Study finds formulas for minimal submanifolds using Möbius transformations.
problem Understanding minimal submanifolds in Euclidean space.
method Monotonicity formulas for minimal submanifolds involving Möbius transformations.
result Proved formulas for minimal submanifolds under Möbius transformations.
Proposes differential and integral invariants under Mobius transformation.
problem Handling non-rigid deformation in 2-D and 3-D shapes.
method Focuses on Mobius transformation, proposes differential and integral invariants.
result Proposes differential and integral invariants under Mobius transformation.
New discretization of Möbius energy invariant under transformations.
problem Discretizing Möbius energy invariantly under Möbius transformations.
method Starting with Doyle and Schramm's cosine formula, we introduce a new discretization.
result Γ-convergence of the new discretized energies to the Möbius energy.
This paper classifies Möbius homogeneous hypersurfaces in a sphere.
problem Classifying hypersurfaces in a sphere under Möbius transformations.
method Using Möbius transformation group to classify hypersurfaces.
result Möbius homogeneous hypersurfaces are completely classified.
Proves constraints on groups extending Möbius transformations on spheres.
problem Constraints on groups extending Möbius transformations on spheres.
method Proved constraints through group transitivity and topological entropy analysis.
result Groups must be 4-transitive or arc 4-transitive, and contain elements of positive topological entropy.
CMC-1 surfaces linked via Möbius transformations between circle patterns.
problem Characterizing and relating CMC-1 surfaces via circle patterns.
method Osculating Möbius transformations between circle patterns induce realizations in hyperbolic space.
result One-to-one correspondence between CMC-1 surfaces under specific conditions.
The paper proves stability for Möbius transformations in high dimensions.
problem Quantifying how close a map is to a Möbius transformation.
method Local average conformal-isoperimetric deficit controls map deviation.
result Optimal bounds on the deviation of maps from Möbius transformations.
New method detects projective equivalences and symmetries in rational 3D curves.
problem Detecting projective equivalences and symmetries in rational 3D curves.
method Using differential invariants and Möbius transformations to avoid solving large polynomial systems.
result Efficient algorithm for detecting projective equivalences and symmetries without solving large polynomial systems.
Geometric structures on quaternionic unit ball for slice regular Möbius transformations.
problem No new problem introduced.
method Introducing Hermitian, Riemannian, and Kähler-like structures on quaternionic unit ball using regular Möbius transformations.
result Geometric structures are natural generalizations of complex setup and solve problems not achieved by other geometries.
Smooth manifold structure on Möbius transformations of quaternionic ball identified.
problem Identifying the manifold structure of Möbius transformations of quaternionic unit ball.
method Realizing M(B) as a quotient of Sp(1,1) and using Lie group properties. result The manifold M(B) is diffeomorphic to R4imesS3. Researchers find explicit Bäcklund transforms for specific quadrics.
problem Isometric deformations of diagonal higher dimensional quadrics without center.
method Explicitly found Bäcklund transforms using the Bianchi Permutability Theorem and 3-moving Möbius configuration.
result Explicit solutions can be iterated with arbitrary constants.
Defines circumcenter of mass for polytopes without triangulation.
problem Defining circumcenter of mass for polytopes without triangulation.
method Investigates how volumes of polytopes change under Möbius transformations.
result Provides a definition of circumcenter of mass independent of triangulation.
We describe the action of the (Mobius) inversion on the data of the Weierstrass representation of surfaces in the three-space and show that the Moutard transformation of two-dimensional Dirac operators has a geometrical meaning: it maps the potential U of a surface S into the potential of its inversion.
The study classifies conformal biharmonic and k-polyharmonic maps between space forms.
problem Classifying conformal biharmonic and k-polyharmonic maps between space forms.
method Proving conditions for proper biharmonic and k-polyharmonic maps between space forms.
result Proper k-polyharmonic conformal maps exist if and only if the dimension is 2k.
