January 01, 1970
We study the local stability of the bridge family \[\Phi(T):=\inf_{u\in\mathcal{A}_T}\|\nabla u\|_{L^2(\mathbb{R}^n_+)}, \qquad T>0,\quad n\ge3,\] where \[\mathcal{A}_T := \Bigl\{ u\in \dot{H}^1(\mathbb{R}^n_+): \|u\|_{L^{\frac{2n}{n-2}}(\mathbb{R}_{+}^n)}=1,\;\|u\|_{L^{\frac{2(n-1)}{n-2}}(\partial\mathbb{R}_{+}^n)}=T \Bigr\},\] and \(\dot{H}^1(\mathbb{R}^n_+)\) is the completion of \(C_c^\infty(\overline{\mathbb{R}^n_+})\) in the norm \(\|\nabla \varphi\|_{L^2(\mathbb{R}^n_+)}\). Let \(\mathcal{M}_T\) denote the set of minimizers of \(\Phi(T)\). We prove that, for every \(T\neq T_E\), there exists \(\alpha_T>0\) such that \[\|\nabla u\|_{L^2(\mathbb{R}_{+}^n)}^2-\Phi(T)^2 \ge \alpha_T\,d_T(u,\mathcal{M}_T)^2 +o\!\bigl(d_T(u,\mathcal{M}_T)^2\bigr) \qquad\text{for all }u\in\mathcal{A}_T,\] where \(T_E\) is the Escobar threshold and \(d_T\) is the distance in \(\dot{H}^1(\mathbb{R}^n_+)\).
Sharp critical Sobolev inequalities play a central role in elliptic PDE, calculus of variations, and conformal geometry. Besides the determination of the optimal constants and the classification of extremals, a fundamental question is quantitative stability: if a function nearly attains equality, must it be close, modulo the natural symmetries, to the manifold of extremals?
In the present paper we study this question for a family of sharp inequalities on the half-space \[\mathbb{H}:=\{x=(x_1,x')\in \mathbb{R}\times \mathbb{R}^{n-1}:x_1>0\}, \qquad \partial\mathbb{H}:=\{x_1=0\}, \qquad n\ge 3,\] We write \(\dot{H}^1(\mathbb{H})\) for the Sobolev space endowed with \(\|u\|_{\dot{H}^1(\mathbb{H})}:=\|\nabla u\|_{L^2(\mathbb{H})}\). The two endpoint models are classical. The first is the sharp Sobolev inequality on \(\mathbb{R}^n\), \[\label{eq:intro-Sobolev} \|\nabla \varphi\|_{L^2(\mathbb{R}^n)}^2 \ge S_n\,\|\varphi\|_{L^{2^*}(\mathbb{R}^n)}^2 \qquad\text{for all }\varphi\in \dot{H}^1(\mathbb{R}^n), \qquad 2^*=\frac{2n}{n-2}.\tag{1}\] The optimal constant and the extremals in 1 were identified by Aubin–Talenti [1], [2]. A quantitative stability result for 1 was later established by Bianchi–Egnell [3]. The second endpoint is the boundary-critical Sobolev trace inequality on \(\mathbb{H}\), \[\label{eq:intro-Escobar} \|\nabla \varphi\|_{L^2(\mathbb{H})}^2 \ge E_n\,\|\varphi\|_{L^{2^\#}(\partial\mathbb{H})}^2 \qquad\text{for all }\varphi\in \dot{H}^1(\mathbb{H}), \qquad 2^\#=\frac{2(n-1)}{n-2}.\tag{2}\] The sharp form of this inequality was proved by Escobar [4]. Quantitative stability for the Escobar inequality 2 and related trace inequalities has also been investigated recently; see for example [5]–[7].
The Sobolev–Escobar bridge inequality couples these two critical quantities 1 –2 . In the Hilbertian case \(p=2\), it is given by the doubly constrained minimization problem \[\label{eq:intro-Phi} \Phi(T)= \inf_{u\in\mathcal{A}_T}\|\nabla u\|_{L^2(\mathbb{H})}, \qquad T>0,\tag{3}\] where \[\label{eq:intro-AT} \mathcal{A}_T = \Bigl\{ u\in \dot{H}^1(\mathbb{H}): \|u\|_{L^{2^*}(\mathbb{H})}=1, \;\|u\|_{L^{2^\#}(\partial\mathbb{H})}=T \Bigr\}.\tag{4}\] The endpoint \(T=0\) corresponds to the sharp Sobolev inequality 1 , \(T=T_E>0\) recovers the sharp Escobar trace inequality 2 . Thus 3 furnishes a one-parameter interpolation between the bulk-critical and boundary-critical problems. The bridge problem 3 is already well understood at the level of existence and classification of extremals. In the general \(p\)-case, this was established by Maggi–Neumayer [8], while in the conformally invariant case \(p=2\) the classification goes back to Carlen–Loss [9]. For further results on the bridge problem, see also Maggi–Neumayer–Tomasetti [10].
We are interested in the quantitative stability of 3 . When \(T\neq T_E\), no Bianchi–Egnell type quantitative stability theorem [3] seems to be available for the bridge problem 3 , even in the Hilbert case \(p=2\). At the special value \(T=T_E\) and for \(p=2\), the bridge problem 3 recovers the Escobar regime 2 , and the corresponding stability was proved in [5]. The present paper is concerned with the case of \(T\neq T_E\).
The main new difficulty is that, away from the endpoint \(T=T_E\), 3 no longer arises as a perturbation of a single scale-invariant inequality, but instead as a doubly constrained family of minimizers. In particular, the second variation must be analyzed on the linearized constraint space 7 , modulo the symmetry directions generated by tangential translations and critical dilations. The key issue, therefore, is to prove that the linearized operator is coercive transverse to the minimizing manifold.
Our approach is to reduce this coercivity problem to a Robin spectral problem on a model ball. After writing \(\psi=U\hat{\phi}\), where \(U\) is a bridge minimizer, the ground-state transform converts the Hessian into a quadratic form 11 for the conformal metric \(g_U=U^{4/(n-2)}|dx|^2\). Because the \(p=2\) bridge extremals are explicitly classified, this conformal metric is isometric to a geodesic ball in \(\mathbb{S}^n\) when \(0<T<T_E\), and to a geodesic ball in \(\mathbb{H}^n\) when \(T>T_E\); see Subsection 2.2. In these variables, the linearized equation becomes a Robin problem on a bounded model space; see Proposition 1, and the symmetry-generated Jacobi fields become explicit \(\ell=1\) modes; see Appendix 4. The proof is then reduced to a kernel identification and spectral-gap argument for the corresponding Robin quadratic form.
