Laplace Radius: Context and Applications

• Laplace radius is a context-dependent measure defined differently in capillarity (as the surface-of-tension radius), spectral geometry, analytic continuation, and privacy mechanisms.
• In nanodroplet mechanics, researchers use proxies like the radius of gyration to estimate the surface-of-tension radius, which yields accurate measures for interface tension.
• In spectral geometry and local differential privacy, the radius parameter governs eigenvalue scaling and support truncation, underscoring its versatile role in controlling operator and analytic properties.

Searching arXiv for papers relevant to the term "Laplace Radius" and its usage across contexts. “Laplace radius” is not a uniformly standardized term in the arXiv literature. In strict capillarity theory, the closest formal meaning is the radius of the dividing surface that enters the Young–Laplace relation, namely the surface-of-tension radius. In other domains, the same phrase or its nearest analogue denotes the radius parameter governing a Laplace operator on balls or distance spheres, the analyticity scale in Laplace–Fourier continuation, the Bohr radius for a discrete Laplace transform, or the truncation radius of a sparse discrete-Laplace mechanism. In the surveyed literature, the expression is therefore context-dependent rather than canonical (Malek et al., 2017, Matos et al., 2013, Kounchev et al., 2011, Ahamed et al., 2024, Zamani et al., 10 May 2026).

1. Terminological status and scope

In capillarity, the relevant radius is the one appearing in

$$\Delta P = P_{\mathrm{in}}-P_{\mathrm{out}}=\frac{2\gamma}{R},$$

and, in the strict thermodynamic formulation, this $R$ is the surface-of-tension radius $R_s$ (Malek et al., 2017). By contrast, several spectral-geometry papers use “radius” to mean the geodesic-ball radius $r$, the geodesic distance-sphere radius $r$, the intrinsic radial variable $t$, or a warping function $r(t)$, without introducing a named invariant called “Laplace radius” (Berge, 2022, Bettiol et al., 2020, Matos et al., 2013). In complex analysis and transform theory, the relevant quantity is instead an analyticity or convergence scale, such as the radius $R$ of the ball $B_R$ in Laplace–Fourier continuation or the Bohr radius $r_\gamma$ for a discrete Laplace transform (Kounchev et al., 2011, Ahamed et al., 2024). In local differential privacy, the radius is a support-truncation parameter $R$0 for a sparse discrete-Laplace channel (Zamani et al., 10 May 2026).

This heterogeneity matters because identically named radius parameters can play sharply different roles: thermodynamic dividing surface, geometric domain size, support cardinality proxy, analyticity scale, or inverse-scattering warping factor. A plausible implication is that any use of “Laplace radius” requires immediate specification of the governing operator or constitutive law.

2. Capillarity: the Laplace radius as surface-of-tension radius

The clearest thermodynamic usage occurs in nanodroplet mechanics. For a spherical droplet, the Young–Laplace equation formally models a droplet with a sharp interface at $R$1, the surface of tension (Malek et al., 2017). In the molecular-dynamics study of TIP4P/2005 water droplets at $R$2, the internal overpressure was extracted from the microscopic pressure tensor, whose radial mean pressure was

$R$3

with $R$4 and $R$5 obtained from coarse-grained Irving–Kirkwood/Schofield–Henderson expressions (Malek et al., 2017).

The key empirical result was that beneath a diffuse interfacial region of about $R$6, roughly two molecular layers, the pressure tensor becomes approximately isotropic and nearly constant with $R$7. This allowed the authors to define an interior liquid pressure $R$8 by averaging over the isotropic core. Because the vapour pressure is negligible on the pressure scale of the droplets, they approximate

$R$9

They then test Young–Laplace scaling against an inferred droplet radius rather than a directly computed $R_s$0 (Malek et al., 2017).

The operational radius used in the fit is

$R_s$1

the uniform-sphere conversion from radius of gyration to physical radius. For the $R_s$2 droplet, the paper reports $R_s$3. This is distinct from

$R_s$4

the radius of the isotropic interior region used only to delimit the averaging domain for $R_s$5; for the same droplet, $R_s$6. The distinction is explicit: $R_s$7 is not the radius inserted into the Young–Laplace test (Malek et al., 2017).

