Constructs metrics with negative curvature on specific manifold types.
problem Creating negatively curved metrics on locally conformally flat manifolds.
method Using Morse functions to construct conformal metrics.
result Successfully constructs conformal metrics with negative sectional curvature.
The paper constructs new bimetric conformal invariants using metric perturbations.
problem Developing new conformal invariants in Riemannian geometry.
method Using linear metric perturbations and conformal invariants.
result New bimetric conformal invariants on 4D manifolds are derived.
Einstein metrics on products are shown to be warped.
problem Characterizing Einstein metrics on conformal products.
method Proving Einstein metrics on conformal products are warped products under natural geometric conditions.
result Einstein metrics on conformal products are proven to be warped products.
Boundary distances determine conformal metrics
problem Determining conformal metrics from boundary distances
method Comparing renormalized boundary distances
result Metrics are equal if distances match
The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.
problem Understanding conformal properties of cubic metrics with isotropic scalar curvature.
method Analyzing the conformal flatness and isotropic scalar curvature of cubic metrics.
result Cubic metrics with weakly isotropic scalar curvature must be Minkowski metrics.
Global obstructions found for conformally Einstein metrics in 6D.
problem Obstructing the existence of conformally Einstein metrics in six dimensions.
method Presentation of global conformal invariants.
result Nontrivial global conformal invariant obstructing conformally Einstein metrics.
Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré-Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conforma…
Constructs conformal metrics with negative curvature on manifolds with boundary.
problem Creating conformal metrics with negative curvature on manifolds with boundary.
method Using Morse functions to construct conformal metrics and proving results for compact 3-manifolds with boundary.
result Any Riemannian metric on compact 3-manifolds with boundary is conformal to a compact metric of negative sectional curvature.
For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conform…
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the spher…
We study complex non-Kähler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kähler metrics, too.
Unique conformal metrics found on certain manifolds.
problem Finding unique conformal metrics with constant Q-curvature.
method Proving uniqueness on manifolds with positive scalar curvature.
result Only metrics of the form λg with λ>0 are constant Q-curvature.
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincare-Lovelock metrics, also have Fefferman-Graham…
Study properties of Kenmotsu manifolds with conformal η-Einstein soliton metrics.
problem Properties of Kenmotsu manifolds with specific soliton metrics.
method Investigated properties and constructed a 3D example.
result Properties and construction of 3D Kenmotsu manifold with conformal η-Einstein soliton.
There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …
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.
Flow stabilizes on non-Kähler metrics near Calabi-Yau.
problem Stability of conformally balanced metrics flow near Calabi-Yau manifolds.
method Proving stability of the anomaly flow around Calabi-Yau metrics.
result The flow can converge on non-Kähler metrics near Calabi-Yau.
Modified construction for conformal structures with twistor spinors.
problem Geometric construction and characterization of conformal structures.
method Geometric construction and characterization of 2n-dimensional split-signature conformal structures. result Explicit geometrically constructed Fefferman-Graham ambient metric with vanishing Q-curvature. 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.
In this paper we show that for a Berger metric g^ on S3, the non-positively curved conformally compact Einstein metric on the 4-ball B1(0) with (S3,[g^]) as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \ci…
We derive some necessary conditions on a Riemannian metric (M,g) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti--self--dual, the self--dual Weyl spinor must be of algebraic type D and satisfy a simple first order conformally invariant condition which is necessar…
Study on filling 3D metrics with 4D Poincaré-Einstein structures.
problem Finding a conformal filling by a Poincaré-Einstein metric in 4D.
method Compactness result for conformally compact Einstein 4-manifolds under invariant conditions, with a rigidity result for hyperbolic metrics.
result Established compactness results and derived existence results for conformal fillings.
Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be contained in the split-real-form of the exceptional group G_2. In this note we show…
In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n≥5 and with Poincarë exponent less than 2n−4, the set of conformal metrics of positive constant Q and positive …
Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.
problem Understanding conformal metrics with finite total Q-curvature.
method Introduces conformal mass and provides necessary and sufficient conditions for normality.
result Derives volume comparison theorems and proves a positive mass type theorem related to Q-curvature.