In this paper, we study conformally flat hypersurfaces of dimension n(≥4) in Sn+1 using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension n(≥4) with constant Möbius scalar curvature under the Möbius transformation group …
Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.
problem Optimizing total σ2-curvature on spheres with positive scalar curvature. method Analyzes metrics conformal to the standard sphere, uses Sobolev norms to measure closeness.
result Near-minimizers of total σ2-curvature are almost the standard metric (up to Möbius transformations). In earlier work we introduced geometrically natural probability measures on the group of all Möbius transformations in order to study "random" groups of Möbius transformations, random surfaces, and in particular random two-generator groups, that is groups where the generators are selected randomly, with a view to estim…
Sharp stability of Möbius group among sphere-valued maps proved in arbitrary dimensions.
problem Proving a sharp quantitative form of Liouville's theorem for sphere-valued maps.
method New arguments and an inequality from Sobolev inequality proof.
result Sharp stability estimate for weakly conformal maps of arbitrary dimensions.
Study projective derivative cocycles for circle diffeomorphisms.
problem Understanding reducibility and almost reducibility in circle diffeomorphisms.
method Computing precise expressions for projective derivative cocycles and extending to 3-torus.
result Generalization of results to diagonal action on 3-torus.
Discrete version of Liouville's theorem for simplicial complexes.
problem Finding equivalent simplicial complexes under discrete conformal equivalence.
method Proving an analogous statement for simplicial complexes, considering combinatorial equivalence and scale factors associated with vertices.
result All discretely conformally equivalent simplicial complexes are combinatorially equivalent.
A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the prob…
In this paper, we study generic conformally flat hypersurfaces in the Euclidean 4-space R4 using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed Möbius form under the Möbius transformation group of R4. Such examples come from …
The study provides a criterion for fractional-linear integrals of geodesics on surfaces.
problem Existence and classification of fractional-linear integrals for geodesic flows on Riemannian surfaces.
method Criterion and analysis of moduli space of local integrals.
result The moduli space of such local integrals is either the 2D projective plane or finite points.
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Study on metric spaces with Möbius self-homeomorphisms and their properties.
problem Characterizing metric spaces with specific homogeneity properties.
method Investigation of homogeneity with Möbius and quasi-Möbius self-homeomorphisms.
result New characterization of snowflakes of boundaries of rank-one symmetric spaces.
Algebraic method reveals criterion for quaternionic Möbius group reversibility.
problem Characterizing reversibility in quaternionic Möbius group elements.
method Purely algebraic approach using matrix entries and conjugacy invariants.
result Explicit criterion for reversibility in terms of matrix entries.
The paper develops a regularity theory for O'hara knot energies, focusing on Möbius energy.
problem Developing a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara.
method Reinterpreting O'hara knot energies as a nonlinear, nonlocal Lp-energy acting on the unit tangent of the knot parametrization, drawing a connection to the theory of (fractional) harmonic maps into spheres. result Proves regularity for minimizers and critical knots of the scale-invariant O'hara knot energies.
Algorithm samples polygons of fixed edge lengths in any dimension.
problem Sampling random closed polygons with fixed edge lengths in any dimension.
method Weighted edge vectors on unit sphere, Möbius transformation, reweighting factors.
result Algorithm samples polygons according to standard probability measures efficiently.
MuRP embeds multi-relational graphs in hyperbolic space for better hierarchical representation.
problem Current hyperbolic models struggle with multi-relational knowledge graphs that exhibit multiple hierarchies.
method MuRP embeds multi-relational graph data in the Poincaré ball model of hyperbolic space, learning relation-specific parameters for entity embeddings.
result MuRP embeddings outperform Euclidean counterparts and other methods on link prediction tasks, especially at lower dimensions.
Let Γ be a 3-dimensional Kleinian punctured torus group with ccidental parabolic transformations. The deformation space of Γ in the group of Möbius transformations on the 2-sphere is well-known as the Maskit slice of punctured torus groups. In this paper, we study deformations Γ′ of Γ in the group of Möbius tra…
New Y-systems for Miquel dynamics are Möbius invariant.
problem Miquel dynamics circle centers are not Möbius invariant.
method Introduced new Y-systems involving only intersection points.
result New Y-systems are Möbius invariant and satisfy the transformation group principle.
A physically natural potential energy for simple closed curves in R3 is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot oc…
The Schwarzian derivative is generalized to Finsler manifolds and its properties studied.
problem Generalizing the Schwarzian derivative to Finsler manifolds.
method Identifying a tensor field and defining Mobius mappings on Finsler manifolds.
result Mobius mappings preserve circles and are conformal on certain Finsler manifolds.
