--- title: Jammes Cheeger Inequalities url: https://www.emergentmind.com/topics/jammes-cheeger-inequalities type: topic --- # Jammes Cheeger Inequalities Jammes Cheeger inequalities are boundary-sensitive Cheeger-type lower bounds in which a Steklov or Steklov-like spectral quantity is controlled by isoperimetric data involving both interior and boundary geometry. In the Riemannian setting, the guiding model is a lower bound for the first nonzero Steklov eigenvalue by a product of an interior Cheeger constant and a boundary Cheeger constant; subsequent work extends this principle to higher Steklov eigenvalues, finite Markov chains and weighted graphs, Carnot–Carathéodory spaces, and related form-valued spectral problems [1705.08643][2509.05667]. ## 1. Canonical Steklov formulation For a compact Riemannian manifold \((M,g)\) with smooth boundary \(\partial M\), the Steklov problem is \[ \begin{cases} \Delta f = 0 & \text{in } M,\\ \partial_\nu f = \sigma f & \text{on } \partial M, \end{cases} \] with spectrum \[ 0 = \sigma_1 < \sigma_2 \le \cdots \le \sigma_k \to +\infty. \] Equivalently, the \(\sigma_k\) are the eigenvalues of the Dirichlet-to-Neumann map \(S\), which sends boundary data to the normal derivative of its harmonic extension [1705.08643]. Within this setting, the Jammes viewpoint is that the correct isoperimetric control is not the classical Cheeger ratio alone. The continuous constants recalled in the discrete Steklov literature are the Escobar Cheeger constant \[ h_E(M) = \inf_{\substack{A\subset M\\ \mathrm{Area}(A\cap \partial M)\le \frac12\mathrm{Area}(\partial M)}} \frac{\mathrm{Area}(\partial A\cap \mathrm{int}(M))}{\mathrm{Area}(A\cap \partial M)} \] and the Jammes Cheeger constant \[ h_J(M) = \inf_{\substack{A\subset M\\ \mathrm{Vol}(A)\le \frac12\mathrm{Vol}(M)}} \frac{\mathrm{Area}(\partial A\cap \mathrm{int}(M))}{\mathrm{Area}(A\cap \partial M)}. \] The numerator is an interior boundary term, while the denominator is a boundary measure on \(\partial M\), so the inequality is intrinsically adapted to the Steklov operator rather than to the ordinary Dirichlet Laplacian [2509.05667]. The higher-order Steklov paper states the classical Jammes principle in product form: for the first nonzero Steklov eigenvalue, Jammes’ result gives, in the manifold case, \[ \sigma_2(M)\ge c\, h(M)\, h'(M), \] or, in the paper’s notation, a lower bound involving the product of an interior Cheeger constant and a boundary Cheeger constant [1705.08643]. This product structure is the defining formal feature of Jammes-type inequalities. ## 2. Discrete graph and Markov-chain versions A recent finite-state formulation places Jammes inequalities in the setting of a finite continuous-time Markov chain \[ G=(V,p,\mu), \] with Laplacian \[ \Delta f(x)=\sum_{y\in V}p_{xy}\bigl(f(x)-f(y)\bigr), \] and, in the reversible case, \[ p_{xy}\mu(x)=p_{yx}\mu(y)\qquad \forall x,y\in V. \] For a weighted graph \(G=(V,w,\mu)\) and another measure \(\nu\in\mathcal M(V)\), the generalized Cheeger constant is \[ h(\mu,\nu) = \inf_{\substack{\emptyset\neq A\subset V\\ \nu(A)\le \nu(V)/2}} \frac{w(A,A^c)}{\mu(A)}. \] The main generalized Laplacian estimate is \[ \lambda_2(\Delta_{w,\mu}) \ge \frac12\, h(\mu,\nu)\, h(\deg,\nu), \] where \[ \deg(x)=\sum_{y\ne x}p_{xy}\mu(x). \] When \(\mu=\nu\), this recovers the standard weighted Cheeger lower bound \[ \lambda_2 \ge \frac12\,h_\mu(G)^2. \] The decoupling of \(\mu\) and \(\nu\) is the mechanism that makes the Steklov limit accessible [2509.05667]. For a designated boundary set \(B\subset V\), the discrete Steklov operator is defined by harmonic extension. If \(f\in C(B,\mathbb C)\), its harmonic extension \(F\in C(V,\mathbb C)\) satisfies \[ \Delta F(x)=0\quad \text{for }x\in B^c,\qquad F|_B=f, \] and the Steklov operator is \[ Tf = \Delta F\big|_B. \] Its eigenvalues are \[ 0=\sigma_1\le \sigma_2\le \cdots \le \sigma_b,\qquad b=|B|. \] The resulting discrete Jammes inequality is \[ 2\,h(\mu_B,\mu_B)\ge \sigma_2 \ge \frac12\, h(\mu_B,\nu)\, h(\deg,\nu), \] where \(\mu_B=1_B\mu\). The lower bound is the discrete Jammes-type estimate, while the upper bound \[ \sigma_2\le 2h(\mu_B,\mu_B) \] is the discrete Escobar-type estimate. Choosing \(\nu=\mu_B\) yields \[ 2\,h(\mu_B,\mu_B)\ge \sigma_2 \ge \frac12\, h(\mu_B,\mu_B)\, h(\deg,\mu_B). \] The paper also states that this improves earlier graph Jammes bounds from Hassannezhad–Miclo and can be sharper than earlier normalized estimates in examples [2509.05667]. The proof proceeds by introducing rescaled Laplacians \[ \Delta_r=(1_B+r1_{B^c})\Delta, \qquad \mu_r=(1_B+\tfrac1r 1_{B^c})\mu, \] applying the generalized Cheeger inequality to each \(\Delta_r\), and sending \(r\to\infty\). In the reversible case this uses Hassannezhad–Miclo’s convergence \[ \lim_{r\to\infty}\lambda_k(r)=\sigma_k,\qquad 1\le k\le b, \] and the same paper extends the convergence mechanism to non-reversible Markov chains by resolvent convergence [2509.05667]. ## 3. Higher-order Jammes inequalities for Steklov spectra The higher-order theory of Hassannezhad and Miclo extends the first-eigenvalue Jammes estimate to all higher Steklov eigenvalues in three parallel settings: finite state spaces, measurable state spaces, and compact Riemannian manifolds with boundary [1705.08643]. In the Riemannian formulation, for an admissible open set \(A\subset M\), \[ n(A)=\frac{L(\partial_i A)}{\mu(A)}, \qquad n'(A)=\frac{L(\partial_i A)}{L(\partial_e A)}, \] where \(\partial_i A := \partial A \cap \operatorname{Int} M\) and \(\partial_e A := \partial A \cap \partial M\). In the finite and measurable settings there are analogous definitions using the boundary measure induced by the generator or kernel. One then defines \[ p(A):=\inf_{B\subset A} n(B), \qquad p'(A):=\inf_{B\subset A} n'(B), \] and the \(k\)-th Cheeger–Steklov constant \[ L_k(M) := \inf_{(A_1,\dots,A_k)\in \mathcal A^k} \max_{i\in[k]} p(A_i)\,p'(A_i). \] The principal higher-order lower bounds are: \[ \sigma_k(M)\ge c_0\,\frac{L_k(M)}{\|L\|_\infty} \quad\text{in finite state spaces}, \] \[ \sigma_k(M)\ge c_1\,L_k(M) \quad\text{in measurable state spaces}, \] and \[ \sigma_k(M)\ge c_2\,L_k(M) \quad\text{in compact Riemannian manifolds with boundary}. \] These inequalities extend the Escobar–Jammes philosophy from \(\sigma_2\) to the full Steklov spectrum [1705.08643]. A central intermediary object is the Dirichlet–Steklov connectivity spectrum \[ K_k := \inf_{(A_1,\dots,A_k)\in\mathcal A^k} \max_{i\in[k]} \sigma_1(A_i), \] where \(\sigma_1(A_i)\) is the first eigenvalue of the Dirichlet–Steklov operator on \(A_i\cap V\). The paper proves \[ K_k \ge L_k(M), \] so that comparison between \(\sigma_k\) and \(K_k\) implies the higher-order Cheeger–Steklov estimates. It also derives a logarithmic refinement of the usual higher-order type, \[ \sigma_{2k} \ge \frac{c}{\log^2(k+1)}\,L_k. \] Methodologically, the Steklov operator is approximated by accelerated operators: sped-up Markov generators in finite and measurable settings, and mass concentration deformations of the Laplace–Beltrami operator in the manifold setting. This creates a uniform framework in which higher-order Cheeger inequalities for ordinary generators can be transferred to Steklov spectra [1705.08643]. ## 4. Boundary-sensitive analogues beyond the classical Steklov problem The Jammes mechanism persists in settings where the operator is not literally the Dirichlet-to-Neumann map but where the geometry of a separating hypersurface still controls a boundary-sensitive spectrum. In rank-varying Carnot–Carathéodory spaces, the geometric sub-Laplacian admits Dirichlet, Neumann, and mixed boundary Cheeger inequalities [2312.13058]. For a bounded connected domain \(\Omega\), the Neumann Cheeger constant is \[ h_N(\Omega)=\inf_\Sigma \frac{\sigma(\Sigma)}{\min\{\omega(\Omega_1),\omega(\Omega_2)\}}, \] where \(\Sigma\subseteq \Omega\) ranges over piecewise smooth hypersurfaces that separate \(\Omega\) into disjoint open sets \(\Omega_1,\Omega_2\). The first nontrivial Neumann eigenvalue then satisfies \[ \lambda_2^N(\Omega) \ge \frac14\, h_N(\Omega)^2. \] The same paper proves the Dirichlet estimate \[ \lambda_1^D(\Omega) \ge \frac14\, h_D(\Omega)^2 \] and the mixed estimate \[ \lambda_1^Z(\Omega,\Gamma) \ge \frac14\, h_Z(\Omega,\Gamma)^2. \] The Neumann formulation is explicitly described as the boundary-sensitive, Jammes-like refinement: the denominator uses the smaller of the two volumes cut by the hypersurface, so the quantity is separator-based rather than subset-based. The proof follows the classical pattern of applying a Cheeger inequality to a nodal domain of a second Neumann eigenfunction, but requires a sub-Riemannian coarea formula, horizontal perimeter, and a generalized Courant nodal domain theorem [2312.13058]. The same work gives a max-flow min-cut criterion. If a horizontal vector field \(V\) satisfies \[ \|V\|\le 1,\qquad \operatorname{div}_\omega(V)\ge h, \] then \[ h_D(\Omega)\ge h, \] and if \(V\) is inward-pointing on \(\partial\Omega\), then \[ h_N(\Omega)\ge h. \] This provides a concrete lower-bounding mechanism for the boundary-sensitive Cheeger constants themselves [2312.13058]. ## 5. Related extensions of the Jammes philosophy A broader class of results adapts the Cheeger/Jammes paradigm to operators for which the relevant geometry is no longer hypersurface separation. For coexact \(1\)-forms on a closed, connected, orientable Riemannian manifold \(M\), the first positive coexact \(1\)-form eigenvalue \(\lambda_1^1\) satisfies \[ \lambda_1^1 \ge \min(1, C\cdot h^1)^2, \] where \(C=C(a,b,D)\) depends on diameter and curvature bounds, and \[ h^1 := \inf_{\gamma \hookrightarrow M} \frac{l(\gamma)}{A(\gamma)}. \] Here \(\gamma\) ranges over real homologically trivial smooth closed curves, and \(A(\gamma)\) is the infimum, over spanning rectifiable currents, of the normalized spanning area. The paper explicitly states that the result is “Cheeger-like” in the same sense as Jammes-type refinements: the geometry is no longer a hypersurface separator but a lower-dimensional object adapted to the coexact \(1\)-form setting, namely a closed curve and its spanning surface [2103.09167]. The same structural migration appears in shape-optimization problems. For \[ \mathcal F_{p,q}(\Omega)=\frac{\lambda_p(\Omega)^{1/p}}{\lambda_q(\Omega)^{1/q}}, \qquad 1\le q