Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.
problem Analyzing the continuity equation on specific complex surfaces.
method Extended La Nave-Tian's continuity equation to Hermitian setting, proving estimates and Gromov-Hausdorff convergence.
result Proved a priori estimates for solutions on Hopf and Inoue surfaces, and convergence of Inoue surfaces to a circle.
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic 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.
New quantum invariant for framed 3-manifolds using ideal triangulations.
problem Quantum invariants of framed 3-manifolds with vanishing first Betti number.
method Based on ideal triangulations and Hopf algebras, using the pentagon equation and graphical representations.
result Construction of a new quantum invariant for closed framed 3-manifolds.
Integrates Lax pair equations for a specific Lie algebra.
problem Low-dimensional Lie algebras of infinitesimal character ildeβ0. method Shows complete integrability of Lax pair equations.
result Proves complete integrability for certain Lie algebras.
Explicitly determined foliated cohomology of affine Reeb flow on Hopf manifold.
problem Determine the foliated cohomology of the affine Reeb flow on the Hopf manifold.
method Explicit calculation and analysis of the cohomology space and its dual.
result The space HF1(M) contains obstructions to solving the cohomological equation. Quantum invariants help distinguish knots via algebra and equations.
problem Determine if two knots are equivalent.
method Use Hopf algebras and solutions to the Yang-Baxter equation.
result Quantum invariants are powerful tools for knot classification.
The exterior algebra of a vector space admits a family of braided Hopf structures.
problem Identifying the exterior algebra with a Nichols algebra and studying its braided Hopf structures.
method Explicit computation of structure constants and construction of solutions to the Yang-Baxter equation.
result The exterior algebra of a vector space admits a one-parameter family of braided Hopf structures.
Method classifies solutions to elliptic problems in disk-like domains.
problem Classifying solutions to overdetermined elliptic problems in topological disks.
method Poincare-Hopf index theorem approach.
result Analogue of Hopf's uniqueness theorem for constant mean curvature spheres in general analytic context.
We identify higher-charge configurations that satisfy Euler-Lagrange equations for the (strong coupling limit of) Faddeev-Hopf model, by means of adequate changes of the domain metric and a reduction technique based on α-Hopf construction. In the last case it is proved that the solutions are local minima for the redu…
Study on solutions to complex equations, proving strong comparison and Liouville theorems.
problem Analyzing continuous viscosity solutions to fully nonlinear elliptic equations.
method Proving strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations.
result Liouville theorem for entire solutions, showing they are either constants or standard bubbles.
Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.
problem Characterizing uniformly elliptic Weingarten spheres in S2xR.
method Proving bounded second fundamental form and applying Hopf's result.
result Rotational uniformly elliptic Weingarten surfaces in S2xR are congruent to the canonical example.
New knot polynomials derived from Nichols algebras and braided Hopf algebras.
problem Developing new knot invariants from algebraic structures.
method Constructing knot invariants from solutions to the Yang--Baxter equation over generalized Yetter--Drinfel'd modules.
result Reproduces known knot polynomials and discovers new multivariable invariants.
Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.
problem Finding constant mean curvature surfaces in specific spacetimes.
method Partial differential equations in the complex plane, generalizing holomorphy.
result Extends Hopf's theorem to new spacetime geometries.
The paper extends Gauss-Bonnet and Hopf-Poincaré theorems to branched sections of fiber bundles.
problem Extending classical theorems to branched sections of fiber bundles.
method Defining index of singularity points, calculating examples, and proving Hopf-Poincaré-Gauss-Bonnet theorem for resolvable branched sections.
result Analog of Hopf-Poincaré-Gauss-Bonnet theorem for resolvable branched sections.
Study new Hopf real hypersurfaces in indefinite complex projective space.
problem Problems posed by H.~Anciaux and K.~Panagiotidou on non-degenerate real hypersurfaces.
method Changed point of view, constructed new families, obtained rigidity results.
result Classified η-umbilical real hypersurfaces and characterized Killing Reeb vector field. Local classification of quasilinear systems with linearizable characteristic webs.
problem Classifying quasilinear systems with linearizable characteristic webs.
method Reciprocal transformations to uncoupled Hopf equations.
result Every quasilinear system with n>3 components can be transformed to n uncoupled Hopf equations.
The paper constructs Yang-Baxter solutions using categorical augmented racks.
problem Solutions to the Yang-Baxter equation in knot theory.
method Interpreting augmented racks in tensor categories and constructing solutions using quantum heaps and Hopf algebra modules.
result Explicit constructions and infinite families of Yang-Baxter solutions are provided.
Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.
problem Finding compact surfaces in Euclidean 3-space with specific curvature properties.
method Representation of solutions to linear elliptic equations with discontinuous coefficients.
result Compact surfaces of genus zero with specific curvature properties are round spheres.
The Laplace equation is reinterpreted in terms of differential geometry and quantum systems.
problem The Laplace equation in Euclidean plane and its symmetries.
method Representation of Lie algebra of conformal group in terms of solutions and derivatives, using Hopf bundle and Fock space.
result Dilations and rotations are coset representatives in a homogeneous space defined by the Lie algebra of the conformal group.