With droplets of $R_s$8, $R_s$9, $r$0, and $r$1 molecules, the fit

$r$2

yielded

$r$3

close to the estimated planar-interface value

$r$4

for TIP4P/2005 water at $r$5 (Malek et al., 2017). In this setting, the “Laplace radius” is therefore best understood as the formal surface-of-tension radius $r$6, while the actual computation uses the proxy $r$7.

3. Spectral geometry: radius as the control parameter of Laplace spectra

In spectral geometry, the decisive object is usually not a named “Laplace radius” but a radius parameter governing a family of Laplace eigenvalue problems. For Dirichlet eigenvalues on geodesic balls in spherically symmetric manifolds, the radius $r$8 indexes eigenvalue branches $r$9, and the small-radius asymptotic is

$r$0

The same work proves that, for small radius, Dirichlet Laplace eigenvalues on the sphere are smaller than the Euclidean same-radius values, whereas the opposite holds in hyperbolic space (Berge, 2022).

A complementary construction replaces the original metric on a geodesic ball $r$1 by an area-preserving rotationally symmetric metric

$r$2

where

$r$3

For radial functions under $r$4,

$r$5

and the first Dirichlet eigenvalue on $r$6 satisfies a sharp upper bound computable entirely from the radius profile $r$7 via a recursive family $r$8 (Gimeno et al., 2021). Equality holds exactly when the inward mean curvature of each geodesic sphere $r$9 is radial (Gimeno et al., 2021).

On homogeneous distance spheres in rank-one symmetric spaces, the radius $t$0 enters the induced metric through explicit scale and anisotropy parameters. The resulting full Laplace–Beltrami spectrum on a geodesic distance sphere $t$1 is

$t$2

and in the compact case the resonant radii are

$t$3

These are precisely the radii at which Jacobi eigenvalues vanish and bifurcation of embedded constant-mean-curvature spheres occurs (Bettiol et al., 2020).

A further distinction appears in the study of complete radial graphs in $t$4. There, the paper explicitly states that it does not introduce a named invariant called “Laplace radius.” Instead, the decisive variables are the Euclidean radius $t$5, the intrinsic geodesic radius

$t$6

and the warping function $t$7, which determines the radial Laplacian and yields

$t$8

for every complete hypersurface that is the graph of a real radial function (Matos et al., 2013).

4. Analytic continuation and transform theory

In Laplace–Fourier continuation, the organizing parameter is the radius $t$9 of the real ball

$r(t)$0

This same $r(t)$1 controls the holomorphic extension of the Laplace–Fourier coefficients to the complex disc $r(t)$2, the transformed coefficients $r(t)$3 on $r(t)$4, and the final $r(t)$5-variable extension domain, the Lie ball

$r(t)$6

In this literature, the radius is therefore a scale of holomorphic continuation rather than a geometric Laplace-law radius (Kounchev et al., 2011).

A different analytic usage appears in Bohr inequalities for the discrete Laplace transform. For

$r(t)$7

the paper studies

$r(t)$8

It proves

$r(t)$9

and then identifies a refined Bohr-type radius $R$0 as the root in $R$1 of

$R$2

for $R$3, where

$R$4

In the unit-disk case $R$5, the paper states

$R$6

Here “Laplace radius” means a Bohr radius for the discrete Laplace transform, not a geometric radius attached to a Laplace operator (Ahamed et al., 2024).

5. Sparse discrete-Laplace mechanisms and radius truncation

In local differential privacy, the radius parameter appears as a sparsity constraint. The radius-truncated sparse discrete-Laplace mechanism on a line metric is

$R$7

with support

$R$8

The support size is

$R$9

so radius and support cardinality are equivalent (Zamani et al., 10 May 2026).

