---
title: Escobar Cheeger Constant
url: https://www.emergentmind.com/topics/escobar-cheeger-constant
type: topic
---

# Escobar Cheeger Constant

The Escobar Cheeger constant is a boundary-sensitive isoperimetric invariant associated with the Steklov problem on manifolds with boundary. In the formulation recalled for a compact connected Riemannian manifold \(M\) with smooth boundary, it is
\[
h_E(M)=\inf_{\substack{ A\subset M\\ \operatorname{Area}(A\cap \partial M)\le \frac12 \operatorname{Area}(\partial M) }} \frac{\operatorname{Area}(\partial A\cap \operatorname{int}(M))}{\operatorname{Area}(A\cap \partial M)}.
\]
The denominator measures how much of \(A\) lies on the boundary \(\partial M\), while the numerator measures the “interior boundary” of \(A\), namely the part of \(\partial A\) lying inside \(M\) [2509.05667]. In contrast with the classical Cheeger constant, which controls Laplace-type spectra through bulk volume and interior boundary, the Escobar constant is adapted to Steklov eigenvalues and boundary geometry; later work extends the same idea to finite graphs, higher-order Steklov inequalities, and planar higher-order Escobar constants [1705.08643, 1905.07634].

## 1. Classical definition and boundary-isoperimetric structure

Escobar’s constant is defined by minimizing an interior-to-boundary ratio over subsets \(A\subset M\) subject to the admissibility constraint
\[
\operatorname{Area}(A\cap \partial M)\le \frac12 \operatorname{Area}(\partial M).
\]
This is the boundary analogue of the half-measure restriction familiar from Cheeger theory, but here the relevant size functional is the amount of boundary captured by \(A\), not the Riemannian volume of \(A\) itself [2509.05667].

The geometric interpretation is intrinsic to the Steklov setting. The numerator
\[
\operatorname{Area}(\partial A\cap \operatorname{int}(M))
\]
measures the new interface created inside the manifold by cutting out \(A\), whereas the denominator
\[
\operatorname{Area}(A\cap \partial M)
\]
measures the portion of the ambient boundary on which Steklov data live [2509.05667]. This is why the Escobar constant is naturally described as a boundary-sensitive isoperimetric quantity.

A related but notationally different framework appears in the study of planar domains. For a bounded planar domain \(M\), one paper defines, for non-empty open subsets \(\Omega\subset M\) with piecewise smooth boundary, the decomposition
\[
\partial \Omega=\partial_I\Omega\cup \partial_E\Omega,
\]
and writes the Escobar ratio as
\[
\eta(\Omega):=\frac{|\Omega\cap M|}{|\Omega\cap \partial M|}=\frac{|\partial_I\Omega|}{|\partial_E\Omega|},
\]
with the convention that \(\eta(\Omega)=\infty\) if \(\partial_E\Omega=\emptyset\) [1905.07634]. That paper further states that \(I_2(M)\) is the classical Escobar constant and that \(I_k(M)\), \(k\ge 3\), are higher-order analogues [1905.07634]. The coexistence of \(h_E(M)\) and \(I_2(M)\) indicates that the literature uses closely related but convention-dependent normalizations.

## 2. Steklov eigenvalues and first-order Cheeger-type inequalities

The principal role of the Escobar Cheeger constant is spectral. In the Steklov problem, the first non-trivial eigenvalue is governed by mixed interior-boundary isoperimetry rather than by the ordinary Cheeger ratio. Hassannezhad and Miclo explicitly place their work as an extension of the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 [1705.08643].

In the form recorded there, Jammes’ inequality is
\[
\sigma_2(M)\ge \frac14\,h_2(M)\,h_2'(M),
\]
where
\[
h_2(M):=\inf_{A\in\mathcal A}\max\{\eta(A),\eta(M\setminus A)\},
\qquad
h_2'(M):=\inf\{\eta'(A): A\in\mathcal A,\ \mu(A)\le \mu(M)/2\}.
\]
Within that framework, \(h_2(M)\) is essentially the classical Cheeger constant, while \(h_2'(M)\) is the boundary or Steklov analogue; their product reflects the fact that Steklov spectra depend simultaneously on bulk connectivity and boundary accessibility [1705.08643].

