We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equatio…
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48 results for “conformal equations”
One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity.…
New conformal geometry method solves Einstein-Weyl equations.
problem Solving Einstein-Weyl equations in 4D spacetimes.
method Combining conformal and complex geometry techniques.
result Reduced Einstein-Weyl equations to a single conformally invariant scalar equation.
We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equ…
Superintegrable systems on surfaces are classified geometrically.
problem Classifying superintegrable systems on conformal surfaces.
method Geometric structures on conformal surfaces, conformal covariant structural equations.
result Explicit set of algebraic equations defining superintegrable systems on all constant curvature surfaces.
Study finds nonexistence and nonuniqueness in Einstein equations, with some exceptions.
problem Vacuum Einstein conformal constraint equations in far-from-CMC cases.
method Analyzes nonexistence and nonuniqueness results, provides exceptions.
result Shows some cases have non-trivial solutions for positive Yamabe metrics.
In 3D, conformal geodesics are variational.
problem Whether conformal geodesics are variational in 3D.
method Demonstrated that the equation for unparametrized conformal geodesics is variational.
result Variationality of conformal geodesics in 3D.
The paper studies equations in conformal geometry with gradient and existence results.
problem Equations in conformal geometry on closed smooth Riemannian manifolds.
method Local gradient and second derivative estimates, existence result.
result Proved local gradient and second derivative estimates for solutions.
Research on invariant equations in 6D spinor geometry.
problem Understanding conformal transformations in 6D spinorial formalism.
method Deriving and analyzing conformally invariant spinorial equations.
result Established integrability conditions for some equations.
We establish Liouville type theorems for degenerate conformally invariant equations.
Reviews recent work on biharmonic conformal maps and their properties.
problem Understanding biharmonic conformal maps and their properties.
method Analyzes recent research on biharmonic conformal maps, immersions, and maps between manifolds.
result Links biharmonic conformal maps to isoparametric functions and Yamabe type equations.
Sharp rates and symmetry for higher order conformally invariant equations near singularities.
problem Understanding solutions near isolated singularities for higher order conformally invariant equations.
method Blow-up analysis for local integral equations, Fowler solutions, Harnack inequality.
result Sharp blow-up rates and asymptotic radial symmetry of solutions near singularities.
A Conformally Invariant Dirac-type Equation on Compact Spin Manifolds: the Effect of the Geometrymath.DG
Study shows strict inequality for Dirac-type equation on spin manifolds, leading to existence results.
problem Investigating strict inequality for a Dirac-type equation on compact spin manifolds.
method Analyzing a generalized conformally invariant equation involving the Dirac operator with a non-linear convolution term.
result Strict inequality holds, except for round sphere conformal cases, providing existence results for a ground state.
Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.
problem Obstructing higher conformal Yang-Mills equations on conformally compact manifolds.
method Formal asymptotics, Dirichlet-to-Neumann maps, higher transverse derivative boundary operators.
result Obstructing current is the variation of a conformally invariant coefficient in the interior Yang-Mills energy expansion.
On a conformal manifold, it is well known that parallel sections of the standard tractor bundle with non-vanishing scale are in 1-1 correspondence with solutions of the conformal Einstein equation. In 2 dimensions conformal geometry carries no local information but one can remedy this by equipping the surface with a Mö…
Characterizes conformal Killing tensors and their Killing scales.
problem Characterizing conformal Killing tensors and their Killing scales.
method Differential prolongation using conformally invariant tractor calculus.
result Provides an invariant characterisation of Einstein Killing scales.
We study solutions to conformally invariant equations with isolated singularties.
Study of Poincare-Lovelock metrics on conformally compact manifolds.
problem Understanding Poincare-Lovelock metrics in conformally compact geometry.
method Fefferman-Graham expansion and Lovelock equation analysis.
result Existence of fillings for conformal classes near round sphere.
Defines a new conformally invariant Yang-Mills type energy for 6-manifolds.
problem No specific problem stated; focuses on defining a new energy.
method Defines a conformally invariant action S on gauge connections on a 6-manifold M, leading to higher-order conformally invariant Yang-Mills equations.
result The Euler-Lagrange equations of S provide a conformally invariant analogue of Yang-Mills equations, with special cases recovering known invariants.
Relates field theory algebras to sigma model geometric structures.
problem Conformal field theory and sigma model geometric structures.
method Formulate conformal invariance conditions as generalized Maurer-Cartan equations.
result Einstein equations with extra fields in the quasi-classical limit.
Study integrability of conformal geodesics on gravitational instantons.
problem Integrability of conformal geodesic flow on gravitational instantons.
method Analyzing conformal geodesic flow equations on -invariant gravitational instantons, separating Hamilton-Jacobi equations, finding commuting first integrals, and using conformal Killing-Yano tensors.
result First example of an integrable conformal geodesic flow on a non-symmetric four-manifold.
