^{- \tfrac{1}{2}} \nabla v_{\rho_i} |^2 \\ \geq & \int_{\Omega} | \nabla | \nabla v|^{\tfrac{1}{2}} - \tfrac{1}{2 - \gamma} | \nabla v|^{- \tfrac{1}{2}} \nabla v |^2 . \end{align}\] Considering the above, the convergence 43 and \(\chi(\Sigma_t) = 0\) in the lines 38 41 , we obtain the positive mass inequality, \[\begin{align} E \geq & (6 - 2 \gamma) \int_{M} \left| \nabla | \nabla v|^{\tfrac{1}{2}} - {\tfrac{1}{ 2 - \gamma}} | \nabla v|^{- \tfrac{1}{2}} \nabla v \right|^2 \mathrm{d} t \\ & \quad + \int_{M} 2 (- \gamma u^{- 1} \Delta_g u + \tfrac{1}{2} R_g +\tfrac{(3-\gamma)(4-\gamma)}{(2-\gamma)^2} ) | \nabla v| \\ & \quad + 2 \int_{\partial_- M} | \nabla v| \left( { \tfrac{4 - \gamma}{2 - \gamma}} + (H_{\partial_- M} + \gamma u^{- 1} u_{\nu_-}) \right) . \end{align}\] Vanishing mass then implies that \[\nabla | \nabla v|^{\tfrac{1}{2}} - {\tfrac{1}{ 2 - \gamma}} | \nabla v|^{- \tfrac{1}{2}} \nabla v = 0\] holds almost everywhere. The rest of the proof is similar to that of Theorem 3 (see Remark 4). Therefore, if the mass \(m=0\), \(M\) is isometric to a hyperbolic cusp. ◻