A later generalized Cheeger framework on finite graphs yields an explicit Escobar-type bound for the first non-trivial discrete Steklov eigenvalue:
\[
2\,h(\mu_B,\mu_B)\ \ge\ \sigma_2\ \ge\ \frac12\, h(\mu_B,\mu_B)\, h(\deg,\mu_B).
\]
Here \(h(\mu_B,\mu_B)\) is the discrete analogue of Escobar’s constant, while \(h(\deg,\mu_B)\) is a Jammes-type factor. The upper estimate
\[
\sigma_2\le 2\,h(\mu_B,\mu_B)
\]
is highlighted as sharp [2509.05667]. This makes the Escobar constant the natural upper-control quantity for \(\sigma_2\), with the lower bound requiring an additional bulk-degree term.

## 3. Discrete analogue on finite weighted graphs

For a finite weighted graph \(G=(V,w,\mu)\) with boundary subset \(B\subset V\), the relevant boundary measure is
\[
\mu_B:=1_B\,\mu\in\mathcal M(V),
\]
so that \(\mu_B(x)=\mu(x)\) for \(x\in B\) and \(0\) otherwise [2509.05667]. The generalized Cheeger constant is extended to measures that may vanish on vertices by
\[
h(\mu,\nu)=\inf_{\substack{A\subset V:\ \mu(A)>0,\ \nu(A)\le \nu(V)/2}} \frac{w(A,A^c)}{\mu(A)},
\]
and the discrete Escobar Cheeger constant is the special case
\[
h(\mu_B,\mu_B).
\]
In this setting, \(\mu_B(A)=\sum_{x\in A\cap B}\mu(x)\) plays the role of boundary measure, while
\[
w(A,A^c)=\sum_{x\in A,\,y\notin A} w(x,y)
\]
plays the role of interior boundary size [2509.05667].

This discrete formulation is embedded in a reversible weighted-graph Steklov theory. The proof strategy first establishes a generalized Cheeger inequality for graph Laplacians,
\[
\lambda_2(\Delta_{w,\mu}) \ge \frac12\, h(\mu,\nu)\, h(\deg,\nu)\qquad \forall \nu\in\mathcal M(V),
\]
and then passes from Laplacian eigenvalues to Steklov eigenvalues by accelerating the dynamics on the interior vertices \(B^c\). With
\[
\Delta_r := (1_B + r1_{B^c})\Delta,
\qquad
\mu_r := (1_B + r^{-1}1_{B^c})\mu,
\]
the relevant Laplacian eigenvalues converge as
\[
\lim_{r\to\infty}\lambda_k(r)=\sigma_k \qquad (1\le k\le b=|B|),
\]
yielding the Steklov inequalities in the limit [2509.05667].

Sharpness is illustrated by explicit examples. For the path graph \(x-y-z\) with boundary \(B=\{x,z\}\), weights \(w\equiv 1\), and measure
\[
\mu(x)=\epsilon,\qquad \mu(y)=\mu(z)=1,\qquad \epsilon\in(0,1),
\]
one has
\[
\sigma_2=\frac12\left(1+\frac{1}{\epsilon}\right),\qquad
h(\mu_B,\mu_B)=\frac{1}{\epsilon},\qquad
h(\deg,\mu_B)=\frac13.
\]
This shows that the estimate has the correct order and is sharp in \(\epsilon\) [2509.05667].