Clarifies relation for solving control-affine Schrödinger bridge problems.
problem Solving control-affine Schrödinger bridge problems via Hopf-Cole transform.
method Applies Hopf-Cole transform to conditions of optimality, resulting in nonlinear PDEs.
result Generic control-affine Schrödinger bridge requires further algorithmic development.
Paper analyzes a new Hopf-Lax semigroup in metric spaces.
problem Analyzing a new Hopf-Lax semigroup in metric spaces.
method Using continuous sections of quotient maps and variational problems.
result The 'symmetrized' Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equation.
We construct and prove a diagrammatic version of the Duflo isomorphism between the invariant subalgebra of the symmetric algebra of a Lie algebra and the center of the universal enveloping algebra. This version implies the original for metrized Lie algebras (Lie algebras with an invariant non-degenerate bilinear form).…
We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined by Tanno.
Paper extends quantum invariant to colored ideal triangulations.
problem Quantum invariants for colored ideal triangulations.
method Uses Heisenberg double and pentagon relation to extend invariant.
result Invariance of colored ideal triangulations under moves.
Deep autoencoder finds linear PDE coordinates for nonlinear equations.
problem Discovering linear coordinates for nonlinear PDEs.
method Residual network architecture for finding intrinsic coordinates.
result Deep learning autoencoder transforms nonlinear PDEs into linear ones.
The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.
problem Analyzing blow-up behavior and extending solutions for a magnetic system.
method Formulated as a magnetic geodesic equation on an infinite-dimensional Lie group, computed Mañé's critical value, established Hopf-Rinow theorem.
result Computed Mañé's critical value for the magnetic two-component Hunter-Saxton system and extended solutions beyond blow-up.
Study Ricci iteration on spheres and projective spaces, proving existence and convergence.
problem Existence and behavior of Ricci iteration on spheres and projective spaces.
method Ricci iteration, Hopf fibration, prescribed Ricci curvature equation.
result Existence and convergence of Ricci iteration on 3-sphere confirmed, partial results in higher dimensions.
This paper explores evolving spheres to Hopf spheres using integer coefficients.
problem Conditions for evolving rotationally symmetric spheres to Hopf spheres.
method Integer linear combination of radii of curvature.
result The fate of an initial sphere is determined by the local geometry of umbilic points.
We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.
Topological method detects Hopf bifurcations from time series.
problem Detecting Hopf bifurcations in nonlinear systems from time series data.
method Persistent homology applied to Takens embedding for phase space reconstructions.
result A simple scalar topological functional identifies critical bifurcation points.
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Ham…
This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from Rn by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 …
The study establishes conditions for quasiregular mappings from Heisenberg groups to link complements.
problem Conditions for nonconstant quasiregular mappings from Heisenberg groups to link complements.
method Translation of a growth condition on fundamental groups into the existence of a supersolution to the 4-harmonic equation.
result A link complement admits a nonconstant quasiregular mapping from the Heisenberg group only if the link is empty, an unknot, or a Hopf link.
Study on unique vortex equations and their geometric implications.
problem Uniqueness of vortex equations involving entire functions.
method Analyzing entire functions on the complex plane and showing geometric applications.
result Uniqueness of harmonic maps and affine spherical immersions with polynomial differential constraints, but failure for non-polynomial entire functions.
The paper studies real hypersurfaces in complex space forms with Miao-Tam critical metrics.
problem Characterizing real hypersurfaces with Miao-Tam critical metrics in complex space forms.
method Analyzing the equation −(Δgλ)g+ablag2λ−λRic=g for real hypersurfaces in complex space forms. result Compact real hypersurfaces in complex Euclidean space with Miao-Tam critical metrics are spheres, and non-flat complex space forms do not admit such metrics.
Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
Study automorphism groups of Reeb components with complex leaves and solve functional equations.
problem Understanding the automorphism groups of Reeb components with complex structure.
method Review Hopf construction, solve Schröder's equation on the half line, analyze leafwise holomorphic automorphisms.
result Automorphism groups of Reeb components have infinite dimensions when leaf holonomy is trivial, finite dimensions otherwise.
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…
Automorphism groups of Hopf manifolds are finite and have a bounded order.
problem Understanding the structure of automorphism groups of Hopf manifolds.
method Proved that the automorphism groups of Hopf manifolds are Jordan.
result Automorphism groups of Hopf manifolds are finite and have a bounded order.
We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the im…
Algorithm calculates Hopf invariant for simplicial mappings.
problem Computing Hopf invariant for simplicial mappings.
method Proposed an algorithm based on Whitehead's integral formula.
result Algorithm successfully computes Hopf invariant.
Classifies Legendrian Hopf links in 3-sphere.
problem Classifying Legendrian Hopf links in the 3-sphere.
method Complete classification up to coarse equivalence.
result Classification of Legendrian Hopf links.
Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.
problem Understanding intrinsic Hopf-Lax semigroup and its relation to intrinsic slope.
method Introduces and proves the link between intrinsic Hopf-Lax semigroup and intrinsic slope.
result Intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.
In this paper, we study the structure of the singular set for a C1 smooth surface in the 3-dimensional Heisenberg group H1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation …
Analyzes properties of Hopf manifolds from analytic and metric perspectives.
problem Properties of Hopf manifolds
method Analytic and metric structure
result Reviews old and new properties of Hopf manifolds
We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometr…