New method solves Beltrami equation using Hodge star.
problem Solving the Beltrami equation.
method Using Hodge star operator and elliptic PDE theory.
result Essentially unique homeomorphic solution found.
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48 results for “Beltrami equations”
The study of harmonic diffeomorphisms reduces to solving a specific Beltrami equation.
problem Classifying harmonic diffeomorphisms between surfaces.
method Reduction to solving a Beltrami equation and an elliptic sinh-Gordon equation.
result Solutions to the sinh-Gordon equation classify harmonic maps.
Holomorphic solutions vary in Sobolev spaces for Beltrami equations.
problem Analyzing the behavior of solutions to Beltrami equations in Sobolev spaces.
method Holomorphic dependence of solutions in Sobolev spaces of varying Beltrami differentials.
result Solutions vary holomorphically in Sobolev spaces of higher order.
The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.
problem Computing conformal mappings between Riemannian surfaces.
method Adapting the conjugate function method to Riemannian surfaces using -adaptive finite element methods.
result Highly accurate numerical computations of conformal mappings on surfaces, including complex geometries.
The article approximates solutions to the Beltrami equation using similarity surfaces.
problem Approximating solutions to the Beltrami equation.
method Constructing similarity surfaces from polygons and analyzing their conformal uniformization.
result Holomorphic dependence of Christoffel symbols on polygons and convergence to a specific affine connection.
Using the symmetry properties of two-dmensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it. We conjecture that the generalized Maurer-Cartan equation for the corresponding subalgebra gives solutions to the…
New Beltrami vector fields with polyhedral symmetries discovered.
problem Finding Beltrami vector fields with specific symmetries.
method Constructing vector fields with polyhedral symmetries using solutions to the Helmholtz equation.
result All simplest Beltrami fields with polyhedral symmetries are calculated.
The paper solves complex structure changes and Beltrami equations using Hodge theory.
problem Changes of complex structures on Kähler manifolds and solving Beltrami equations.
method Global geometric methods using Hodge theory and -Hodge theory.
result Closed explicit extension formulas for holomorphic canonical forms and pluricanonical forms.
Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.
problem Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
method Using open books, proved existence of non-vanishing steady solutions to the Euler equations for vector fields in odd dimensions.
result Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point cloud…
The paper proves Schauder estimates for Laplace-Beltrami on manifolds with fibered boundaries.
problem Analyzing heat-type equations on manifolds with specific boundary conditions.
method Proving Schauder estimates for the Laplace-Beltrami operator on manifolds with fibered boundaries and a Φ-metric.
result The proof of parabolic Schauder estimates for the Laplace-Beltrami operator.
This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
This paper solves a Calderón problem for Beltrami fields on manifolds.
problem Reconstructing a 3D manifold from boundary measurements of Beltrami fields.
method Defined a normal-to-tangential map for Beltrami fields and used it to reconstruct the manifold.
result A real-analytic 3-manifold can be reconstructed from its normal-to-tangential map.
This paper, focusing on the growth rate of the measure, gives pointwise bounds of solutions of eigenvalue equations of the Laplace-Beltrami operator on noncompact Riemannian manifolds.
Study examines conditions for entire bounded or large solutions to elliptic equations under radiality.
problem Conditions for existence of entire bounded or large solutions to elliptic equations.
method Analyzes Laplace operator and Laplace Beltrami operator on harmonic groups and Euclidean spaces, with focus on radiality of the equation.
result Necessary and sufficient conditions for existence of entire bounded or large solutions are provided.
Solves sigma model on U(3)/U(1)^3, describing geodesics and spectrum.
problem Classical and quantum problems for 1D sigma model with specific target space.
method Mapping to Gaudin model, solving polynomial equations.
result Explicit description of geodesics and spectrum found.
GNPs learn operators on non-Euclidean geometries using neural networks.
problem Learning operators on complex geometries like manifolds.
method Geometric Neural Operators (GNPs) that incorporate geometric properties.
result GNPs can estimate metrics, solve PDEs, and learn LB operators on manifolds.
