Optimizes maps with controlled distortion for geometric tasks.
problem Free-boundary diffeomorphism optimization in geometric modeling.
method Least-squares quasiconformal (LSQC) operator and Spectral Beltrami Network (SBN).
result LSQC minimizer well-posed under mild conditions, stable under mesh refinement.
Leibniz cohomology reveals connections on manifolds.
problem Understanding connections on Riemannian manifolds using Leibniz cohomology.
method Expressing Levi-Civita connection as a cochain in Leibniz cohomology of vector fields.
result Vanishing of Leibniz coboundary implies eigenfunctions of the Laplacian.
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 L2-Hodge theory. result Closed explicit extension formulas for holomorphic canonical forms and pluricanonical forms.
Develops QC theory for 3D point clouds, extending existing algorithms.
problem No existing QC theories and algorithms for point cloud data.
method Introduces PCQC maps and PCBCs, proving convergence to continuous setting.
result PCBCs effectively measure local geometric distortions in PC deformations.
The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace…
Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.
problem Establishing bounds for eigenfunctions on product spaces.
method Combining asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.
result Sharp Lp bounds for eigenfunctions on products of rank-one symmetric spaces. Registration, which aims to find an optimal one-to-one correspondence between different data, is an important problem in various fields. This problem is especially challenging when large deformations occur. In this paper, we present a novel algorithm to obtain diffeomorphic image or surface registrations with large def…
Study elliptic operators on weighted manifolds using H-convergence.
problem Asymptotic behavior of elliptic operators on weighted Riemannian manifolds.
method H-convergence theory applied to uniformly elliptic operators with measurable coefficients.
result Established H-compactness result for elliptic operators on weighted Riemannian manifolds.
We generalize the Omori-Yau almost maximum principle of the Laplace-Beltrami operator on a complete Riemannian manifold M to a second-order linear semi-elliptic operator L with bounded coefficients and no zeroth order term. Using this result, we prove some Liouville-type theorems for a real-valued C2 function …
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.
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.
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.
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.
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.
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.
Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct bou…
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.
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 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.
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
New proof of Zelditch's generalization using Riemannian geometry.
problem Asymptotic formula for eigenfunction sums on compact manifolds.
method Riemannian geometry methods.
result Proof of Zelditch's generalization without Fourier integral operators.
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.
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…
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.
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.
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.
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.
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.
Study surfaces of finite Chen-type using Beltrami operators.
problem Characterize surfaces with finite Chen-type.
method Analyze Beltrami operators for surface fundamental forms.
result Identify surfaces of finite type in II and III forms.
Due to the isotropy d-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the R-radius hyperboloid model of d-dimensional hyperbolic geometry with R>0 and d≥2, we compute azimuthal Fourier expansions for a fundamental so…
The two main topics of this text are as follows: Firstly, three modifications of the theorem of Beltrami will be presented for diffeomorphisms between Riemannian manifolds and a space form which preserve the geodesic circles, the geodesic hyperspheres, or the minimal surfaces, respectively. Secondly, it is defined what…
The paper studies the spectrum of Laplace-Beltrami operators on complex spaces.
problem Analyzing the spectrum of Laplace-Beltrami operators on compact complex spaces.
method Examined the Friedrichs extension of Laplace-Beltrami and Hodge-Kodaira Laplacians, providing estimates for eigenvalues and trace-class properties.
result Discrete spectrum and trace-class properties of Laplace-Beltrami operators on compact complex spaces.
In this paper, classical isometric helicoidal and rotational surfaces are studied, and generalized by Bour's theorem in three dimensional Euclidean space. Moreover, the third Laplace-Beltrami operators of two classical surfaces are obtained.
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differ…
Study eigenvalues of Laplace-Beltrami under Ricci flow on 3-manifolds.
problem Eigenvalue behavior under Ricci flow on 3-manifolds.
method Monotonic quantities and bounds constructed for the first eigenvalue.
result Eigenvalue tends to zero in converging cases after rescaling.
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.
Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on Rn endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to [0,+∞).
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.
We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.
We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami field…