0.0.0.1 Keywords.

Heisenberg-type inequality; quantum mechanics on curved spacetime; spacelike hypersurfaces; spectral geometry

0.0.0.2 PACS numbers:

03.65.Ta, 04.62.+v, 04.20.-q, 04.20.Cv, 02.30.Tb.

1 Introduction↩︎

On a Riemannian manifold \((\Sigma,h)\), strict localization of a quantum state to a bounded domain \(\Omega\subset\Sigma\) with Dirichlet boundary conditions entails a non-vanishing spectral cost for the momentum observable: once \(\psi|_{\partial\Omega}=0\), the \(L^{2}\)-norm of its gradient cannot be arbitrarily small. In spectral terms, the intrinsic uncertainty is governed by the first Dirichlet eigenvalue of the Laplace–Beltrami operator on \(\Omega\) [2]. This paper applies this principle to canonical data on Cauchy slices of curved spacetimes, making it directly usable from a physics perspective. On any spacelike hypersurface \((\Sigma,h)\) in a curved spacetime, we prove the following lower bound for states strictly localized to geodesic balls \(B_{\Sigma}(p,r)\)² \[\sigma_{p}(\psi)\;\ge\;\hbar\,\sqrt{\lambda_1},\] where \(\sigma_{p}(\psi)\) denotes the momentum standard deviation (Theorem 1). The estimate is coordinate- and foliation-invariant: it depends only on the induced Riemannian data of the slice and is manifestly insensitive to lapse, shift, and extrinsic curvature. In particular, the variance decomposition 12 and the kinetic-energy window \[\hbar\|\nabla^{h}|\psi|\|_{L^{2}}\;\le\;\sigma_{p}(\psi)\;\le\;\hbar\|\nabla^{h}\psi\|_{L^{2}},\] cf. 14 , separate the geometric contribution carried by \(|\psi|\) from the fluctuations of the phase gradient in an orthonormal frame. Operationally, this gives a clean, gauge-invariant bridge between spectral geometry on \((\Sigma,h)\) and momentum uncertainty defined by canonical commutators on the slice.

Beyond the main bound, we obtain a universal Heisenberg-type inequality \[\sigma_{p}(\psi)\,r\ge \frac{\hbar}{2}\] on weakly mean-convex geodesic balls of radius \(r\), see Corollary 1. The argument relies solely on the boundary-distance Hardy inequality, available under the distributional superharmonicity of the distance to \(\partial B_{\Sigma}(p,r)\) [3]–[5]. This baseline is never attained and its constant is optimal, hence it can be used as a robust, geometry-free floor whenever only coarse boundary information is accessible.

The results here abstract and extend the framework introduced for black-hole slices in [1]: no symmetry, stationarity, or vacuum assumption is required, and the statement is purely Riemannian on the chosen slice. When curvature information is available, classical comparison techniques immediately transfer to quantitative control of \(\lambda_{1}\) and thus of \(\sigma_{p}\) : see, e.g., Cheng-type eigenvalue comparison theorems [6], [7], small-ball asymptotics [8], and explicit radial analyses on spherically symmetric manifolds [9]. In constant positive curvature, our universal product floor coexists with stronger model-specific bounds derived in [10].

Organization. Section 2 introduces the minimal geometric setting and the precise Dirichlet localization; in particular, Assumption 1 records the standing conditions for \(B_{\Sigma}(p,r)\). Section 3 establishes the intrinsic Heisenberg bound (Theorem 1). The subsection containing Corollary 1 derives the universal Hardy-Heisenberg baseline 15 . Section 4 closes with a discussion of scope, extensions (including spinors/forms), and computational aspects.