The paper proves that pure $B_R$0-local differential privacy is incompatible with genuinely input-dependent sparse supports: if $B_R$1 for some inputs, then some output has positive probability under one input and zero under the other, giving infinite privacy loss (Zamani et al., 10 May 2026). For approximate $B_R$2-LDP, the exact privacy defect depends on the separation $B_R$3. In the radius-truncated case,

$B_R$4

because the supports are disjoint (Zamani et al., 10 May 2026).

This yields a sharp feasibility threshold. Nontrivial $B_R$5-LDP on range $B_R$6 with $B_R$7 is impossible unless

$B_R$8

equivalently

$B_R$9

Under the clean sufficient regime

$r_\gamma$0

the overlap term vanishes and the privacy defect is controlled purely by support leakage, with

$r_\gamma$1

after identifying $r_\gamma$2 in the paper’s support-size notation (Zamani et al., 10 May 2026).

The same work shows that distortion moments

$r_\gamma$3

are nondecreasing in support size, hence in $r_\gamma$4. The design principle is therefore to choose the smallest support size, equivalently the smallest radius, that satisfies the target privacy constraint (Zamani et al., 10 May 2026).

6. Potential theory, inverse geometry, and recurrent distinctions

In classical potential theory near a sphere, the sphere radius can become the decisive Laplace-analytic scale. For a point source outside a dielectric sphere of radius $r_\gamma$5, the Kelvin transformation

$r_\gamma$6

uses $r_\gamma$7 as the inversion radius and sends the source at $r_\gamma$8 to the image point

$r_\gamma$9

The paper shows that the interior solution of Laplace’s equation is most efficiently represented by radially inverted irregular spheroidal harmonics whose singularity matches the semi-infinite image line of the analytically continued solution (Majić et al., 2017).

For multipole matrix elements of the Laplace Green function between two spheres of equal radius $R$00, the same radius sets the scale of the surface distributions, the Fourier-space factors $R$01, and the regime change at the overlap threshold

$R$02

In the non-overlapping regime $R$03, the coupled coefficients are proportional to $R$04; in the overlapping regime $R$05, they are expressed through generalized hypergeometric functions and often reduce to polynomials in $R$06 (Makuch et al., 2015).

For toroidal shells and solid tori, the paper explicitly states that it does not define any quantity literally called “Laplace radius.” The closest analogue is the major radius $R$07, with thickness controlled by

$R$08

The zeroth-order exterior potential of the torus is exactly the potential of a circular loop of radius $R$09 and the same mass, and for a homogeneous circular torus the first nontrivial correction is $R$10 (Huré et al., 2020).

In inverse resonance scattering on rotationally symmetric manifolds,

$R$11

the relevant geometric quantity is the rotation radius $R$12. The paper shows that, under compact support assumptions on the perturbation, this radius is uniquely determined by eigenvalues and resonances through the reduction of the Laplacian to one-dimensional Schrödinger operators (Isozaki et al., 2019). Again, the literature does not name this quantity a “Laplace radius,” but it is a radius profile encoded by Laplace spectral data.

Several recurrent misconceptions follow from these distinctions. First, in the nanodroplet literature the isotropic-core cutoff $R$13 is not the radius entering the Young–Laplace fit; the fit uses $R$14 as a proxy for $R$15 (Malek et al., 2017). Second, in radial Laplacian problems the radial coordinate itself is not a named “Laplace radius.” For example, the singular analysis of the radial Schrödinger reduction concerns the behavior of the Laplacian at $R$16 and the necessity of

$R$17

not a separate radius invariant (Khelashvili et al., 2015). Third, in radial-graph spectral geometry the decisive objects are $R$18, $R$19, and $R$20, rather than a distinguished “Laplace radius” (Matos et al., 2013).

Taken together, these usages show that “Laplace radius” denotes a family of radius notions attached to the Laplace operator, Laplace law, or discrete-Laplace kernel. The strict thermodynamic sense is the surface-of-tension radius of Young–Laplace theory; beyond that setting, the term functions mainly as a contextual shorthand for whichever radius parameter controls the corresponding Laplace problem.