For \(\lambda>0\) and \(z\in\mathbb{R}^{n-1}\), define the natural action \[\bigl((\lambda,z)\cdot u\bigr)(x_1,x') := \lambda^{\frac{n-2}{2}} u\bigl(\lambda x_1,\lambda(x'-z)\bigr).\] This action preserves the Dirichlet norm, the \(L^{2^*}(\mathbb{H})\)-norm, and the \(L^{2^\#}(\partial\mathbb{H})\)-norm. Fix \(T>0\), and let \(U_T\in\mathcal{A}_T\) be a positive minimizer. By the classification of the half-space bridge minimizers due to Carlen–Loss [9] in the conformal case \(p=2\), and Maggi–Neumayer [8] in the general case, the full minimizing set is \[\mathcal{M}_T := \{\,\pm(\lambda,z)\cdot U_T:\;\lambda>0,\;z\in\mathbb{R}^{n-1}\,\}.\] For \(u\in\mathcal{A}_T\), we define the distance to the minimizing manifold by \[d_T(u) := \inf_{v\in\mathcal{M}_T}\|\nabla(u-v)\|_{L^2(\mathbb{H})},\] and the bridge deficit by \[\delta_T(u) := \|\nabla u\|_{L^2(\mathbb{H})}^2-\Phi(T)^2.\]
Our main result is a local Bianchi–Egnell type estimate.
Theorem 1. Fix \(T>0\) with \(T\neq T_E\). Then there exists \(\alpha_T>0\) such that \[\delta_T(u) \ge \alpha_T\,d_T(u)^2 + o\!\bigl(d_T(u)^2\bigr) \qquad \text{as } u\in\mathcal{A}_T,\;d_T(u)\to0 .\]
Remark 1. In the first version of this paper, we left open the question whether there exists a constant \(C_T>0\) such that \[\label{eq:stability-global} \delta_T(u)\ge C_T d_T(u)^2 \qquad\text{for all }u\in\mathcal{A}_T\qquad{(1)}\] This question has since been answered affirmatively by Neumayer [11]. More precisely, Neumayer proved a qualitative stability theorem for the Sobolev–Escobar bridge inequality: if \(u_k\in\mathcal{A}_T\) and \(\delta_T(u_k)\to0\), then \(d_T(u_k)\to0\). In the conformal case \(p=2\), combining this qualitative compactness result with Theorem 1 gives ?? for some constant \(C_T>0\).
In this section \(U\) denotes a positive bridge minimizer. The negative component is obtained by the sign change \(U\mapsto -U\), and will not be distinguished in the notation.
Let \(0<U\in\mathcal{A}_T\) be a minimizer of \(E(u)=\int_{\mathbb{H}}|\nabla u|^2\,dx\). Set \(G(u)=\int_{\mathbb{H}}|u|^{2^*}\,dx\), \(H(u)=\int_{\partial\mathbb{H}}|u|^{2^\#}\,dS\). Since \(U\) is a constrained minimizer, there exist \(\mu,\eta\in\mathbb{R}\) such that \(U\) is a critical point of \[\label{eq:L-equation} \mathcal{L}(u):=E(u)-\mu G(u)-\eta H(u).\tag{5}\] Moreover, \(U\) satisfies \[\label{eq:EL-bridge} \left\{ \begin{align} -\Delta U &= \lambda\,U^{2^*-1} &&\text{in }\mathbb{H},\\ \partial_\nu U &= \sigma\,U^{2^\#-1} &&\text{on }\partial\mathbb{H}, \end{align} \right.\tag{6}\] where \(\lambda=\frac{\mu\,2^*}{2}\), \(\sigma=\frac{\eta\,2^\#}{2}\). The linearized constraint space at \(U\) is \[\label{eq:XU-def} \mathcal{X}_U := \left\{ \psi\in \dot{H}^1(\mathbb{H}): \int_{\mathbb{H}}U^{2^*-1}\psi\,dx=0,\; \int_{\partial\mathbb{H}}U^{2^\#-1}\psi\,dS=0 \right\}.\tag{7}\]
The Hessian of the Lagrangian at \(U\) is the quadratic form \[\label{eq:QU-def} Q_U(\psi) = \int_{\mathbb{H}}|\nabla\psi|^2\,dx -\lambda(2^*-1)\int_{\mathbb{H}}U^{2^*-2}\psi^2\,dx -\sigma(2^\#-1)\int_{\partial\mathbb{H}}U^{2^\#-2}\psi^2\,dS .\tag{8}\]
We next use the standard ground-state transform \(\psi=U\hat{\phi}\); see, for example, [12]. Since \[|\nabla(U\hat{\phi})|^2 = U^2|\nabla\hat{\phi}|^2+\nabla U\cdot \nabla(U\hat{\phi}^2),\] an integration by parts, together with 6 , yields \[\label{eq:ground-trans} \int_{\mathbb{H}}|\nabla\psi|^2\,dx = \int_{\mathbb{H}}U^2|\nabla\hat{\phi}|^2\,dx +\lambda\int_{\mathbb{H}}U^{2^*}\hat{\phi}^2\,dx +\sigma\int_{\partial\mathbb{H}}U^{2^\#}\hat{\phi}^2\,dS .\tag{9}\] Substituting 9 into 8 and using \(\psi=U\hat{\phi}\), we obtain \[\label{eq:QU-conj} Q_U(\psi) = \int_{\mathbb{H}}U^2|\nabla\hat{\phi}|^2\,dx -(2^*-2)\lambda\int_{\mathbb{H}}U^{2^*}\hat{\phi}^2\,dx -\beta\int_{\partial\mathbb{H}}U^{2^\#}\hat{\phi}^2\,dS, \quad \beta=(2^\#-2)\sigma.\tag{10}\]
Assume from now on that \(T\neq T_E\), and define \(g_U:=U^{\frac{4}{n-2}}|dx|^2\) on \(\mathbb{H}\). If \(\psi\in C_c^\infty(\overline{\mathbb{H}})\) and \(\hat{\phi}=\psi/U\), then \[dV_{g_U}=U^{2^*}\,dx, \qquad dS_{g_U}=U^{2^\#}\,dS, \qquad \int_{\mathbb{H}}U^2|\nabla\hat{\phi}|^2\,dx = \int_{\mathbb{H}}|\nabla_{g_U}\hat{\phi}|^2\,dV_{g_U}.\] Hence 10 becomes \[\label{eq:QU-metric} Q_U(\psi) = \int_{\mathbb{H}}|\nabla_{g_U}\hat{\phi}|^2\,dV_{g_U} -(2^*-2)\lambda\int_{\mathbb{H}}\hat{\phi}^2\,dV_{g_U} -\beta\int_{\partial\mathbb{H}}\hat{\phi}^2\,dS_{g_U}.\tag{11}\]
We now recall the explicit conformal models in the two nondegenerate branches.