The article contains a brief description on the study of conformal scalar curvature equations, and discusses selected topics and questions concerning the equations in open spaces.
Variationality of conformal geodesics fails in higher dimensions.
problem The variationality of conformal geodesics in higher dimensions.
method Analysis of conformal geodesics in three and higher dimensions.
result Variationality fails in both parametrized and un-parametrized conformal geodesics in higher dimensions.
Researchers solve field equations for special gravitational instantons.
problem Solving field equations for conformally Kähler Riemannian four-manifolds.
method Developed a framework to solve the field equations for generalised gravitational instantons using conformal self-duality and cosmological Einstein-Maxwell.
result Found conformally self-dual and Einstein-Maxwell generalisations of specific geometries.
We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal t…
This article is a survey of results involving conformal deformation of Riemannian metrics and fully nonlinear equations.
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of …
New equations define frustrated conformal transformations generating maps on ambitwistor space.
problem Defining new conformal transformations in twistor theory.
method Introducing frustrated conformal transformations involving instantons.
result Solutions to the new equation generate maps on ambitwistor space.
The constraint equations of general relativity can in many cases be solved by the conformal method. We show that a slight modification of the equations of the conformal method admits no solution for a broad range of parameters. This suggests that the question of existence or non-existence of solutions to the original e…
New solutions found for Einstein-Maxwell equations on complex manifolds.
problem Finding solutions to Einstein-Maxwell equations on complex manifolds.
method Constructing a family of conformally Kähler solutions that deform the Page metric.
result Displaying infinitely many geometrically distinct families of solutions.
We study some conformally invariant integral equations using the method of moving spheres.
We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold …
New variational principles found for conformal geodesics.
problem Challenges in Lagrangian formulation for conformal geodesics.
method Enlarging the class of variations leads to a variational formulation with a third-order conformally invariant Lagrangian.
result Some integral curves of the fourth-order ODE system are spirals.
A gluing construction of singular solutions for a fully non-linear equation in conformal geometrymath.DG
Paper studies solutions to a specific equation in conformal geometry with singular sets.
problem Singular solutions to a fully non-linear equation in conformal geometry.
method Uses a classical gluing method adapted to the fully non-linear setting.
result Shows the classical gluing method can be applied to the --Yamabe equation.
Researchers solve metric curvature equations on manifolds with boundary.
problem Finding complete conformal metrics with specific curvature functions.
method Revealed algebraic structure of fully nonlinear equations; used topological obstructions.
result Solved a class of fully nonlinear equations for conformal metrics.
The paper develops a comprehensive theory of submanifolds in conformal geometries.
problem Understanding submanifolds in conformal geometries of arbitrary dimension.
method Using conformal tractor calculus, the paper provides a new framework for studying submanifolds.
result The theory includes a new notion of distinguished submanifolds and characterizes them in various dimensions.
We investigate fourth order Paneitz equations of critical growth in the case of -dimensional closed conformally flat manifolds, . Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the -cu…
Novel approach classifies conformally equivariant spinor operators.
problem Classifying conformally equivariant differential operators on spinors.
method Classification based on vector-valued partial differential equations and spin Howe duality.
result Solutions for a vector-valued system of PDEs associated with -modules.
Lp almost conformal isometries of Sub-Semi-Riemannian metrics and solvability of a Ricci equationmath.DG
Study shows almost conformal isometry between Sub-Semi-Riemannian metrics and solves a Ricci equation.
problem Finding almost conformal isometry between Sub-Semi-Riemannian metrics.
method Analyzes conditions for Lp almost conformal isometry and solvability of Ricci equation.
result Shows solvability of a Ricci equation without closeness assumptions.
Classification of ground state solutions to critical Dirac equation on spheres.
problem Classifying ground state solutions of the critical Dirac equation.
method Exploiting conformal covariance and relating to the Yamabe equation.
result Ground state solutions are given by Killing spinors up to conformal diffeomorphisms.
Paper introduces p-Laplace equations for curvature in conformal geometry.
problem Study of geometry and topology of manifolds.
method Introducing p-Laplace equations for intermediate Schouten curvature.
result Nonnegative intermediate Schouten curvature leads to estimates on Hausdorff dimension of singular sets and vanishing of homotopy groups.
Further results on Liouville type theorems for some conformally invariant fully nonlinear equationsmath.AP
We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.
We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the geometry of the center of mass of such solutions. We also review methods to con…
Study classifies half conformally flat GQE manifolds of signature (2,2).
problem Classifying half conformally flat generalized quasi-Einstein manifolds.
method Analysis and examples provided.
result Natural affine quasi-Einstein equation derived.
We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…
The article classifies solutions to a nonlocal equation in conformal geometry.
problem The study of conformal metrics with constant Q-curvature in \(\mathbb{R}^n\).
method Analyzes a nonlocal equation involving the fractional Laplacian.
result Classifies all solutions to the equation based on their behavior at infinity.