2 Minimal geometric setting↩︎

Let \((\mathcal{M},g)\) be a time-oriented \(C^{2}\) Lorentzian manifold solving the Einstein field equations (with arbitrary stress-energy tensor and possibly \(\Lambda\)). Let \(\Sigma\subset\mathcal{M}\) be a \(C^{2}\) spacelike hypersurface with induced Riemannian metric \(h\) and volume element \(\mathrm d\mu_{h}\). Fix \(p\in\Sigma\) and \(r>0\). Since \(\Sigma\) is spacelike, it admits a unique future-directed unit normal field \(n\) along \(\Sigma\) with \(g(n,n)=-1\). The induced metric is \(h(X,Y):=g(X,Y)\) for \(X,Y\in T\Sigma\), and the Levi–Civita connection \(\nabla^{h}\) on \((\Sigma,h)\) is the tangential projection of the spacetime connection \(\nabla\). A tubular neighborhood of \(\Sigma\) is obtained by the normal exponential map \(\exp^{g}\): for some \(\varepsilon>0\), the map \[\Phi:(-\varepsilon,\varepsilon)\times\Sigma\to\mathcal{M},\qquad (t,x)\mapsto \exp^{g}_{x}\!\big(t\,n(x)\big)\] is a diffeomorphism onto its image; the level sets \(\Sigma_{t}:=\Phi(\{t\}\times\Sigma)\) define a local foliation by spacelike slices with \(\Sigma_{0}=\Sigma\). In these (Gaussian normal) coordinates, the metric takes the block form \[g=-\mathrm dt^{2}+h_{t},\] where \(h_{t}\) is a \(C^{1}\) family of Riemannian metrics on \(\Sigma\) with \(h_{0}=h\). More generally, for an arbitrary local time³ function \(t\) with \(\Sigma=\{t=0\}\), the \((3+1)\)-decomposition reads \[g=-N^{2}\mathrm dt^{2}+h_{ij}(t,x)\big(\mathrm dx^{i}+\beta^{i}\mathrm dt\big)\big(\mathrm dx^{j}+\beta^{j}\mathrm dt\big),\] with lapse \(N>0\) and shift vector field \(\beta\). The extrinsic curvature (second fundamental form) of \(\Sigma\) is \[K(X,Y):=-\,g(\nabla_{X}n,Y)= -\tfrac12\,(\mathcal{L}_{n}h)(X,Y),\] and, in Gaussian normal coordinates, \(\partial_{t}h\!\mid_{t=0}=-2K\). None of the extrinsic data \((K,N,\beta)\) will be used in our estimates below; the bound depends only on the intrinsic geometry of \((\Sigma,h)\) and is therefore coordinate- and foliation-independent.

2.0.0.1 Geodesic balls on the slice.

All domains are taken within \((\Sigma,h)\). We write \(B_{\Sigma}(p,r)\) for the \(h\)-geodesic ball of radius \(r\) centered at \(p\). Throughout we choose \(r\) strictly below the injectivity radius at \(p\) so that \(B_{\Sigma}(p,r)\) is a bounded Lipschitz domain; in particular, the Dirichlet Laplacian on \(B_{\Sigma}(p,r)\) has compact resolvent. These minimal hypotheses are recorded in Assumption 1 and will be in force for the main intrinsic estimate.

2.1 Hilbert space on geodesic balls↩︎

We work on an \(n\)-dimensional (or \(n=3\)) Riemannian manifold \((\Sigma,h)\) and restrict attention to the geodesic ball \(B_\Sigma(p,r)\subset(\Sigma,h)\). Let \(\{\partial_i\}\) be a local coordinate basis and let \(\{X_a\}_{a=1}^n\) be an \(h\)-orthonormal frame. Indices \(i,j,k,\dots\) refer to coordinates (raised/lowered with \(h\)), while \(a,b,c,\dots\) refer to the orthonormal frame (raised/lowered with \(\delta\)). The frames are related by a vielbein \(e_a{}^{\,i}\) and its inverse \(e^a{}_{\,i}\): \[X_a=e_a{}^{\,i}\,\partial_i,\qquad \partial_i=e_i{}^{\,a}\,X_a,\qquad e_i{}^{\,a}e_a{}^{\,j}=\delta_i^{\,j},\qquad e_a{}^{\,i}e_i{}^{\,b}=\delta_a^{\,b},\] so that \[h_{ij}=e_i{}^{\,a}e_j{}^{\,b}\,\delta_{ab},\qquad h^{ij}=e^i{}_{\,a}e^j{}_{\,b}\,\delta^{ab},\qquad d\mu_h=\sqrt{|h|}\,d^n x .\] All integrals and \(L^2\) norms below are taken over \(B_\Sigma(p,r)\) with respect to \(d\mu_h\).

For \(v=\sum_{a=1}^n v^a X_a(x)\in T_x\Sigma\) we set \[\|v(x)\|_h^2 := h(v,v) = \sum_{a=1}^n |v^a(x)|^2 .\] The pointwise gradient norm is \[\label{eq:gradnorm} \|\nabla^h \psi(x)\|_h^2 := h(\nabla^h \psi,\nabla^h \psi) = \sum_{a=1}^n |X_a \psi(x)|^2 ,\qquad \psi:\Sigma\to\mathbb{C}.\tag{1}\] The \(L^2\) inner product and norms are \[\langle f,g\rangle_{L^2} := \int_{B_\Sigma(p,r)} \overline{f}\,g\,d\mu_h,\qquad \|f\|_{L^2} := \Big(\int_{B_\Sigma(p,r)} |f|^2\,d\mu_h\Big)^{1/2},\] and, for vector fields \(V(x)\) on \(B_\Sigma(p,r)\), \[\begin{align} \|V\|_{L^2} &:=& \Big(\int_{B_\Sigma(p,r)} \|V(x)\|_h^2\,d\mu_h\Big)^{1/2},\\ \|\nabla^h \psi\|_{L^2} &:=& \Big(\int_{B_\Sigma(p,r)} \|\nabla^h \psi\|_h^{2}\,d\mu_h\Big)^{1/2} = \Big(\sum_{a=1}^n \!\int_{B_\Sigma(p,r)} |X_a\psi|^2\,d\mu_h\Big)^{\!1/2}. \end{align}\] For normalized states \(\psi\in L^2(B_\Sigma(p,r),d\mu_h)\) with \(\|\psi\|_{L^2}=1\), we write expectation values as \(\langle A\rangle := \langle \psi, A\psi\rangle_{L^2}\).

2.2 Momentum operator and variance decomposition↩︎

In an \(h\)-orthonormal frame the (DeWitt–symmetrized) momentum components are \[\label{eq:Pa} P_a:=-\,i\hbar\Big(\nabla^h_{X_a}+\tfrac12\,\mathrm{div}_h X_a\Big),\tag{2}\] which are symmetric on the natural Dirichlet form domain (e.g.\(H^1_0(\Omega)\) on a domain \(\Omega\subset\Sigma\)). The momentum variance with respect to \(\psi\) is the quadratic form \[\begin{align} \label{eq:VarP} \mathrm{Var}_\psi(P) := \Big\langle \big\| P-\langle P\rangle \big\|_h^2 \Big\rangle \end{align}\tag{3}\] with \(P:=(P_a)_{a=1}^n\) and \(\langle P\rangle=(\langle P_a\rangle)_{a=1}^n\). For clarity, we set \(\big\langle\|P\|_{h}^{2}\big\rangle := \sum_{a=1}^{n} \langle \psi,\, P_{a}^{\dagger} P_{a}\, \psi \rangle\); hence (by Green’s formula with Dirichlet data), we obtain the index‑free identity \[\big\langle \,\|P\|^2_h\,\big\rangle = -\,\hbar^2 \langle \psi,\,\Delta_h \psi\rangle =\hbar^{2}\,\|\nabla^h \psi\|_{L^2}^{2},\] which leads to \[\label{eq:variance1} \mathrm{Var}_{\psi}(P)=\hbar^{2}\,\|\nabla^h \psi\|_{L^2}^{2}\;-\;\|\langle P\rangle\|^{2}_h.\tag{4}\] We measure momentum uncertainty by the standard deviation \[\label{eq:sigmadef} \sigma_{p}(\psi):=\sqrt{\mathrm{Var}_{\psi}(P)}.\tag{5}\]

Geometric content. Eqs. 2 –5 are framed entirely in the induced Riemannian geometry \((\Sigma,h)\): they are coordinate- and foliation-independent, use only bulk Dirichlet data on \(B_\Sigma(p,r)\), and prepare the ground for comparisons with the Dirichlet spectrum via Rayleigh–Ritz.

From a physicist’s viewpoint, 2 –5 say: “variance of momentum on a curved, bounded region equals kinetic energy (in units \(\hbar^{2}\)) with the global drift subtracted.” Because 4 depends only on the intrinsic geometry of \((\Sigma,h)\) via \(\nabla^{h}\) and \(\mu_{h}\), it is coordinate-independent and well adapted to comparisons with spectral data.