The spherical branch \(0<T<T_E\). After a tangential translation and a critical dilation, \[\label{eq:sphere} U(x)=C_t^{\mathrm S}\,(1+|x-te_1|^2)^{-\frac{n-2}{2}}\tag{12}\] for some \(t\in\mathbb{R}\) and \(C_t^{\mathrm S}>0\), where \(t\) and \(C_t^{\mathrm S}\) are determined by the two constraints 4 ; see Carlen–Loss [9] and Maggi–Neumayer [8]. Writing \(y:=x-te_1\), one has \[g_U = \bigl(C_t^{\mathrm S}\bigr)^{\frac{4}{n-2}}(1+|y|^2)^{-2}|dy|^2.\] If \(\Pi_S^{-1}:\mathbb{R}^n\to \mathbb{S}^n\setminus\{N\}\subset\mathbb{R}^{n+1}\) denotes the inverse stereographic projection from the north pole \(N\), namely \[\Pi_S^{-1}(y) = \left( \frac{2y}{1+|y|^2}, \frac{|y|^2-1}{1+|y|^2} \right), \qquad y\in\mathbb{R}^n,\] see for example [13], then \[(\Pi_S^{-1})^*g_{\mathbb{S}^n} = \frac{4}{(1+|y|^2)^2}|dy|^2.\] Thus, \[g_U=\alpha_t^{\mathrm S}(\Pi_S^{-1})^*g_{\mathbb{S}^n}, \qquad \alpha_t^{\mathrm S}=\frac{(C_t^{\mathrm S})^{4/(n-2)}}{4}.\] Therefore \(\Pi_S^{-1}\) identifies \((\{y_1>-t\},g_U)\) with a geodesic ball in \((\mathbb{S}^n,\alpha_t^{\mathrm S}g_1)\). After composing with a suitable rotation \(O_t\in O(n+1)\), we obtain an isometry \[F_t^{\mathrm S}:(\mathbb{H},g_U)\longrightarrow (B_1(R_t),\,\alpha_t^{\mathrm S}g_1),\] where \(B_1(R_t)\subset \mathbb{S}^n\) denotes the geodesic ball of radius \(R_t\) centered at the north pole.
The hyperbolic branch \(T>T_E\). After a tangential translation and a critical dilation, \[\label{eq:hyperbolic} U(x)=C_t^{\mathrm H}\,(|x-te_1|^2-1)^{-\frac{n-2}{2}}\tag{13}\] for some \(t<-1\) and \(C_t^{\mathrm H}>0\), again determined by the two constraints 4 ; see [8], [9]. Writing \(y=x-te_1\), \[g_U = \bigl(C_t^{\mathrm H}\bigr)^{\frac{4}{n-2}}(|y|^2-1)^{-2}|dy|^2.\] If \(\Pi_H^{-1}:\{y\in\mathbb{R}^n:\;|y|>1\}\to \mathbb{H}^n\subset\mathbb{R}^{n+1}\) denotes the inverse hyperbolic stereographic projection, namely \[\Pi_H^{-1}(y) = \left( \frac{2y}{|y|^2-1}, \frac{|y|^2+1}{|y|^2-1} \right), \qquad |y|>1,\] see for example [14], then \[(\Pi_H^{-1})^*g_{\mathbb{H}^n} = \frac{4}{(|y|^2-1)^2}|dy|^2.\] Thus, \[g_U=\alpha_t^{\mathrm H}(\Pi_H^{-1})^*g_{\mathbb{H}^n}, \qquad \alpha_t^{\mathrm H}=\frac{(C_t^{\mathrm H})^{4/(n-2)}}{4}.\] Therefore \(\Pi_H^{-1}\) identifies \((\{y_1>-t\},g_U)\) with a geodesic ball in \((\mathbb{H}^n,\alpha_t^{\mathrm H}g_{-1})\). After composing with a suitable Lorentz isometry \(L_t\), we obtain an isometry \[F_t^{\mathrm H}:(\mathbb{H},g_U)\longrightarrow (B_{-1}(R_t),\,\alpha_t^{\mathrm H}g_{-1}),\] where \(B_{-1}(R_t)\subset\mathbb{H}^n\) denotes the geodesic ball of radius \(R_t\) centered at a fixed base point \(o=e_{n+1}\).
In both cases we write \[\vartheta= \begin{cases} 1,&0<T<T_E,\\ -1,&T>T_E, \end{cases} \qquad \tilde{g}_t:=\alpha_t g_{\vartheta}, \qquad \kappa_t:=\frac{\vartheta}{\alpha_t},\] and let \[\label{eq:reduction-map} F:(\mathbb{H},g_U)\longrightarrow (B_{\vartheta}(R_t),\tilde{g}_t)\tag{14}\] denote the corresponding isometry. Since \(U\) solves 6 , the scalar-curvature formula for the conformal metric \(g_U\) yields \[\label{eq:lambda-k} \lambda=\frac{n(n-2)}{4}\,\kappa_t, \qquad (2^*-2)\lambda=n\kappa_t.\tag{15}\] Substituting 15 into 11 , we obtain \[\label{eq:QU-metric-k} Q_U(\psi) = \int_{\mathbb{H}}|\nabla_{g_U}\hat{\phi}|^2\,dV_{g_U} - n\kappa_t\int_{\mathbb{H}}\hat{\phi}^2\,dV_{g_U} - \beta\int_{\partial\mathbb{H}}\hat{\phi}^2\,dS_{g_U}.\tag{16}\]
Proposition 1. Assume \(T\neq T_E\). For \(\psi\in C_c^\infty(\overline{\mathbb{H}})\), set \(\hat{\phi}=\frac{\psi}{U}, \phi=\hat{\phi}\circ F^{-1}.\) Then \[Q_U(\psi)=Q_t(\phi),\] where \[\label{eq:QU-ball} Q_t(\phi)= \int_{B_{\vartheta}(R_t)}|\nabla_{\tilde{g}_t}\phi|^2\,dV_{\tilde{g}_t} - n\kappa_t\int_{B_{\vartheta}(R_t)}\phi^2\,dV_{\tilde{g}_t} - \beta\int_{\partial B_{\vartheta}(R_t)}\phi^2\,dS_{\tilde{g}_t}.\qquad{(2)}\] Moreover, the associated bilinear form \(\mathcal{B}_t(\phi,\zeta)=0\) for all \(\zeta\in H^1(B_{\vartheta}(R_t))\) if and only if \(\phi\) is a weak solution of \[\label{eq:Robin} \left\{ \begin{align} -\Delta_{\tilde{g}_t}\phi &= n\kappa_t\,\phi &&\text{in }B_{\vartheta}(R_t),\\ \partial_{\nu_{\tilde{g}_t}}\phi &= \beta\,\phi &&\text{on }\partial B_{\vartheta}(R_t). \end{align} \right.\qquad{(3)}\] Moreover, the linearized constraints 7 are transformed into \[\label{eq:ball-constraints} \int_{B_{\vartheta}(R_t)}\phi\,dV_{\tilde{g}_t}=0, \qquad \int_{\partial B_{\vartheta}(R_t)}\phi\,dS_{\tilde{g}_t}=0.\qquad{(4)}\]
Proof. The identity \(Q_U(\psi)=Q_t(\phi)\) follows immediately from 16 by transporting the three terms through the isometry \(F\). The weak Robin ?? is exactly the Euler–Lagrange equation associated with the bilinear form \(\mathcal{B}_t\). Finally, since \(\psi=U\hat{\phi}\), the linearized constraints become \[\label{eq:const-F} \int_{\mathbb{H}}\hat{\phi}\,dV_{g_U}=0, \qquad \int_{\partial\mathbb{H}}\hat{\phi}\,dS_{g_U}=0.\tag{17}\] Transporting 17 by \(F\) gives ?? . ◻
For a fixed \(U\in\mathcal{M}_T\), one has \[T_U\mathcal{M}_T=\operatorname{span}\{Z_0,Z_2,\dots,Z_n\},\] where \[\label{eq:tangent-all} Z_0=\frac{n-2}{2}U+x\cdot\nabla U, \qquad Z_i=\partial_{x_i}U,\quad i=2,\dots,n.\tag{18}\]
Lemma 1. There exists \(\varepsilon_T>0\) such that whenever \(u\in\mathcal{A}_T\), \(d_T(u)<\varepsilon_T\), the infimum \[d_T(u)^2=\inf_{U\in\mathcal{M}_T}\|\nabla(u-U)\|_{L^2(\mathbb{H})}^2\] is attained at some \(U\in\mathcal{M}_T\). Moreover, if \(u=U+\psi\) with \(U\) a nearest point, then \[\label{eq:orth-D12} \psi\perp_{\dot{H}^1} T_U\mathcal{M}_T,\tag{19}\] where \(T_U\mathcal{M}_T\) denotes the tangent space to the connected component of \(\mathcal{M}_T\) containing \(U\).
