Study on Palais-Smale sequences for conformal Dirac-Einstein problem, proving existence of solutions.
problem Existence of solutions to conformal Dirac-Einstein problem.
method Characterization of Palais-Smale sequences, proving Aubin type result, showing existence of infinitely many solutions.
result Existence of positive solutions and infinitely many solutions under certain symmetries.
Paper finds singular solutions for a specific physics problem on a sphere.
problem Existence of singular solutions to the conformal Dirac-Einstein system.
method Constructs a family of singular solutions on the 3D sphere.
result Constructs solutions with exactly two singularities.
Paper proves existence of solutions for a specific system.
problem Existence of solutions for a conformal Dirac-Einstein system.
method Perturbation methods to prove existence of solutions.
result Existence of solutions for the conformal Dirac-Einstein system.
Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.
problem Analyzing solutions of Dirac-Einstein equations on R3. method Proving smoothness and asymptotic behavior, classifying ground state solutions.
result Scalar part is given by Aubin-Talenti functions, spinorial part is conformal image of −21-Killing spinors on S3. Study Dirac-Einstein equations on manifolds with boundary, focusing on constant volume and chiral conditions.
problem Analyzing Dirac-Einstein equations on manifolds with boundary conditions.
method Characterizing bubbling phenomena and classifying ground state bubbles, proving an Aubin-type inequality.
result Proved an Aubin-type inequality and existence result.
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.
New flow deforms Riemannian metrics smoothly.
problem Deforming Riemannian metrics on spin manifolds.
method Parabolic flow based on Dirac-Einstein functional.
result Local well-posedness of smooth solutions proved.
The paper proves compactness for Dirac-Einstein spin manifolds.
problem Compactness of Dirac-Einstein spin manifolds under specific conditions.
method Study of the Hilbert-Einstein-Dirac functional, proving compactness for critical points.
result Compactness result for Dirac-Einstein spin manifolds in dimensions three and four.
Researchers solve a complex problem about determining Riemannian manifolds.
problem Determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map.
method Uses conformal invariance and introduces a new coordinate system to solve the problem on a real-analytic Riemannian manifold.
result Locally conformally real-analytic manifolds in dimensions ≥ 3 can be determined from the Dirichlet-to-Neumann map.
New invariant connects boundary PDEs and conformal geometry.
problem Global conformal invariants of boundary PDEs.
method Variational considerations and conformal invariants.
result Compact Bach-flat manifolds with umbilic boundary admit Poincaré-Einstein metrics.
Study prescribed curvature tensor in locally conformally flat manifolds.
problem Solving the Prescribed Curvature Tensor problem in locally conformally flat manifolds.
method Explicit solutions provided for special cases of the tensor R, including complete metrics on Rn.
result Explicit examples of metrics g and conformal metrics g that solve the problem are exhibited.
Two new methods improve efficiency of conformal predictive systems.
problem Efficiency of conformal predictive systems in regression problems.
method Split conformal predictive systems and cross-conformal predictive systems.
result Cross-conformal predictive systems are more efficient but not guaranteed valid.
Develops calculus for conformal hypersurfaces and new Willmore energy functionals.
problem Invariant theory for conformal hypersurfaces.
method Solving singular Yamabe problem, developing calculus of differential operators, computing asymptotics.
result New higher Willmore energy functionals for embedded surfaces.
Optimal controls for conformal Laplacian obstacle problems on spheres and manifolds.
problem Optimal control of conformal metrics with constant scalar curvature.
method Analysis of optimal control problem on Riemannian manifolds with positive Yamabe invariant.
result Existence of smooth optimal controls inducing metrics with constant scalar curvature.
Study on spin-zero rest-mass fields using conformal geometric method.
problem Wellposedness of Cauchy and Goursat problems for spin-n/2 zero rest-mass equations. method Conformal geometric method, energy equalities, partial conformal compactification.
result Proves wellposedness of Cauchy and Goursat problems and establishes field decays.
Solves Dirichlet problem for harmonic maps to give geodesic insights.
problem Asymptotic Dirichlet problem for harmonic maps.
method Holographic characterization using conformal geodesics.
result Characterizes conformal geodesics on the boundary.
New metric proves uniqueness of solutions to a specific 4-manifold problem.
problem Uniqueness of solutions to the σ2-Yamabe problem on 4-manifolds. method Defined a new formal Riemannian metric on conformal classes.
result Solutions are unique unless the manifold is round.
Study area minimizing currents in conformal cones, solving Dirichlet problems.
problem Area minimizing currents in conformal cones.
method Minimizing problem of area functionals, Dirichlet problem of minimal surface equations.
result Existence and uniqueness of area minimizing currents in a wide class of conformal manifolds.
Solves modified Schouten tensor problems in conformal metric classes.
problem Prescribed problems for modified Schouten tensors in conformal classes of metrics.
method Uniform ellipticity confirmation under topological and functional constraints.
result Extends results from previous work on smooth complete metrics.
The paper solves area minimizing problems in special geometric cones.
problem Area minimizing problems in conformal cones.
method Defining NCM condition, proving existence of minimal graphs, solving in specific cones.
result Existence of minimal graphs in mean convex conformal cones.
Teichmüller discusses generalizing extremal problems in conformal geometry.
problem Generalizing extremal problems in conformal geometry.
method Wide generalization across function theory, topology, and algebra.
result Final chapter in the Handbook of Teichmüller theory.
Singular Yamabe problems involve changing sign solutions with interesting geometric properties.
problem Solving Yamabe problems with changing sign solutions and their geometric implications.
method Analyzing the behavior of conformal factors and zero loci in various dimensions.
result Zero loci of solutions are critical for conformal functionals and can be Willmore energy minimizers.