3 An intrinsic Heisenberg-type lower bound↩︎

The identity 4 – i.e. the variance decomposition \(\mathrm{Var}_{\psi}(P)=\hbar^{2}\,\|\nabla^h \psi\|_{L^2}^{2}-\|\langle P\rangle\|_h^{2}\) – is derived in the Dirichlet setting with \(\psi\in H^{1}_{0}(B_{\Sigma}(p,r))\), where Green’s formula yields \(\langle\psi,P^{\dagger}P\,\psi\rangle=\hbar^{2}\|\nabla^h \psi\|_{L^2}^{2}\). For Neumann data (\(\partial_{n}\psi=0\)) the boundary term in Green’s identity also vanishes, so the same representation holds on the Neumann form domain \(H^{1}(B_{\Sigma}(p,r))\); however, the first Neumann eigenvalue is \(0\) on connected domains, which makes our spectral lower bound degenerate. For Robin data (\(\partial_{n}\psi+\kappa\psi=0\)) a surface contribution proportional to \(\int_{\partial B_{\Sigma}(p,r)}|\psi|^{2}\) survives, so the variance cannot be expressed purely by the bulk Dirichlet form. Accordingly, in the following we impose Dirichlet boundary conditions throughout and use the Dirichlet Laplacian with its Rayleigh–Ritz characterization. Write \(\lambda_{1}(\Omega;h)\) for the first Dirichlet eigenvalue of \(-\Delta_{h}\) on a bounded domain \(\Omega\subset\Sigma\): \[\label{eq:lambda1} \lambda_{1}(\Omega;h)\;=\;\inf\Bigg\{\frac{\int_{\Omega}\|\nabla^{h}u\|^{2}\,\mathrm d\mu_{h}}{\int_{\Omega}|u|^{2}\,\mathrm d\mu_{h}}: \; 0\neq u\in H^{1}_{0}(\Omega)\Bigg\}.\tag{6}\]

Proof. Write \(\psi = u\,e^{i\phi}\) with \(u = |\psi|\ge 0\), and fix an \(h\)-orthonormal frame \(\{X_a\}_{a=1}^n\) on \(B_\Sigma(p,r)\). For Dirichlet data (\(u|_{\partial B_\Sigma(p,r)}=0\)), integration by parts yields the components of the mean momentum \[\langle P_a\rangle = \hbar \int_{B_\Sigma(p,r)} u^{2}\, X_a \phi \, d\mu_h, \qquad a=1,\dots,n .\] Using the Madelung identity \(\|\nabla^{h}\psi\|_h^{2}=\|\nabla^{h}u\|_h^{2}+u^{2}\|\nabla^{h}\phi\|_h^{2}\), the variance formula 4 gives the exact decomposition \[\label{eq:VarX} \mathrm{Var}_{\psi}(P) = \hbar^{2}\!\int_{B_\Sigma(p,r)} \big(\|\nabla^{h}u\|_h^{2} + u^{2}\|\nabla^{h}\phi\|_h^{2}\big)\,d\mu_h \;-\; \hbar^{2}\sum_{a=1}^{n} \Big(\,\int_{B_\Sigma(p,r)} u^{2}\, X_a \phi \, d\mu_h\Big)^{\!2}.\tag{8}\] Since \(\|\psi\|_{L^2(B_\Sigma(p,r),d\mu_h)}=1\), the weight \(u^{2}\,d\mu_h\) is a probability measure on \(B_\Sigma(p,r)\). Applying the Cauchy–Schwarz inequality with respect to this weight, we obtain \[\sum_{a}\!\Big(\int u^{2} X_a\phi \, d\mu_h\Big)^{\!2} \;\le\; \int u^{2}\sum_{a}(X_a\phi)^{2}\,d\mu_h \;\equiv\; \int u^{2}\|\nabla^{h}\phi\|_h^{2}\,d\mu_h .\] With \(u=|\psi|\) this yields \[\label{eq:VarBound} \mathrm{Var}_{\psi}(P) \;\ge\; \hbar^{2}\,\|\nabla^{h}|\psi|\,\|_{L^{2}}^{2}.\tag{9}\] Setting \(d\nu := u^{2}\,d\mu_h\) and, for any scalar field \(X\), \[E_{\nu}[X] := \int_{B_\Sigma(p,r)} X \, d\nu ,\] we rewrite 8 in terms of \(\nu\)-expectations as \[\label{eq:diff0} \mathrm{Var}_\psi(P) = \hbar^2 \|\nabla^h u\|_{L^2}^2 + \hbar^2\Big( E_\nu[\|\nabla^h\phi\|_h^2] - \sum_{a=1}^n (E_\nu[X_a\phi])^2 \Big).\tag{10}\] Invoking 1 , the difference of expectations simplifies to \[\begin{align} \label{eq:diff} E_\nu[\|\nabla^h\phi\|_h^2] - \sum_{a=1}^n (E_\nu[X_a\phi])^2 = \sum_{a=1}^n E_\nu\!\left[\, \big(X_a\phi -E_\nu[X_a\phi]\big)^{\!2}\, \right]. \end{align}\tag{11}\] In 11 the fluctuation term is written as a sum of variances of the scalar fields \(X_a\phi\) with respect to the probability measure \(d\nu\). One may rewrite this sum as the squared \(h\)-norm of a centered gradient vector and then take the \(\nu\)–expectation. Concretely, \[\sum_{a=1}^n E_\nu\Big[\big(X_a\phi - E_\nu[X_a\phi]\big)^{\!2}\Big] = E_\nu\!\left[\sum_{a=1}^n \big(X_a\phi - E_\nu[X_a\phi]\big)^{\!2}\right] = E_\nu\!\left[\;\|\,\nabla^h\phi - E_\nu[\nabla^h\phi]\,\|_h^{2}\,\right].\] The first equality is linearity of \(E_\nu[\,\cdot\,]\). The second uses the orthonormality of the frame \(\{X_a\}_{a=1}^n\) together with the definition of the pointwise \(h\)-norm in 1 . With 10 and 11 the variance admits the exact decomposition \[\label{eq:VarE2} \mathrm{Var}_\psi(P) = \hbar^2 \|\nabla^h u\|_{L^2}^2 + \hbar^2 E_\nu\!\left[\,\|\,\nabla^h \phi - E_\nu[\nabla^h \phi]\,\|_h^{2}\,\right],\tag{12}\] for \(\psi= u\, e^{i\phi}\). Hence, equality in 9 holds precisely when the gradient \(\nabla^h\phi\) is \(\nu\)-a.e.constant. This is the phase part of the equality condition in Theorem 1; together with \(|\psi|\) being a first Dirichlet eigenfunction it yields equality in 7 .

By the Dirichlet Rayleigh–Ritz characterization 6 (see [2]), for any \(0\neq u\in H^{1}_{0}(B_{\Sigma}(p,r))\), \[\label{eq:Rayleigh} \int_{B_{\Sigma}(p,r)}\|\nabla^{h}u\|_h^{2}\,\mathrm d\mu_{h} \;\ge\; \lambda_{1}\!\left(B_{\Sigma}(p,r);h\right)\int_{B_{\Sigma}(p,r)}|u|^{2}\,\mathrm d\mu_{h}.\tag{13}\] Taking \(u=|\psi|\) with \(\|\psi\|_{L^{2}}=1\) (hence \(\|u\|_{L^{2}}=1\)) and combining this with the lower bound 9 , we obtain \[\sigma_{p}(\psi)^{2} = \mathrm{Var}_{\psi}(P) \;\ge\;\hbar^{2}\!\int_{B_{\Sigma}(p,r)}\|\nabla^{h}|\psi|\,\|_h^{2}\,\mathrm d\mu_{h} \;\ge\;\hbar^{2}\lambda_{1}\!\left(B_{\Sigma}(p,r);h\right).\] Taking square roots yields 7 .\(\square\)

3.1 Universal Hardy baseline on weakly mean-convex balls↩︎

Informally, the boundary \(\partial B_{\Sigma}(p,r)\) of the geodesic ball is weakly mean-convex if it never “bulges outward” with negative mean curvature: the mean curvature with respect to the outward unit normal is nonnegative in the distributional sense, which in this paper is exactly equivalent to the boundary-distance⁵ function \(d(x)=\operatorname{dist}_{h}(x,\partial B_{\Sigma}(p,r))\) being superharmonic on \(B_{\Sigma}(p,r)\). This is the only extra hypothesis beyond Assumption 1 used to obtain the sharp boundary-distance Hardy estimate 16 , and – since \(d(x)\le r\) a.e. – the universal baseline product 15 recorded in Corollary 1.

3.1.0.1 Compatibility with curvature-dependent bounds on spheres.

Consider a slice \((\Sigma,h)\) of constant sectional curvature \(\kappa>0\) (e.g.the round \(S^3\)). For geodesic balls \(B_\Sigma(p,r)\) one has weak mean-convexity for \(0<r\le \pi/(2\sqrt{\kappa})\); hence Corollary 1 applies and yields the universal baseline 15 . By contrast, the model-specific estimate in [10], for \(\kappa>0\), \[\label{eq:Schuermann19} \sigma_p(\psi)\, r \;\ge\; \pi \hbar \,\sqrt{\,1 - \frac{\kappa\,}{\pi^{2}}\, r^{2}\,}\,,\tag{17}\] is valid on the larger interval \(0<r<\pi/\sqrt{\kappa}\) and decreases to \(0\) as \(r\uparrow \pi/\sqrt{\kappa}\) (antipodal radius), where mean-convexity fails. On the overlap \(0<r\le \pi/(2\sqrt{\kappa})\) the bound 17 is strictly stronger than 15 ; for instance, at \(r=\pi/(2\sqrt{\kappa})\) it gives \(\sigma_p\,r\ge \pi\hbar\sqrt{3}/2\gg \hbar/2\). Thus there is no contradiction: 15 is a foliation-independent, boundary-driven floor available under mean-convexity, while 17 is a curvature-sensitive refinement that extends beyond the convexity radius.

4 Discussion, scope, and extensions↩︎

The intrinsic lower bound of Theorem 1 shows that momentum uncertainty on a spacelike slice is governed solely by the Dirichlet spectrum of the induced Riemannian metric. The estimate is therefore insensitive to the lapse, the shift, and the extrinsic curvature, and it remains stable under changes of foliation. In combination with the exact variance decomposition, the kinetic-energy window 14 separates two contributions to \(\sigma_{p}\): the geometric cost encoded in the modulus \(|\psi|\) and the fluctuations of the phase gradient. The lower endpoint is controlled by the ground-state Dirichlet mode of \(-\Delta_{h}\), while the upper endpoint is attained precisely when the mean momentum vanishes. In this sense the result provides a coordinate-independent bridge between spectral geometry and an operational notion of uncertainty.

Under Assumption 1 and weak mean-convexity of the geodesic boundary, the sharp boundary-distance Hardy inequality yields the universal product estimate 15 . This floor is scale-invariant in \(r\) and depends only on the boundary distance \(d(x)\); neither interior curvature nor extrinsic spacetime data enter. Consequently, even without detailed geometric information one obtains a robust, geometry-free lower bound on \(\sigma_{p}\) that is available on sufficiently small balls (below the convexity radius) and in many natural situations. The constant is optimal and never attained, mirroring classical features of Hardy-type inequalities.

Beyond the Hardy floor, curvature hypotheses convert geometric control into spectral control of \(\lambda_{1}\) and hence into quantitative control of \(\sigma_{p}\). Classical eigenvalue comparison theorems under sectional or Ricci curvature bounds (Cheng-type inequalities) yield effective upper and lower bounds for the first Dirichlet eigenvalue of geodesic balls; see [6], [7]. In the small-radius regime, asymptotic expansions quantify the departure of \(\lambda_{1}\) from its Euclidean value in terms of curvature at the center point [8]. On spherically symmetric manifolds, the radial structure can be exploited to compute or bound spectra explicitly [9]. Each of these inputs transfers directly to \(\sigma_{p}\) via Theorem 1.

The Dirichlet framework enforces strict localization and expresses the variance purely through the bulk Dirichlet form, which makes spectral control transparent. On connected domains with Neumann data the ground state is constant and the spectral gap closes, so any induced lower bound degenerates. For Robin data a nonzero surface term persists in the variance, and the uncertainty can no longer be written purely in terms of the bulk form; any lower bound must then incorporate explicit boundary contributions. This clarifies why Dirichlet data are the natural modeling choice for the intrinsic statement proved here.

The argument is not restricted to scalar fields. Replacing \(-\Delta_{h}\) by the Dirac operator or the Hodge Laplacian yields analogous intrinsic lower bounds for spinors or differential forms, controlled by the corresponding first Dirichlet eigenvalues (up to operator-dependent constants). Foliation independence is retained because only the induced Riemannian data on the slice enter. Additional structures – such as gauge potentials or symmetry constraints – can be incorporated via standard variational methods together with diamagnetic or Hardy-type tools.

From a computational standpoint, standard finite-element or spectral discretizations of \(\lambda_{1}\) make the estimate numerically actionable for concrete geometries (for instance, in numerical relativity). Physically, the window 14 clarifies near-saturation behavior: equality at the upper endpoint corresponds to vanishing mean momentum, whereas equality at the lower endpoint forces the frame derivatives \(X_{a}(\arg\psi)\) to be almost everywhere constant. An explicit Euclidean model with ground-state modulus and affine phase exhibits saturation of the intrinsic bound with nonzero mean momentum.

Taken together, the universal Hardy baseline and curvature-sensitive refinements yield a two-tier structure: a geometry-free minimum available under weak mean-convexity and, when curvature information is present, sharper controls tailored to the ambient geometry. This template extends naturally to other strictly local observables, to anisotropic comparison results adapted to symmetry classes or foliations, and to measurement-theoretic formulations of quantum mechanics on curved backgrounds.

Acknowledgments↩︎

The author thanks the spectral geometry community for many classical tools; any errors are the author’s responsibility.

References↩︎