Abstract In quantum theory on curved backgrounds, Heisenberg's uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann--L"uders projection onto a geodesic ball B(r) of radius r on a Cauchy slice, with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation σ p and show that strict confinement to B(r) enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue λ 1 of the Laplace--Beltrami operator on the ball, σ p ≥ √λ 1 , and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus |ψ| from phase-gradient fluctuations and clarifies how the spectral geometry controls momentum uncertainty. 
 Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, σ p r ≥ \pi\hbar/2, depending only on the proper radius r.

We prove resolvent estimates in Schatten spaces for Laplace- Beltrami operators on compact manifolds at the critical exponent. Our proof only uses known bounds for the Hadamard parametrix.

3-D Shape Identity Matching Across Domains Using Shared Kernel Transform Learning With Laplacian–Beltrami Operator

Abstract In this paper, we use the nonhomogeneous Beltrami equation to give an optimal solution to the nonhomogeneous Cauchy–Riemann equation for continuous or smooth families of complex structures and (0, 1)-forms of a Hölder class on a smooth open orientable surface. As an application, we obtain the Oka–Grauert principle for complex line bundles on families of open Riemann surfaces.

The symmetrized Asymptotic Mean Value Laplacian \tilde{\Delta} , obtained as limit of approximating operators \tilde{\Delta}_{r} , is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as r \downarrow 0 , the operators \tilde{\Delta}_{r} eventually admit isolated eigenvalues defined via min-max procedure on any compact uniformly locally doubling metric measure space. Then we prove L^{2} and spectral convergence of \tilde{\Delta}_{r} to the Laplace–Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary.

Abstract Bases from the eigenfunctions of the Laplace–Beltrami operator (LBO), called LBO bases, are popularly used to construct functional mappings for shape correspondence. Although many efforts have been made to improve LBO basis construction and their application in shape correspondence, they often overlook the role of shape feature scales in determining the suitability of LBO bases. This mismatch between the selected LBO bases and shape features results in poor representation, hindering shape correspondence and requiring more iterations for convergence, ultimately reducing efficiency. In this paper, we present an attention‐based module that adaptively learns weights based on the scales of shape features to better utilise LBO bases. This ensures that the selected LBO bases have frequencies that align with the scales of shape features, addressing the mismatch problem and improving feature representation. By filtering out LBO bases with incompatible frequencies, our approach enhances shape correspondence while reducing the number of iterations required for convergence, thereby improving efficiency. Additionally, the selected LBO bases can be easily integrated with existing methods, such as the test‐time adaptation strategy, to further enhance shape correspondence. Experimental results demonstrate that our method achieves higher‐quality results than state‐of‐the‐art methods while at a high efficiency.

Problems associated with data visualization in multidimensional space are considered. One option discussed is the use of Riemannian space with variable curvature in magnitude and sign for modeling the visualization space. The Hilbert-Einstein, Winslow, and Beltrami equations are considered for modeling the visualization and perception space using geometry. The Beltrami equations can, to some extent, mitigate the problems associated with visualizing multidimensional functions, but are limited by two-dimensionality. The use of Hilbert-Einstein equations is complicated by both the ambiguity of interpreting a priori information and technical difficulties. The most promising approach appears to be the use of Winslow-type equations, which correspond to the construction of harmonic coordinates for the Hilbert-Einstein equations.

The article is devoted to the Dirichlet problem with continuous data for a semi-linear Beltrami equation from the complex analysis. One of our main results is a theorem on existence and regularity of its solution in an arbitrary bounded simply connected domain with a representation through the corresponding generalized analytic function with a source. Similarly, we prove the existence of its multi-valued solutions in the spirit of multi-valued analytic functions for the Dirichlet problem in arbitrary bounded domains D with no boundary component degenerated to a single point. On this basis, we derive existence of a regular solution of the Dirichlet problem in such domains for semi-linear Poisson type equations div [ A ( z ) grad U ( z ) ] = G ( z ) Q ( U ( z ) ) with its representation through suitable generalized harmonic functions with sources.

We investigate the global existence and exponential decay of mild solutions for the Boussinesq systems in L p -phase spaces on the framework of a real hyperbolic manifold H d ( R ) , where d ⩾ 2 and 1 < p ⩽ d . We consider a couple of Ebin–Marsden’s Laplace and Laplace–Beltrami operators associated with the corresponding linear system which provides a vectorial matrix semigroup. First, we show the existence and the uniqueness of the bounded mild solution for the linear system by using dispersive and smoothing estimates of the vectorial matrix semigroup. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the bounded mild solutions. By using Gronwall’s inequality, we establish the exponential stability of such solutions. Finally, we give an application of stability to the existence of periodic mild solutions for the Boussinesq systems.