Proof. Since \(\mathcal{M}_T=\mathcal{M}_T^+\cup(-\mathcal{M}_T^+)\), where \[\mathcal{M}_T^+:= \{(\lambda,z)\cdot U_T:\lambda>0,\;z\in\mathbb{R}^{n-1}\},\] and since \(u\mapsto -u\) preserves \(\mathcal{A}_T\) and \(d_T\), we may, after replacing \(u\) by \(-u\) if necessary, assume that the positive component realizes the distance to \(\mathcal{M}_T\). Let \[\Xi:(0,\infty)\times\mathbb{R}^{n-1}\to \dot{H}^1(\mathbb{H}), \qquad \Xi(\lambda,z):=(\lambda,z)\cdot U_T,\] so that \(\mathcal{M}_T^+=\Xi((0,\infty)\times\mathbb{R}^{n-1})\). For fixed \(u\in\mathcal{A}_T\), define \[\mathscr F_u(\lambda,z):=\|\nabla(u-\Xi(\lambda,z))\|_{L^2(\mathbb{H})}^2.\] Then, in the present sign component, \[d_T(u)^2=\inf_{(\lambda,z)\in(0,\infty)\times\mathbb{R}^{n-1}}\mathscr F_u(\lambda,z).\] Choose \(U_*=\Xi(\lambda_*,z_*)\in\mathcal{M}_T^+\) such that \(\|\nabla(u-U_*)\|_{L^2(\mathbb{H})}\le 2d_T(u)\). Using the invariance of \(\mathcal{A}_T\), \(\mathcal{M}_T\), and \(d_T\) under tangential translations and critical dilations, we may replace \(u\) by \((\lambda_*,z_*)^{-1}\cdot u\). Thus it is enough to treat the case \(\|\nabla(u-U_T)\|_{L^2(\mathbb{H})}\le 2d_T(u)\), and hence \[\label{eq:F604d2} \mathscr F_u(1,0)\le 4d_T(u)^2.\tag{20}\]
Let \((\lambda_k,z_k)\) be a minimizing sequence for \(\mathscr F_u\). We claim that, provided \(d_T(u)\) is sufficiently small, the sequence \((\lambda_k,z_k)\) remains in a compact subset of \((0,\infty)\times\mathbb{R}^{n-1}\). Indeed, if \(|z_k|\to\infty\), then the translated profiles drift away tangentially, and therefore \[\label{eq:vanish} \int_{\mathbb{H}}\nabla u\cdot \nabla \Xi(\lambda_k,z_k)\,dx\to0.\tag{21}\] Since \(\|\nabla\Xi(\lambda_k,z_k)\|_{L^2(\mathbb{H})}=\Phi(T)\), it follows that \[\mathscr F_u(\lambda_k,z_k) = \|\nabla u\|_{L^2(\mathbb{H})}^2+\Phi(T)^2 -2\int_{\mathbb{H}}\nabla u\cdot \nabla \Xi(\lambda_k,z_k)\,dx \to \|\nabla u\|_{L^2(\mathbb{H})}^2+\Phi(T)^2.\] Since \(u\in\mathcal{A}_T\), one has \(\|\nabla u\|_{L^2(\mathbb{H})}\ge \Phi(T)\), and if \(d_T(u)\to0\) then \(\|\nabla u\|_{L^2(\mathbb{H})}\to\Phi(T)\). Hence, for \(d_T(u)\) sufficiently small, \[\|\nabla u\|_{L^2(\mathbb{H})}^2+\Phi(T)^2 > 4d_T(u)^2,\] which contradicts 20 . Therefore \((z_k)\) is bounded. Similarly, if \(\lambda_k\to0\) or \(\lambda_k\to\infty\), then 21 again holds. The same contradiction shows that \((\lambda_k)\) stays in a compact subinterval of \((0,\infty)\).
Thus, after passing to a subsequence, \((\lambda_k,z_k)\to(\lambda_0,z_0)\) for some \((\lambda_0,z_0)\in(0,\infty)\times\mathbb{R}^{n-1}\). Since \(\Xi\) is continuous as a map into \(\dot{H}^1(\mathbb{H})\), the function \(\mathscr F_u\) is continuous, and therefore the minimum is attained at \(U:=\Xi(\lambda_0,z_0)\in\mathcal{M}_T\).
Finally, if \(u=U+\psi\) and \(U\) is a nearest point, then \(U\) is a critical point of the restriction of \(W\mapsto \|u-W\|_{\dot{H}^1(\mathbb{H})}^2\) to the connected component of \(\mathcal{M}_T\) containing \(U\). Hence its differential vanishes on \(T_U\mathcal{M}_T\), which is exactly 19 . This proves the lemma. ◻
Lemma 2. Let \(T>0\), and let \(U\) be a positive bridge minimizer. Then there exist \(\varepsilon_U,C_U>0\) such that, whenever \[\psi\in \dot{H}^1(\mathbb{H}),\qquad \psi\perp_{\dot{H}^1}T_U\mathcal{M}_T,\qquad \|\nabla\psi\|_{L^2(\mathbb{H})}\le \varepsilon_U,\qquad U+\psi\in\mathcal{A}_T,\] there exists \(W\in \mathcal{X}_U\cap (T_U\mathcal{M}_T)^{\perp_{\dot{H}^1}}\) such that such that \[\label{eq:quadratic-est} \|\nabla(\psi-W)\|_{L^2(\mathbb{H})} \le C_U\,\|\nabla\psi\|_{L^2(\mathbb{H})}^2.\tag{22}\] In particular, \[\label{eq:W-psi} \|\nabla W\|_{L^2(\mathbb{H})}^2 = \|\nabla\psi\|_{L^2(\mathbb{H})}^2 + o\!\bigl(\|\nabla\psi\|_{L^2(\mathbb{H})}^2\bigr) \qquad\text{as~ }\|\nabla\psi\|_{L^2(\mathbb{H})}\to0.\tag{23}\]
Proof. Set \(\mathcal{C}_U=(T_U\mathcal{M}_T)^{\perp_{\dot{H}^1}}\), and define \(\mathfrak D:\dot{H}^1(\mathbb{H})\to\mathbb{R}^2\) by \[\mathfrak D(\varphi) := \left( \int_{\mathbb{H}}U^{2^*-1}\varphi\,dx,\; \int_{\partial\mathbb{H}}U^{2^\#-1}\varphi\,dS \right).\] Then \(\mathcal{X}_U=\ker\mathfrak D\). We claim that \[\label{eq:defect} |\mathfrak D(\psi)|\le C\,\|\nabla\psi\|_{L^2(\mathbb{H})}^2.\tag{24}\] Since \(u=U+\psi\in\mathcal{A}_T\) and \(U\in\mathcal{A}_T\), one has \(\|U+\psi\|_{L^{2^*}(\mathbb{H})}=\|U\|_{L^{2^*}(\mathbb{H})}\). Set \(p=2^*\). The pointwise expansion \[\label{eq:remainder-est} \bigl||a+b|^p-a^p-pa^{p-1}b\bigr| \le C\bigl(a^{p-2}|b|^2+|b|^p\bigr), \qquad a\ge0,\quad b\in\mathbb{R},\quad p>2 .\tag{25}\] yields \[\left|\,p\int_{\mathbb{H}}U^{p-1}\psi\,dx\right| \le C\int_{\mathbb{H}}U^{p-2}\psi^2\,dx + C\int_{\mathbb{H}}|\psi|^p\,dx.\] By Hölder inequality and the Sobolev inequality, \[\int_{\mathbb{H}}U^{p-2}\psi^2\,dx =O(\|\nabla\psi\|_{L^2(\mathbb{H})}^2),\quad \int_{\mathbb{H}}|\psi|^p\,dx= o(\|\nabla\psi\|_{L^2(\mathbb{H})}^2).\] Therefore \[\label{eq:bulk-remainder} \int_{\mathbb{H}}U^{2^*-1}\psi\,dx = O(\|\nabla\psi\|_{L^2(\mathbb{H})}^2).\tag{26}\] Similarly, by 25 with \(q=2^\#\) and the trace inequality, we obtain \[\label{eq:boundary-remainder} \int_{\partial\mathbb{H}}U^{2^\#-1}\psi\,dS = O(\|\nabla\psi\|_{L^2(\mathbb{H})}^2).\tag{27}\] Thus, 26 and 27 imply 24 .