## 4. Higher-order generalizations

The higher-order theory replaces a single test region by \(k\)-tuples of disjoint regions. Hassannezhad and Miclo define the \(k\)-th Cheeger-Steklov constant by
\[
l_k(M):=\inf_{(A_1,\dots,A_k)\in\mathcal A^k} \ \max_{j\in[k]} p(A_j)p'(A_j),
\]
and prove, in finite spaces, measurable spaces, and Riemannian manifolds, that
\[
\sigma_k(M)\ \ge\ c\, l_k(M),\qquad \forall k\in\mathbb N,
\]
for a universal constant \(c>0\) [1705.08643]. In the same work they also obtain improved logarithmic versions such as
\[
\sigma_{2k}(M)\ \ge\ c\,\log^2(k+1)\, l_k(M)
\]
in the measurable and manifold settings, with an additional \(|L|^{-1}\) factor in the finite-state case [1705.08643].

In the manifold case, the Steklov-adapted isoperimetric quantities are defined from an open set \(A\subset M\) by
\[
\eta(A):=\frac{L(d_iA)}{\mu(A)},
\qquad
\eta'(A):=\frac{L(\partial_e A)}{\mu(A)},
\]
where \(d_iA=\partial A\cap \operatorname{Int}M\) and \(\partial_eA=\partial A\cap \partial M\) [1705.08643]. The paper states that Escobar’s original idea is encoded in the quantity \(p'(A)\) or, in the manifold notation, \(\eta'(A)\): a boundary-to-volume ratio measuring how strongly a set \(A\) interacts with the Steklov boundary [1705.08643].

The proof architecture is itself part of the modern theory. In finite and measurable settings it is based on accelerated Markov operators; in the manifold setting it uses mass concentration deformations of the Laplace-Beltrami operator converging to the Steklov operator. An intermediary object, the Dirichlet-Steklov connectivity spectrum,
\[
K_k := \min_{(A_1,\dots,A_k)\in \mathcal A^k}\ \max_{j\in[k]}\sigma_1(A_j),
\]
or its manifold analogue, provides the bridge between spectral convergence and the final lower bound [1705.08643].

## 5. Higher-order Escobar constants on planar domains

A distinct but closely related development studies higher-order Escobar constants \(I_k(M)\) for bounded planar domains. For \(k\in\mathbb N_+\),
\[
I_k(M):=\inf_{(\Omega_1,\dots,\Omega_k)\in A_k(M)}\max_{1\le j\le k}\eta(\Omega_j),
\]
where \(A_k(M)\) is the family of mutually disjoint \(k\)-tuples of non-empty open sets in \(M\) with piecewise smooth boundary [1905.07634]. In that terminology,
\[
I_1(M)=0,\qquad I_2(M)\ \text{is the classical Escobar constant},
\]
and \(I_k(M)\), \(k\ge 3\), are the higher-order Escobar constants [1905.07634].

These constants are boundary-isoperimetric analogues of higher Cheeger constants. The paper states that \(I_k(M)\) is scaling invariant, that \(I_k(M)\le 1\) for bounded planar domains, and that \(I_{k+1}(M)\ge I_k(M)\). It also remarks that no positive universal lower bound exists, since for thin rectangles \(I_k(M)\to 0\) as the width tends to \(0\) [1905.07634].

For the unit disk \(D\subset\mathbb R^2\), the exact formula is
\[
I_k(D)=\frac{\sin(\pi/k)}{\pi/k}, \qquad k\in \mathbb{N},
\]
obtained by matching upper and lower bounds through a regular \(k\)-partition by congruent sectors or curvilinear triangular pieces [1905.07634]. For a regular \(n\)-gon \(D_n\), the same paper proves
\[
I_k(D_n)=\cos(\pi/n)\le I_k(D)\qquad \text{for all }k\ge n,
\]
and, when \(n=mk\),
\[
I_k(D_n)=\sin(\pi/k)\cot(\pi/n)\,\frac{k}{n}\le I_k(D).
\]
For an arbitrary Euclidean \(n\)-gon \(M\) with smallest interior angle \(\theta_1\), it proves
\[
I_k(M)\le \sin(\theta_1/2)\le \cos(\pi/n), \qquad k\ge 3,
\]
and for a fixed polygon there exists \(n_0\ge n\) such that
\[
I_k(M)=\sin(\theta_1/2)\qquad\text{for all }k\ge n_0
\]
[1905.07634]. A recurring geometric theme is that extremal or near-extremal \(k\)-tuples concentrate near the sharpest corner.