ABSTRACT This article will study the heat equation , with dynamical boundary condition , where is a bounded domain with smooth boundary and denote the Laplace (Beltrami) operators on and , respectively, denotes the outward normal derivative in the trace sense on , and . The analyticity of the associated semigroups is proved, based on the semigroup theory of one‐sided coupled operator matrices.

Spectral estimate for the Laplace–Beltrami operator on the hyperbolic half-plane

The Ahlfors-Beurling transform is one of the important operators in complex analysis. It is the "Hilbert transform" on complex plane. This transform plays an essential role in applications to the theory of quasiconformal mappings and to the Beltrami equation with discontinuous coeffcients. Therefore, approximations of the Ahlfors-Beurling transform are of great interest. This article is devoted to the approximation of the Ahlfors- Beurling transform in the Lebesgue spaces. We prove that the approximating operators are bounded maps in the Lebesgue spaces and converges strongly to the Ahlfors-Beurling transform

Atlas-free continuous structural connectivity has garnered increasing attention due to the limitations of atlas-based approaches, including the arbitrary selection of brain atlases and potential information loss. Typically, continuous structural connectivity is represented by a probability density function, with kernel density estimation as a common estimation method. However, constructing an appropriate kernel function on the cortical surface poses significant challenges. Current methods often inflate the cortical surface into a sphere and apply the spherical heat kernel, introducing distortions to density estimation. In this study, we propose a novel approach using the Riemannian diffusion kernel derived from the Laplace–Beltrami operator on the cortical surface to smooth streamline endpoints into a continuous density. Our method inherently accounts for the complex geometry of the cortical surface and exhibits computational efficiency, even with dense tractography datasets. Additionally, we investigate the number of streamlines or fiber tracts required to achieve a reliable continuous representation of structural connectivity. Through simulations and analyses of data from the Adolescent Brain Cognitive Development (ABCD) Study, we demonstrate the potential of the Riemannian diffusion kernel in enhancing the estimation and analysis of continuous structural connectivity.

Generalized Beltrami Fields. Exact Solutions

Here, we study boundary value problems for the Laplace–Beltrami operator on a three-dimensional sphere with a circular cut, obtained by removing a smooth closed geodesic from S3 embedded in R4. The presence of the cut introduces singular perturbations of the domain, and we develop an analytical framework to characterize well-posed problems in this setting. Our approach combines Green’s functions, spectral analysis, and Sobolev space methods to establish solvability criteria and uniqueness results. In particular, we identify explicit conditions for the existence of solutions with data supported near the cut, and extend the formulation to include delta-type perturbations supported on the removed circle. These results generalize earlier work on punctured two-dimensional spheres and provide a foundation for the study of PDEs on manifolds with localized singularities.

The challenge in weak gravitational lensing caused by galaxies and clusters is to infer the projected mass density distribution from gravitational lensing measurements, known as the inversion problem. We introduce a novel theoretical approach to solving the inversion problem. The cornerstone of the proposed method lies in a complex formalism that describes the lens mapping as a quasi-conformal mapping with the Beltrami coefficient given by the negative of the reduced shear, which can, in principle, be observed from the image ellipticities. We propose an algorithm called QCLens that is based on this complex formalism. QCLens computes the underlying quasi-conformal mapping using a finite element approach by reducing the problem to two elliptic partial differential equations that solely depend on the reduced shear field. Experimental results for both the Schwarzschild and the singular isothermal lens demonstrate the agreement of our proposed method with the analytically computable solutions.

The article is a review of our results on the compactness theorems for the families of solutions of the Beltrami equation and the corresponding Dirichlet problem. The review includes the following results: 1) theorems on the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation in some Jordan domain whose characteristics have a compact support and satisfy certain constraints of an integral nature; 2) theorems on the compact classes of solutions of the Dirichlet problem for the Beltrami equation with integral constraints, which is considered in some Jordan domain; 3) results on the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation, whise characteristics have a compact support and satisfy certain constraints of set-theoretic type; 4) results on the compact classes of solutions of the Dirichlet problem for the Beltrami equation, which are considered in some Jordan domain and whose characteristics satisfy certain set-theoretic-type constraints; 5) results on the compactness of solutions of the Dirichlet problem in a simply connected domain, which are obtained in terms of simple ends in the case when the maximal dilations of these solutions satisfy certain integral constraints; 6) theorems on the existence of homeomorphic ACL-solutions of the quasilinear Beltrami equation with two characteristics under certain conditions on complex coefficients; 7) theorems on the existence of continuous ACL-solutions of the Beltrami equation, which are logarithmically Holder in a given domain; and 8) results on the existence of continuous solutions of the Beltrami equations with two characteristics, which satisfy a certain condition on the dilation of inverse mappings, and the solutions themselves satisfy the condition of hydrodynamic normalization in the vicinity of infinity. The vast majority of the results were included in the Ph.D. thesis of the first co-author.