We claim that the restriction \(\mathfrak D|_{\mathcal{C}_U}:\mathcal{C}_U\to\mathbb{R}^2\) is surjective. By duality, it is enough to show that if a linear functional \(\ell\in(\mathbb{R}^2)^*\) vanishes on \(\mathfrak D(\mathcal{C}_U)\), then \(\ell=0\). Write \(\ell(\xi_1,\xi_2)=\alpha \xi_1+\beta \xi_2\) for some \((\alpha,\beta)\in\mathbb{R}^2\), and assume that \[\alpha \int_{\mathbb{H}}U^{2^*-1}\varphi\,dx + \beta \int_{\partial\mathbb{H}}U^{2^\#-1}\varphi\,dS =0 \qquad\text{for all } \varphi\in\mathcal{C}_U.\] Since \(T_U\mathcal{M}_T\subset \mathcal{X}_U=\ker\mathfrak D\), the same identity holds trivially for every \(\varphi\in T_U\mathcal{M}_T\). Because \(\dot{H}^1(\mathbb{H})=\mathcal{C}_U\oplus T_U\mathcal{M}_T,\) it follows that \[\alpha \int_{\mathbb{H}}U^{2^*-1}\varphi\,dx + \beta \int_{\partial\mathbb{H}}U^{2^\#-1}\varphi\,dS =0 \qquad\text{for all } \varphi\in\dot{H}^1(\mathbb{H}).\] Now take \(\varphi\in C_c^\infty(\mathbb{H})\). Then the boundary term vanishes, and we obtain \[\alpha \int_{\mathbb{H}}U^{2^*-1}\varphi\,dx=0 \qquad\text{for all } \varphi\in C_c^\infty(\mathbb{H}).\] Since \(U>0\) in \(\mathbb{H}\), this implies \(\alpha=0\). Therefore \[\beta \int_{\partial\mathbb{H}}U^{2^\#-1}\varphi\,dS=0 \qquad\text{for all } \varphi\in\dot{H}^1(\mathbb{H}).\] Choosing \(\varphi\) with nontrivial trace on \(\partial\mathbb{H}\), we conclude that \(\beta=0\). Hence \(\ell=0\), and therefore \(\mathfrak D|_{\mathcal{C}_U}\) is surjective.
Since \(\mathfrak D|_{\mathcal{C}_U}\) is surjective and \(\mathbb{R}^2\) is finite-dimensional, there exists a bounded linear right inverse \(\mathcal{R}_U:\mathbb{R}^2\to\mathcal{C}_U\). Let \(\tau=\mathcal{R}_U(\mathfrak D(\psi))\) and \(W=\psi-\tau\). Then \(W\in\mathcal{C}_U\) and \(\mathfrak D(W)=0\), so \(W\in \mathcal{X}_U\cap (T_U\mathcal{M}_T)^{\perp_{\dot{H}^1}}\). Furthermore, by 24 , \[\|\nabla(\psi-W)\|_{L^2(\mathbb{H})} = \|\nabla\tau\|_{L^2(\mathbb{H})} \le C_U\,|\mathfrak D(\psi)| \le C_U\,\|\nabla\psi\|_{L^2(\mathbb{H})}^2.\] This proves 22 , while 23 follows immediately. ◻
Proposition 2. Assume \(T\neq T_E\). Then there exists \(\Lambda_T>0\) such that, for every positive minimizer \(U\in\mathcal{M}_T\) and every \(W\in \mathcal{X}_U\cap (T_U\mathcal{M}_T)^{\perp_{\dot{H}^1}}\), one has \[\label{eq:spectral-gap} Q_U(W)\ge \Lambda_T\,\|\nabla W\|_{L^2(\mathbb{H})}^2.\qquad{(5)}\]
Proof. By symmetry invariance, it is enough to work at one fixed positive minimizer \(U\in\mathcal{M}_T\). Set \[\mathcal{H}_{0,t} := \left\{ \phi\in H^1(B_{\vartheta}(R_t)): \eqref{eq:ball-constraints}~ \text{holds} \right\}, \quad \mathcal{T}_U(W):=\Bigl(\frac{W}{U}\Bigr)\circ F^{-1}.\] By Proposition 1, \(\mathcal{T}_U\) identifies \(\mathcal{X}_U\) with \(\mathcal{H}_{0,t}\), and \(Q_U\) with \(\mathcal{Q}_t\). By Proposition 3, \[\label{eq:kernel-phi} \ker(\mathcal{Q}_t|\mathcal{H}_{0,t}) = \operatorname{span}\{\phi_0,\phi_2,\dots,\phi_n\}.\tag{28}\] We now set \[\mathcal{Z}_{t,U} = \left\{ \phi\in \mathcal{H}_{0,t}: \Lambda_j(\phi)=0,\;j\in\{0,2,\dots,n\} \right\},\] where \(\Lambda_j(\phi):=\bigl\langle U(\phi\circ F),\,Z_j\bigr\rangle_{\dot{H}^1(\mathbb{H})}\). Since each \(\Lambda_j\) is a continuous linear functional on \(H^1(B_{\vartheta}(R_t))\), the space \(\mathcal{Z}_{t,U}\) is weakly closed. It is therefore enough to prove that there exists \(\mu_T>0\) such that \[\label{eq:coercive-z} \mathcal{Q}_t(\phi)\ge \mu_T\|\nabla_{\tilde{g}_t}\phi\|_{L^2(B_{\vartheta}(R_t))}^2 \qquad \qquad\text{for all } \phi\in \mathcal{Z}_{t,U}.\tag{29}\] Assume by contradiction that 29 fails. Then there exists \((\varphi_m)\subset \mathcal{Z}_{t,U}\) such that \[\label{eq:contradict-setting} \|\nabla_{\tilde{g}_t}\varphi_m\|_{L^2(B_{\vartheta}(R_t))}=1, \qquad \mathcal{Q}_t(\varphi_m)\to 0.\tag{30}\] Since \(\varphi_m\in \mathcal{H}_{0,t}\), the Poincaré inequality on the bounded connected Riemannian domain \((B_{\vartheta}(R_t),\tilde{g}_t)\), see [15], gives \[\|\varphi_m\|_{L^2(B_{\vartheta}(R_t),dV_{\tilde{g}_t})} \le C_T\|\nabla_{\tilde{g}_t}\varphi_m\|_{L^2(B_{\vartheta}(R_t),dV_{\tilde{g}_t})} = C_T.\] Thus \((\varphi_m)\) is bounded in \(H^1(B_{\vartheta}(R_t))\). Passing to a subsequence, \[\varphi_m\rightharpoonup \varphi_\infty \quad\text{weakly in }H^1(B_{\vartheta}(R_t)),\] \[\label{eq:compact-strong} \varphi_m\to \varphi_\infty \quad\text{strongly in }L^2(B_{\vartheta}(R_t)) \cap L^2(\partial B_{\vartheta}(R_t)).\tag{31}\] Since \(\mathcal{Z}_{t,U}\) is weakly closed in \(\mathcal{H}_{0,t}\), \(\varphi_\infty\in \mathcal{Z}_{t,U}\). By weak lower semicontinuity of the gradient term and 31 , \(\mathcal{Q}_t(\varphi_\infty)\le \liminf_{m\to\infty}\mathcal{Q}_t(\varphi_m)=0\). Because \(U\) is a constrained minimizer, one has \(\mathcal{Q}_t\ge 0\) on \(\mathcal{H}_{0,t}\). Hence \(\mathcal{Q}_t(\varphi_\infty)=0\), so \(\varphi_\infty\in \ker(\mathcal{Q}_t|\mathcal{H}_{0,t})\).