The purpose of this paper is to give a simpler proof to the problem of controllability of a Hilbert snake \cite{PeSa}. Using the action of the Möbius group of the unit sphere on the configuration space, in the context of a separable Hilbert space. We give a generalization of the Theorem of accessibility contained in \c…
Study of immersions with Willmore energy leading to spherical and catenoid bubbles.
problem Classifying immersions with specific energy properties.
method Analyzing sequences of weak immersions with diverging conformal classes, applying Möbius transformations, and strong Wloc2,2-limits. result Obtaining spherical and catenoid bubbles as limits of immersions.
Study of circle homeomorphisms with square summable diamond shears.
problem Characterizing circle homeomorphisms with specific summability properties.
method Analysis of homeomorphisms in modular coordinates and comparison to Weil-Petersson class.
result Sharp results comparing new class to Weil-Petersson class and Hölder classes.
In this paper, we generalise the first Klein-Maskit combination theorem to discrete groups of Möbius transformations in higher dimensions. As a simple application of the main theorem, some examples will be constructed.
Analytic curves linked to algebraic ones via Schottky groups.
problem Moving between analytic and algebraic representations of Riemann surfaces.
method Identifying Riemann surfaces with Schottky groups and constructing families of non-hyperelliptic surfaces.
result Construction of families of non-hyperelliptic surfaces with specific properties.
Constructs minimal surfaces over Pitot quadrilaterals using harmonic diffeomorphisms.
problem Construct minimal surfaces over Pitot quadrilaterals.
method Develops a fully explicit framework using harmonic diffeomorphisms and Weierstrass data.
result Constructs a unique minimal surface \(Σ^\diamond\) that maximizes Gaussian curvature.
Unified treatment of spacelike and timelike minimal surfaces via Liouville equation.
problem Investigating minimal surfaces in Lorentz-Minkowski space.
method Complex and paracomplex analysis, Möbius-type transformations, pseudo-isometries.
result Unified approach to both spacelike and timelike minimal surfaces.
A new Möbius invariant discretization and decomposition of the Möbius energy is proposed.
problem Lack of Möbius invariant discretization and decomposition in existing discrete Möbius energy.
method Proposed a new discretization of Möbius energy that is Möbius invariant and can be decomposed into Möbius invariant components.
result The proposed discretization and decomposition maintain Möbius invariance and converge to the original components in the continuum limit.
The paper studies Clifford-Bianchi groups acting on hyperbolic spaces and their properties.
problem Understanding the actions of Clifford-Bianchi groups on hyperbolic spaces.
method Developed the abstract and computational theory for determining fundamental domains and generators for orders in low dimensions.
result Found that Clifford-Bianchi groups are arithmetic subgroups of SO(1, n+1) and their Möbius action.
The X-ray transform on the periodic slab [0,1]×Tn, n≥0, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless n=0. We characterize t…
The Willmore energy of a closed surface in R^n is the integral of its squared mean curvature, and is invariant uner Möbius transformations of R^n. We show that any torus in R^3 with energy at most 8π−delta has a representative under the Möbius action, for which the induced metric and a conformal metric of constant (…
We show that conformal transformations on the generalized Minkowski space Rp,q map hyperboloids and affine hyperplanes into hyperboloids and affine hyperplanes. We also show that this action on hyperboloids and affine hyperplanes is transitive when p or q is 0, and that this action has exactly three…
The boundary of a Möbius manifold carries a canonical Möbius structure. This enables one to define the cobordism group of n-dimensional (closed) Möbius manifolds. The purpose of this note is to show that the cobordism group of Möbius circles is zero, i.e., every Möbius circle bounds a Möbius surface. We also complete…
A new invariant captures geometric features of circle embeddings.
problem Capturing geometric features of circle embeddings invariantly.
method Chordal distance transform and persistent homology.
result Persistent homology of chordal distance transform is invariant.
There exists and is unique up to multiplication by a constant function a form of the highest dimension on the manifold of n-dimensional continued fractions in the sense of Klein, such that the form is invariant under the natural action of the group of projective transformations PGL(n+1). A measure corresponding to the …