Extremal spectral properties of Lawson tau-surfaces are investigated. The Lawson tau-surfaces form a two-parametric family of tori or Klein bottles minimally immersed in the standard unitary three-dimensional sphere. A Lawson tau-surface carries an extremal metric for some eigenvalue of the Laplace-Beltrami operator. U…
Unified geometric framework for Brownian motion on various manifolds.
problem Modeling Brownian motion on complex Riemannian manifolds.
method Constructing stochastic differential equations with noise and drift terms aligned with Laplace-Beltrami operators.
result Geometrically transparent and mathematically consistent foundation for diffusion processes.
Physics-based method approximates mean curvature on surface meshes.
problem Estimating mean curvature on triangulated surfaces.
method Derives approximation from Young-Laplace equation and force balance.
result Approximation equivalent to discrete Laplace-Beltrami operator.
We consider the eigenvalue equation for the Laplace-Beltrami operator acting on scalar functions on the non-compact Eguchi-Hanson space. The corresponding differential equation is reducible to a confluent Heun equation with Ince symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their asymptotic s…
We introduce a general notion of twistorial map and classify twistorial harmonic morphisms with one-dimensional fibres from self-dual four-manifolds. Such maps can be characterised as those which pull back Abelian monopoles to self-dual connections. In fact, the constructions involve solving a generalised monopole equa…
Simplified Beltrami's theorem using geometry.
problem Beltrami's theorem
method Parabolic differential geometry
result Simplified approach to Beltrami's theorem
New proof of Finslerian Beltrami theorem in 2D.
problem Generalizing Beltrami's theorem to Finslerian geometry.
method Presented a new proof of the theorem.
result Validated the theorem in Finslerian geometry, including 2D.
New Skyrme model for contact geometry with topological solutions.
problem Finding BPS solutions for maps between contact 3-manifolds.
method Defined a new Skyrme energy functional for maps between contact 3-manifolds and showed existence of solutions to a first-order self-duality equation.
result Existence of solutions to the Beltrami maps equation, generalizing the original Ferreira-Zakrzewski model.
The paper constructs Laplace-Beltrami operators on noncommutative tori.
problem Developing Laplace-Beltrami operators for noncommutative tori.
method Construction of Laplace-Beltrami operators with consideration of non-trivial modular automorphisms.
result Laplace-Beltrami operators on noncommutative tori have properties similar to those on ordinary Riemannian manifolds.
Graph neural network using Beltrami flow for feature and topology evolution.
problem Efficient feature learning and topology evolution on graphs.
method Discretized Beltrami flow applied to graph neural networks with positional encodings.
result Achieves state-of-the-art results on various benchmarks.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
Two unique Beltrami vector fields with icosahedral symmetry are constructed.
problem Constructing Beltrami vector fields with specific symmetries.
method Explicit construction of vector fields with icosahedral symmetry.
result Two unique Beltrami vector fields with icosahedral symmetry are found.
Bounds on Multiplicities of Laplace-Beltrami Operator Eigenvalues on the Real Projective Planemath.DG
Improved bounds on Laplace-Beltrami operator eigenvalues on real projective plane.
problem Improving upper bounds for Laplace-Beltrami operator eigenvalues.
method Analyzing eigenvalues with even indexes and providing bounds for Dirichlet, Neumann, and Steklov eigenvalues.
result Enhanced upper bounds for Laplace-Beltrami operator eigenvalues on the real projective plane.
Formula derived for Laplace-Beltrami on Stiefel manifold.
problem Finding Laplace-Beltrami operator on Stiefel manifold.
method Using the general framework of Laplace operators on constraint manifolds, derived the explicit formula in terms of ambient Euclidean coordinates.
result Extended previously known formulas for sphere and special orthogonal group.