By 28 , \(\varphi_\infty=\sum_{j\in\{0,2,\dots,n\}} a_j\phi_j\). Set \(W_\infty:=U(\varphi_\infty\circ F)\). Since \(\phi_j=\bigl(\frac{Z_j}{U}\bigr)\circ F^{-1}\), we have \[W_\infty=\sum_{j\in\{0,2,\dots,n\}} a_j Z_j \in T_U\mathcal{M}_T.\] But \(\varphi_\infty\in \mathcal{Z}_{t,U}\) implies that \(W_\infty\in (T_U\mathcal{M}_T)^{\perp_{\dot{H}^1}}\). Therefore \(W_\infty=0\), and so \(\varphi_\infty=0\). Returning to ?? , together with 31 and \(\|\nabla_{\tilde{g}_t}\varphi_m\|_{L^2(B_{\vartheta}(R_t))}=1\), gives \(\mathcal{Q}_t(\varphi_m)\to 1\), contradicting 30 . This proves 29 .
Finally, let \(W\in \mathcal{X}_U\cap (T_U\mathcal{M}_T)^{\perp_{\dot{H}^1}}\), \(\phi:=\mathcal{T}_U(W)\in \mathcal{Z}_{t,U}\). Then by 29 , \[Q_U(W)=\mathcal{Q}_t(\phi)\ge \mu_T\|\nabla_{\tilde{g}_t}\phi\|_{L^2(B_{\vartheta}(R_t))}^2.\] Moreover, by 9 and 14 , \[\label{eq:W-ground} \|\nabla W\|_{L^2(\mathbb{H})}^2 = \int_{B_{\vartheta}(R_t)}|\nabla_{\tilde{g}_t}\phi|^2\,dV_{\tilde{g}_t} +\lambda\int_{B_{\vartheta}(R_t)}\phi^2\,dV_{\tilde{g}_t} +\sigma\int_{\partial B_{\vartheta}(R_t)}\phi^2\,dS_{\tilde{g}_t}.\tag{32}\] Since \(\phi\in\mathcal{H}_{0,t}\), Poincaré and the trace inequality yield \[\label{eq:pincare-trace} \int_{B_{\vartheta}(R_t)}\phi^2\,dV_{\tilde{g}_t} +\int_{\partial B_{\vartheta}(R_t)}\phi^2\,dS_{\tilde{g}_t} \le C_T \int_{B_{\vartheta}(R_t)}|\nabla_{\tilde{g}_t}\phi|^2\,dV_{\tilde{g}_t}.\tag{33}\] Together 32 and 33 , with \(\lambda,\sigma\) fixed by \(T\), prove ?? . This completes the proof. ◻
Proof. By the sign reduction discussed above, we may assume that the nearest point \(U\) is positive. Let \(u\in\mathcal{A}_T\) with \(d_T(u)\) sufficiently small, and choose such a nearest point \(U\in\mathcal{M}_T\) as in Lemma 1. Writing \(u=U+\psi\), we have \[\|\nabla\psi\|_{L^2(\mathbb{H})}=d_T(u), \qquad \psi\perp_{\dot{H}^1}T_U\mathcal{M}_T.\] Since \(u,U\in\mathcal{A}_T\), the constraint terms in the Lagrangian cancel. Hence Taylor expansion at \(U\), together with 5 and 8 , gives \[\label{eq:Taylor-delta} \delta_T(u)=Q_U(\psi)+o(\|\nabla\psi\|_{L^2(\mathbb{H})}^2).\tag{34}\] By Lemma 2, there exists \(W\in \mathcal{X}_U\cap (T_U\mathcal{M}_T)^{\perp_{\dot{H}^1}}\) such that \[\|\nabla(\psi-W)\|_{L^2(\mathbb{H})}=O(\|\nabla\psi\|_{L^2(\mathbb{H})}^2), \qquad \|\nabla W\|_{L^2(\mathbb{H})}^2 = \|\nabla\psi\|_{L^2(\mathbb{H})}^2 + o(\|\nabla\psi\|_{L^2(\mathbb{H})}^2).\] If \(\mathcal{B}_U\) is the bilinear form associated with \(Q_U\), then \[Q_U(\psi)-Q_U(W)=\mathcal{B}_U(\psi-W,\psi+W) =o(\|\nabla\psi\|_{L^2(\mathbb{H})}^2).\] Therefore, by Proposition 2, \[\delta_T(u) = Q_U(W)+o(\|\nabla\psi\|_{L^2(\mathbb{H})}^2) \ge \Lambda_T\|\nabla W\|_{L^2(\mathbb{H})}^2 + o(\|\nabla\psi\|_{L^2(\mathbb{H})}^2).\] Using 23 and \(\|\nabla\psi\|_{L^2(\mathbb{H})}=d_T(u)\), we obtain \[\delta_T(u) \ge \Lambda_T d_T(u)^2+o(d_T(u)^2).\] This proves the theorem. ◻
Throughout this appendix, \(U\in\mathcal{A}_T\) is a positive \(p=2\) bridge minimizer with \(T\neq T_E\). By Proposition 1, the kernel problem on \(\mathcal{X}_U\) is reduced to the Robin kernel problem on a model ball. For simplicity we write this model as \((B_\kappa(R),g_\kappa)\), where \(\kappa=\kappa(U)\neq0\) is the constant sectional curvature. More precisely, under the change of variables \(\phi=\Bigl(\frac{\psi}{U}\Bigr)\circ F^{-1}\), where \(F\) is given by 14 , the quadratic form \(Q_U\) is transformed into the reduced quadratic form \[\label{eq:def-Qphi} \mathcal{Q}(\phi) = \int_{B_\kappa(R)}|\nabla_{g_\kappa}\phi|^2\,dV_{g_\kappa} - n\kappa\int_{B_\kappa(R)}\phi^2\,dV_{g_\kappa} - \beta\int_{\partial B_\kappa(R)}\phi^2\,dS_{g_\kappa},\tag{35}\] and the linearized constraints become \[\mathcal{H}_0 := \left\{ \phi\in H^1(B_\kappa(R)): \int_{B_\kappa(R)}\phi\,dV_{g_\kappa}=0,\; \int_{\partial B_\kappa(R)}\phi\,dS_{g_\kappa}=0 \right\}.\] Thus it is enough to identify \(\ker(\mathcal{Q}|\mathcal{H}_0)\).