The definition of a $Q$-homeomorphism with respect to $p$-modulus is a natural generalization of the geometric definition of a quasiconformal mapping: if $Q(z)\leqslant K<\infty$ a.e. (almost everywhere), then $f$ is quasiconformal as $p=2$ on the complex plane $\mathbb{C}$, see Definition A, p. 21--22 in \cite{A}, $f$ has local Lipschitz property as $1<p<2$, and $f^{-1}$ is local Lipschitz continuous as $p>2$, moreover the bounds for $p$ are sharp, see \cite{Ge}. The class of $Q$-homeomorphisms with respect to $p$-modulus as $p=2$ was first considered in the work \cite{MRSY3}. The definition of a ring $Q$-homeomorphism with respect to $p$-modulus generalizes and localizes the definition of a $Q$-homeomorphism with respect to $p$-modulus and is motivated by the ring definition of quasiconformal mappings in the sense of Gehring, see \cite{Ger2}, introduced originally by V.~Ryazanov, U.~Srebro, and E.~Yakubov on the complex plane as $p=2$, see \cite{DegBe}. The concept of lower $Q$-homeomorphisms with respect to $p$-modulus has also proven to be a very useful tool for studying planar and spatial mappings. This definition has also geometric character and is motivated by Gehring's ring definition of quasiconformity. The theory of lower and ring $Q$-homeomorphisms is applied in the study of boundary value problems for Beltrami equations on the complex plane, as well as spatial homeomorphisms of Sobolev classes and more general Orlicz-Sobolev classes, see \cite{GSSCAOT,RSSY,SalAASF,ARSUMB2018,ARSDOP2019,ASZbPrIM,RSSUMZ2013,RSSUMZ2016,SalZbPrI2015,SalMStud2015,SalUMBOS2016}. The theory of lower and ring $Q$-homeomorphisms with respect to $p$-modulus can also be applied to mappings that are quasiconformal in the mean, see \cite{Golberg2,Kru}. Recently, lower and ring $Q$-homeomorphisms with respect to nonconformal modulus have been applied to studying the properties of regular solutions of nonlinear Beltrami equations, see \cite{GolSal,SalStef23,PSSTM}. Furthermore, the theory of ring $Q$-homeomorphisms can be applied to the study of mappings with finite distortion that belong to Orlicz-Sobolev classes $W^{1,\varphi}_{\rm loc}$ under a Calderon-type condition, and, in particular, to Sobolev classes $W^{1,p}_{\rm loc}$ as $p>n-1$, див. \cite{G1}. In the paper, asymptotic behavior at infinity of lower $Q$-homeomorphisms with respect to the $p$-module as $1<p<2$ in the complex plane has been investigated. A sufficient condition on the norm of the function $Q$ under which the mapping exhibits exponential growth at infinity has been found. An example to demonstrate the sharpness of the obtained results has been constructed.

In this paper, using the isoperimetric method, the local and asymptotic properties of regular homeomorphisms at an arbitrary point in a domain of complex plane have been studied. The results were applied to regular solutions of the nonlinear Beltrami equation.

This study investigates three-dimensional two-phase flows in complex geometries found in industrial process equipment design using finite-element numerical simulations. The governing equations are formulated in three-dimensional Cartesian coordinates and solved on unstructured meshes employing the Taylor–Hood “Mini” element, selected for its numerical stability and convergence properties. The convective term in the momentum equation is discretized using a first-order semi-Lagrangian scheme. The two fluid phases are separated by an interface mesh composed of triangular surface elements, which is independent of the primary volumetric fluid mesh. Surface tension effects are incorporated as a source term using the continuum surface force (CSF) model, with the curvature computed via the Laplace–Beltrami operator. At each time step, the positions of the interface mesh nodes are updated according to the local fluid velocity field. The results show that the methodology is stable and can be used to accurately model two-phase flows in complex geometries found in several engineering solutions.