Study improves Beltrami fields with nonconstant proportionality factors.
problem Identify nonconstant functions as proportionality factors for Beltrami fields.
method Applied Cartan's method of moving frames and exterior differential systems.
result The space of associated Beltrami fields depends on the geometry of the level surfaces of the proportionality factor.
Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.
problem Computing Laplace-Beltrami on constrained submanifolds.
method Embedded gradient vector field method, explicit formula derivation.
result Explicit formula for Laplace-Beltrami on orthogonal group.
Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.
problem Determining the domain of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds with tangency points.
method Using tools from Lie groupoids, natural domains of perturbations are found.
result Method allows treatment of geometries with tangency points.
Solve Beltrami problem in dimension two
problem Geodesically equivalent metric pairs in dimension two
method Provide a complete local classification
result Full solution of the problem in dimension two
Combines non-Euclidean and de Sitter geometries on the plane.
problem Exploring Penrose's Conformal Cyclic Cosmology.
method Geometric model combining Beltrami-Klein and de Sitter spaces.
result Discovery of hidden ${f G}_2$ symmetry in de Sitter spaces.
Paper provides integral representation for fractional Laplace-Beltrami operators.
problem Fractional Laplace-Beltrami operators on general Riemannian manifolds.
method Two different proofs for the same result: compact manifolds with or without boundary, and riemannian manifolds without boundary with bounded Ricci curvature.
result Integral representation of fractional Laplace-Beltrami operators for general Riemannian manifolds.
We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1, where the L-B operator is for the metric g = e^{2u}g_0, with the standard Euclidean metric on R^2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order, Delta u +K_g e^(2u)=0, D…
We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct steady nonsingular solutions to the Euler equations on a Riemannian whose flowlines trace out closed curves of all possible knot and link types simultaneously. Using careful contact-topological …
Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as 3D coordinates each vertex is mapped to, which often requires lots of storage memory. This causes inconvenien…
The First Eigenvalue for the Bi-Beltrami-Laplacian on Minimal Isoparametric Hypersurfaces of math.DG
Paper finds first eigenvalues on minimal isoparametric hypersurfaces.
problem Finding first eigenvalues of bi-Beltrami-Laplacian on minimal isoparametric hypersurfaces.
method Variational argument to overcome difficulties in proving limit theorem.
result Determine the first eigenvalues of bi-Beltrami-Laplacian on isoparametric hypersurfaces.
The Navier-Stokes equations on certain manifolds can perform universal computation.
problem Computational universality in viscous fluids.
method Cosymplectic geometry and harmonic 1-forms.
result Stationary Navier-Stokes solutions exhibit Turing completeness.
Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.
problem Geodesics and Laplace-Beltrami spectrum on flag manifolds.
method Invariant metrics and finite-dimensional approximations.
result Explicit solutions for geodesics and spectrum.
The paper finds solutions to a curvature equation using maximum/minimum points of a metric function.
problem Finding solutions to a specific curvature equation on Riemannian manifolds.
method Analyzes the constant -curvature equation and uses the function to generate solutions.
result Positive solutions are generated by maximum or minimum points of the function .
We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…
Due to the isotropy -dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the -radius hyperboloid model of -dimensional hyperbolic geometry with and , we compute azimuthal Fourier expansions for a fundamental so…
Finsler metrics of constant curvature characterized, with a Finslerian Beltrami Theorem.
problem Characterizing Finsler metrics of constant curvature.
method Defined a Weyl-type curvature tensor to characterize Finsler metrics of constant flag curvature.
result Provided a Finslerian version of Beltrami Theorem.
Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.
problem Estimating eigenvalues of the Beltrami-Laplacian on manifolds with Bakry-Émery Ricci curvature.
method Using Bakry-Émery Ricci curvature conditions, the paper derives lower bounds for eigenvalues of the Beltrami-Laplacian.
result Lower bounds for eigenvalues of the Beltrami-Laplacian depend on curvature, gradient bounds, dimension, and diameter of the manifold.