We write \[\phi_j:=\left(\frac{Z_j}{U}\right)\circ F^{-1}, \qquad j\in\{0,2,\dots,n\},\] where \(Z_j\) are defined in 18 . Since these fields \(Z_j\) arise from tangential translations and critical dilations, they belong to \(\ker(Q_U|\mathcal{X}_U)\), and hence \[\label{eq:phi-in-ker} \phi_0,\phi_2,\dots,\phi_n\in \ker(\mathcal{Q}|\mathcal{H}_0).\tag{36}\] The reduced kernel equation is \[\label{eq:Robin-kernel} \left\{ \begin{align} -\Delta_{g_\kappa}\phi-\kappa n\,\phi&=0 &&\text{in }B_\kappa(R),\\ \partial_\nu\phi-\beta\phi&=0 &&\text{on }\partial B_\kappa(R), \end{align} \right. \qquad \phi\in\mathcal{H}_0 .\tag{37}\]
In geodesic polar coordinates, one has, see for example [16], \[\label{eq:geodesic-corr} g_\kappa=dr^2+s_\kappa(r)^2g_{\mathbb{S}^{n-1}},\tag{38}\] where \[s_\kappa(r):= \begin{cases} \dfrac{1}{\sqrt{\kappa}}\sin(\sqrt{\kappa}\,r),&\kappa>0,\\ r,&\kappa=0,\\ \dfrac{1}{\sqrt{-\kappa}}\sinh(\sqrt{-\kappa}\,r),&\kappa<0. \end{cases}\]
We analyze the reduced Robin problem 37 by separation of variables in the geodesic polar coordinates 38 . Expanding \(\phi\) in spherical harmonics on \(\mathbb{S}^{n-1}\), see for example [17], \[\phi(r,\theta)=\sum_{\ell=0}^\infty\sum_{m=1}^{d_\ell}f_{\ell,m}(r)Y_{\ell,m}(\theta), \qquad -\Delta_{\mathbb{S}^{n-1}}Y_{\ell,m}=\ell(\ell+n-2)Y_{\ell,m},\] where \(d_\ell\) denotes the dimension of the space of spherical harmonics of degree \(\ell\) on \(\mathbb{S}^{n-1}\), namely \[\label{eq:dimension-sph} d_\ell=\frac{(2\ell+n-2)(\ell+n-3)!}{\ell!(n-2)!}.\tag{39}\] By orthogonality, 35 decomposes as \(\mathcal{Q}(\phi)=\sum_{\ell,m}\mathcal{Q}_\ell(f_{\ell,m})\), where \[\mathcal{Q}_\ell(f) = \int_0^R \left( |f'|^2+\frac{\ell(\ell+n-2)}{s_\kappa(r)^2}f^2-\kappa n\,f^2 \right) s_\kappa(r)^{n-1}\,dr -\beta s_\kappa(R)^{n-1}f(R)^2.\] Moreover, \[\label{eq:increasing-Q} \mathcal{Q}_{\ell+1}(f)-\mathcal{Q}_\ell(f) = (2\ell+n-1)\int_0^R s_\kappa(r)^{n-3}f(r)^2\,dr>0\tag{40}\] for every nonzero \(f\).
Lemma 3. Each \(\phi_j\) lies in the \(\ell=1\) angular sector. Moreover, \(\phi_0,\phi_2,\dots,\phi_n\) are linearly independent.
Proof. We treat the two nondegenerate branches separately.
Spherical branch \(0<T<T_E\). Writing \(y=x-te_1\), 12 gives \[\frac{Z_0}{U} = \frac{n-2}{2}\,\frac{1-|y|^2-2ty_1}{1+|y|^2}, \qquad \frac{Z_i}{U}=-(n-2)\frac{y_i}{1+|y|^2}, \qquad i=2,\dots,n.\] Let \(X_1',\dots,X_n'\) denote the first \(n\) ambient coordinate functions on \(\mathbb{S}^n\) after the rotation \(O_t\), that is, \[X_\alpha'(\xi):=(O_t^{-1}\xi)_\alpha, \qquad \alpha=1,\dots,n.\] By the definition of the spherical reduction map \(F=O_t\circ \Pi_S^{-1}(\,\cdot\,-te_1)\), one has \[X_1'\circ F = -\frac{1-|y|^2-2ty_1}{\sqrt{1+t^2}\,(1+|y|^2)}, \qquad X_i'\circ F=\frac{2y_i}{1+|y|^2}, \qquad i=2,\dots,n.\] Hence \[\phi_0=-\frac{n-2}{2}\sqrt{1+t^2}\,X_1', \qquad \phi_i=-\frac{n-2}{2}\,X_i', \qquad i=2,\dots,n.\]
Since \(F(\mathbb{H})\) is a centered geodesic ball, in the geodesic polar coordinates of \(g_\kappa\) the first ambient coordinate functions are constant multiples of \(s_\kappa(r)\theta_\alpha\), \(\alpha=1,\dots,n\). Thus each \(\phi_j\) is of the form \(c\,s_\kappa(r)\theta_\alpha\), and hence belongs to the \(\ell=1\) sector.
Hyperbolic branch \(T>T_E\). Again writing \(y=x-te_1\), 13 yields \[\frac{Z_0}{U} = -\frac{n-2}{2}\,\frac{|y|^2+1+2ty_1}{|y|^2-1}, \qquad \frac{Z_i}{U}=-(n-2)\frac{y_i}{|y|^2-1}, \qquad i=2,\dots,n.\] Let \(Y_1',\dots,Y_n'\) denote the first \(n\) ambient coordinate functions on \(\mathbb{H}^n\) after the Lorentz isometry \(L_t\), that is, \(Y_\alpha'(\xi):=(L_t^{-1}\xi)_\alpha\), \(\alpha=1,\dots,n\).