## 6. Relation to other constants and terminological boundaries

The term “Escobar Cheeger constant” sits inside a broader family of isoperimetric quantities, but several nearby notions are not the same object.

| Quantity | Defining feature | Spectral role |
|---|---|---|
| Classical Cheeger constant \(h\) | Bulk/interior isoperimetry | Laplacian, \(\lambda_2\) |
| Escobar constant \(h_E(M)\) | Interior boundary over captured ambient boundary | Steklov, \(\sigma_2\) |
| Jammes-type constant | Boundary-sensitive quantity with bulk admissibility constraint | Lower bound factor for \(\sigma_2\) |
| Higher-order Escobar constants \(I_k(M)\) | \(k\)-tuple boundary-isoperimetric optimization | Higher Steklov bounds on planar domains |

In the generalized graph framework, the authors explicitly say that \(h(\mu_B,\mu_B)\) is the discrete analogue of Escobar’s \(h_E(M)\), whereas \(h(\mu_B,\deg)\) is the discrete analogue of Jammes’ constant; their generalized quantity \(h(\mu,\nu)\) encompasses both by decoupling the denominator measure from the admissibility measure [2509.05667]. In the higher-order theory, the boundary-sensitive component appears through \(p'(A)\) or \(\eta'(A)\), and the final lower bounds for \(\sigma_k\) involve products of interior and boundary isoperimetric factors rather than a single ratio [1705.08643].

Several works clarify what the Escobar Cheeger constant is not. The standard Cheeger constant on hyperbolic manifolds and graphs,
\[
h(M)=\inf_A \frac{\operatorname{Vol}_{n-1}(\partial A)}{\operatorname{Vol}_n(A)},
\qquad
h(T)=\inf_A\frac{|\partial A|}{|A|},
\]
is a classical global isoperimetric constant, and one paper states explicitly that it does not discuss Escobar’s version [1605.04394]. The graph-theoretic Cheeger constant for distance-regular graphs is likewise the standard spectral graph quantity and “not an Escobar-specific variant” [1811.00230]. The Cheeger-like graph constant
\[
Q=\max_{v\sim w}\left(\frac1{\deg v}+\frac1{\deg w}\right)
\]
for controlling the largest normalized Laplacian eigenvalue is introduced as a new graph analogue, not as an Escobar-type constant [1910.12233]. The \(N\)-cluster constant
\[
H_N(\Omega)=\inf\left\{\sum_{i=1}^N\frac{P(E_i)}{|E_i|}\ \Big|\ \mathcal{E}=\{E_i\}_{i=1}^{N}\subseteq \Omega,\ \text{is an \(N\)-cluster}\right\}
\]
is an \(N\)-partition generalization of the classical Cheeger constant and does not mention Escobar [1501.05923]. Conversely, the weighted Escobar constant introduced on smooth metric measure spaces with boundary arises from a generalized Sobolev trace inequality and is explicitly described as not involving any Cheeger-type constant or terminology [1805.03694].

In that sense, the Escobar Cheeger constant is best understood not as a generic name for any boundary variational constant, but as the Steklov-adapted boundary isoperimetric invariant whose discrete avatar is \(h(\mu_B,\mu_B)\), whose first-eigenvalue role is coupled to Jammes-type lower bounds, and whose higher-order descendants include the Cheeger-Steklov constants \(l_k\) and the planar higher-order Escobar constants \(I_k\) [2509.05667, 1705.08643, 1905.07634].

Source: https://www.emergentmind.com/topics/escobar-cheeger-constant