Study identifies obstructions for solving a 4th-order boundary problem.
problem Solving a 4th-order boundary problem with specific curvature conditions.
method Derived Kazdan-Warner type identities using variational formulation and conformal variations.
result Obtained nontrivial integral obstructions to solvability.
The paper solves fractional scalar curvature problems on conformal infinities.
problem Prescribed fractional scalar curvature on conformal infinities.
method Introduced and solved the fractional scalar curvature problem on conformal infinities.
result Existence of smooth solutions to the fractional Yamabe problem in the endpoint case.
Study on 3-manifolds finds regular conformal metrics for rough metrics.
problem Characterize conformal metrics for rough Riemannian metrics on 3-manifolds.
method Analogous to the Yamabe problem, study conformal classes and regularity.
result Characterize when a more regular representative exists in the conformal class.
Study of conformally compact metrics and Lovelock tensors in even dimensions.
problem Understanding conformally compact metrics satisfying Lovelock equations.
method Polyhomogeneous expansions and formal solutions to singular Yamabe-(2q) problem.
result Identification of a boundary obstruction in even dimensions that generalizes the ambient obstruction tensor.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
The Willmore energy, alias bending energy or rigid string action, and its variation-the Willmore invariant-are important surface conformal invariants with applications ranging from cell membranes to the entanglement entropy in quantum gravity. In work of Andersson, Chrusciel, and Friedrich, the same invariant arises as…
We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when k=n/2.
The paper explores projective structures on curves and their applications in conformal geometry.
problem Finding qualitative information about solutions of Hill equations.
method Detailed description of projective structures and their isomorphism classes, correcting previous inaccuracies.
result The Yamabe problem for curves has no general solutions in a conformal/Möbius ambient space.
Researchers solve Yamabe problems for specific operators, finding both uniqueness and nonuniqueness.
problem Prescribing scalar, Q-, or σ₂-curvatures in conformal classes.
method Formally self-adjoint, conformally covariant, polydifferential operators.
result Uniqueness results on the sphere, nonuniqueness in general.
Study conformal invariants from nodal sets on manifolds with boundary.
problem Understanding conformal invariants from nodal sets and eigenvalues on manifolds with boundary.
method Analysis of conformal covariant operators and eigenvalues on manifolds with boundary.
result Relate Dirichlet and Neumann eigenvalues and apply results to curvature prescription problems.
Paper connects Dirac operators and automorphic forms on conformally flat manifolds.
problem Understanding the relationship between Dirac operators and automorphic forms.
method Analyzes joint work with John Ryan on conformally flat manifolds.
result Summarizes the connection between Dirac operators and automorphic forms.
Sharp gap theorem found for Yang-Mills connections in 4D.
problem Yang-Mills connections in 4D.
method Exploiting an associated Yamabe-type problem.
result Sharp conformally invariant gap theorem proved.
Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.
problem Yamabe-type problems and Sobolev spaces on the sphere.
method Detailed spectral analysis, conformal invariance, and Hilbert space introduction.
result Established precise connection between sphere and \(\mathbb{R}^N\) logarithmic Laplacian.
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship genera…
We look at several problems in even dimensional conformal geometry based around the de Rham complex. A leading and motivating problem is to find a conformally invariant replacement for the usual de Rham harmonics. An obviously related problem is to find, for each order of differential form bundle, a ``gauge'' operator …
Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, Q-curvature, Yang-Mills fields, etc... We shall be concer…
New metric defines variational structure for $v_{ rac{n}{2}}$-Yamabe problem.
problem Unique solutions to $v_{ rac{n}{2}}$-Yamabe problem in a conformal class.
method New Riemannian metric and variational characterization.
result Unique solutions in a given conformal class.
Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.
problem Determining the metric on an AH manifold from scattering data.
method Relating eigenvalue problem to Conformal Laplacian and using Guillarmou--Guillopé and Chang--González results.
result Scattering matrix at energy 2n+1 determines jet of the metric on the boundary up to diffeomorphism and conformal factor. Study on non-Berwaldian Landsberg spaces using conformal transformation.
problem Existence and properties of non-Berwaldian Landsberg spaces.
method Using conformal transformation to analyze Berwald and Landsberg tensors.
result Necessary condition for a Landsberg space to be Berwaldian is given.
Study variational problem for time-like curves in Einstein universe.
problem Variational problem for time-like curves in Einstein universe.
method Conformally invariant variational problem, analysis of stationary curves, integration by quadratures.
result Stationary curves are trapped into Einsetin universes of dimension 2, 3, or 4.
Constructs conformal boundary operators and fractional Laplacians.
problem Developing conformally invariant boundary operators and fractional Laplacians.
method Constructs continuously parametrised families of conformally invariant boundary operators on densities.
result Constructs odd-order conformally invariant fractional Laplacian pseudo-differential operators.
The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.
problem Proving the geometric stability conjecture for scalar curvature on manifolds conformal to tori.
method Reduction via the Yamabe problem and handling sequences of manifolds conformal to flat tori or constant negative scalar curvature.
result Proves the geometric stability conjecture for certain sequences of manifolds conformal to flat tori.
New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
Study connects vortices in abelian Higgs models to discrete conformal maps.
problem Understanding vortices in abelian Higgs models.
method Relating vortices to discrete conformal maps via curvature and volume form.
result Constructing discrete vortex solutions using discrete conformal theory.
In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound (…
The paper extends the study of conformally flat spaces to four dimensions.
problem Understanding conformal transformations and flatness in four-dimensional Finsler spaces.
method Extending the study of three-dimensional conformally flat Landsberg and Berwald spaces to four dimensions.
result Conditions for a four-dimensional conformally flat Landsberg space to become a Berwald space.