By the definition of the hyperbolic reduction map \(F=L_t\circ \Pi_H^{-1}(\,\cdot\,-te_1)\), one has \[Y_1'\circ F = \frac{|y|^2+1+2ty_1}{\sqrt{t^2-1}\,(|y|^2-1)}, \qquad Y_i'\circ F=\frac{2y_i}{|y|^2-1}, \qquad i=2,\dots,n.\] Hence \[\phi_0=-\frac{n-2}{2}\sqrt{t^2-1}\,Y_1', \qquad \phi_i=-\frac{n-2}{2}\,Y_i', \qquad i=2,\dots,n.\] Since \(F(\mathbb{H})\) is a centered geodesic ball, in the geodesic polar coordinates of \(g_\kappa\) the first ambient coordinate functions are constant multiples of \(s_\kappa(r)\theta_\alpha\), \(\alpha=1,\dots,n\). Thus each \(\phi_j\) is of the form \(c\,s_\kappa(r)\theta_\alpha\), and hence belongs to the \(\ell=1\) sector.
In both branches, \(\phi_0,\phi_2,\dots,\phi_n\) are nonzero multiples of \(\theta_1,\theta_2,\dots,\theta_n\) times the same radial factor \(s_\kappa(r)\), which is not identically zero. Their linear independence therefore follows from the linear independence of the first spherical harmonics. ◻
Lemma 4. Fix \(\ell\ge1\). In the \(\ell\)-sector, the space of \(H^1\)-solutions of the reduced kernel equation 37 is either trivial or \(d_\ell\)-dimensional.
Proof. Let \[\phi(r,\theta)=f(r)Y(\theta), \qquad -\Delta_{\mathbb{S}^{n-1}}Y=\ell(\ell+n-2)Y .\] Then by 37 , \(f\) satisfies \[\label{eq:sector-ode-appendix} f''(r) + (n-1)\frac{s_\kappa'(r)}{s_\kappa(r)}f'(r) - \frac{\ell(\ell+n-2)}{s_\kappa(r)^2}f(r) + \kappa n\,f(r)=0 \qquad (0<r<R).\tag{41}\] Since \[s_\kappa(r)=r+O(r^3), \qquad \frac{s_\kappa'(r)}{s_\kappa(r)}=\frac{1}{r}+O(r), \qquad \frac{1}{s_\kappa(r)^2}=\frac{1}{r^2}+O(1) \quad\text{as }r\downarrow0,\] equation 41 has a regular singular point at \(r=0\), and can be written in the form \[f''+\frac{n-1}{r}f'-\frac{\ell(\ell+n-2)}{r^2}f+a(r)f'+b(r)f=0,\] with \(a,b\) continuous near \(0\). By the classical Frobenius theory for regular singular equations, the indicial equation is \[\alpha(\alpha-1)+(n-1)\alpha-\ell(\ell+n-2)=0,\] whose roots are \(\alpha_+=\ell\), \(\alpha_-=-(\ell+n-2)\). Hence there is a basis of local solutions of the form \[f_{\rm reg}(r)=r^\ell(1+o(1)), \qquad f_{\rm sing}(r)=r^{-(\ell+n-2)}(1+o(1)) \qquad (r\downarrow0).\]
We claim that the singular branch is not \(H^1\)-admissible. Indeed, \(f'_{\rm sing}(r)\sim r^{-(\ell+n-1)}\), and since \(dV_{g_\kappa}\sim r^{n-1}\,dr\,d\theta\) near \(r=0\), we get \[\int_0^\varepsilon |f'_{\rm sing}(r)|^2\,r^{n-1}\,dr \sim \int_0^\varepsilon r^{-2(\ell+n-1)}r^{n-1}\,dr = \int_0^\varepsilon r^{-2\ell-n+1}\,dr = \infty\] for every \(\ell\ge0\) and \(n\ge3\). Thus only the regular branch can belong to \(H^1\).
It follows that the local \(H^1\)-solution space near \(r=0\) is one-dimensional. Hence any two global \(H^1\)-solutions of 41 are proportional, since on every interval \([r_0,R]\subset(0,R]\) the equation 41 is a regular second-order linear ODE and uniqueness for the Cauchy problem applies. Therefore the admissible radial profile is unique up to a multiplicative constant.
Consequently, if the \(\ell\)-sector is nontrivial, its full \(H^1\)-solution space is \[\{\,f_\ell(r)Y(\theta): Y\in\mathcal{Y}_\ell\,\},\qquad \text{where~~} \mathcal{Y}_{\ell}=\left\{Y:-\Delta_{\mathbb{S}^{n-1}} Y=\ell(\ell+n-2) Y\right\},\] and therefore has dimension \(d_\ell\). ◻
For the radial sector \(\ell=0\), no nontrivial element of \(\ker(\mathcal{Q}|\mathcal{H}_0)\) exists. Indeed, if \[\phi(x)=f(r)\in \ker(\mathcal{Q}|\mathcal{H}_0)\] is radial, then \(\phi\) is constant on \(\partial B_\kappa(R)\). Since \(\phi\in\mathcal{H}_0\), \(f(R)=0\). The Robin boundary condition 37 gives \(f'(R)=\beta f(R)=0\). By uniqueness for the Cauchy problem for the radial ODE, it follows that \(f\equiv0\). Thus \[\label{eq:non-ell610} \ker(\mathcal{Q}|\mathcal{H}_0)\cap\{\ell=0\}=\{0\}.\tag{42}\]
Proposition 3. Assume \(T\neq T_E\). Then \[\label{eq:kernel-appendix} \ker(\mathcal{Q}|\mathcal{H}_0) = \operatorname{span}\{\phi_0,\phi_2,\dots,\phi_n\}.\qquad{(6)}\]
Proof. By 36 and Lemma 3, \(\phi_j\), \(j\in\{0,2,\dots,n\}\), are linearly independent elements of \(\ker(\mathcal{Q}|\mathcal{H}_0)\), and each lies in the \(\ell=1\) sector. In particular, the \(\ell=1\) kernel is nontrivial, so its lowest Rayleigh level is \(0\).
Since \(\mathcal{Q}\ge0\) on \(\mathcal{H}_0\), the \(\ell=1\) sector has lowest level \(0\). By 40 , for every \(\ell\ge2\) and every nonzero radial profile \(f\), one has \(\mathcal{Q}_\ell(f)>\mathcal{Q}_1(f)\ge0\). Hence no sector \(\ell\ge2\) can contain a kernel element. By 42 , the radial sector also contributes no kernel element. Therefore \[\ker(\mathcal{Q}|\mathcal{H}_0)\subset \{\ell=1\}.\]
By Lemma 4, in the \(\ell=1\) sector the space of \(H^1\)-solutions of the reduced kernel equation 37 is either trivial or \(d_1\)-dimensional. Since this sector already contains the nonzero kernel elements \(\phi_0,\phi_2,\dots,\phi_n\), it is nontrivial; hence its dimension is exactly \(d_1=\dim\mathcal{Y}_1=n\) by 39 .
Because \(\phi_0,\phi_2,\dots,\phi_n\) are \(n\) linearly independent kernel elements in the \(\ell=1\) sector, they form a basis of the entire kernel. This is exactly ?? . This completes the proof. ◻
Conflict of interest: Authors state no conflict of interest.
Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
The authors thank Robin Neumayer for helpful correspondence concerning the global stability problem. G-D. Li was supported by NSFC (No.12561019).