The paper studies special solitons on specific contact metric manifolds.
problem Characterizing solitons on N(k)-contact metric manifolds.
method Analyzing ∗-conformal Einstein solitons and gradient solitons on N(k)-contact metric manifolds. result Conditions for solitons to be expanding, steady, or shrinking are determined.
The study finds a metric that maximizes the second eigenvalue of the Conformal Laplacian.
problem Maximizing the second eigenvalue of the Conformal Laplacian over conformal metrics.
method Analyzes properties of the Conformal Laplacian and constructs metrics to maximize eigenvalues.
result Existence of a metric that maximizes the second eigenvalue of the Conformal Laplacian.
Study on new metrics on para-Kähler-Norden manifolds with conformal deformation.
problem Exploring geometric and harmonic properties of new metrics.
method Conformal deformation of Berger-type metric, analysis of Levi-Civita link, study of curvature varieties, and harmonic maps.
result Detailed examination of curvature varieties and harmonic maps on the manifold.
The Fefferman metric connects CR manifolds to conformal geodesics in 3D.
problem Understanding the Fefferman metric on CR manifolds.
method Explicit description of the Fefferman metric and variational characterization of conformal geodesics.
result Conformal geodesics have lifts to chains and null chains, and are characterized by total torsion.
New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.
problem Existence of essential conformal transformations in pseudo-Riemannian manifolds.
method Construction of compact locally conformally pseudo-Kähler manifolds with essential conformal transformations.
result Found compact examples of pseudo-Kähler manifolds with essential conformal transformations that are not conformally flat.
New proof of instability for certain Einstein metrics.
problem Einstein metrics on specific 4-manifolds.
method Proving instability of conformally Kähler, Einstein metrics.
result Proven instability of certain Einstein metrics.
The paper characterizes ambient metrics using conformal completion and null infinity properties.
problem Characterizing ambient metrics from a conformal perspective.
method Proving conformal completion and analyzing null infinity properties.
result Identifying conformally covariant conditions to characterize ambient metrics.
The conformal Fefferman-Graham ambient metric construction is one of the most fundamental constructions in conformal geometry. It embeds a manifold with a conformal structure into a pseudo-Riemannian manifold whose Ricci tensor vanishes up to a certain order along the original manifold. Despite the general existence re…
Establishes a connection between Kähler metrics and vector bundle sections.
problem Finding Kähler metrics in a conformal class.
method One-to-one correspondence between Kähler metrics and parallel sections of a vector bundle with conformally invariant connection.
result Obstructions for a Riemannian metric to be conformal to a Kähler metric.
An indecomposable Lie group with Riemannian bi-invariant metric is always simple and hence Einstein. For indefinite metrics this is no longer true, not even for simple Lie groups. We study the question of whether a semi-Riemannian bi-invariant metric is conformal to an Einstein metric. We obtain results for all three c…
The study finds points on surfaces where a tensor is conformal to a metric.
problem Existence of conformal points on surfaces.
method Analyzes symmetric bilinear two-tensor fields and Riemannian metrics.
result Provides conditions for the existence of conformal points.
In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. W…
Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.
problem Understanding special non-Kähler metrics on complex nilmanifolds.
method Analyzing locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced metrics on compact complex manifolds. result Compact complex nilmanifolds with balanced or k-Gauduchon metrics are tori, extending previous results. A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and tra…
In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles…
Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.
problem Characterize Weyl-Einstein structures on conformal solvmanifolds.
method Analyzing left-invariant metrics and using conformal Lie group structures.
result Every conformal solvmanifold with Weyl-Einstein structure is Einstein.
Adapted metrics found on complex manifolds.
problem Finding metrics suitable for complex manifolds.
method Characterizing adapted metrics as critical points of a functional.
result Gauduchon metric is adapted on locally conformally product manifolds.
This article classifies closed G2-structures such that the induced metric is conformally flat. It is shown that any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples. In particular, it follows from the classification that any closed G2-structure inducing a metric t…
Synthetic approach to conformal transformations in metric and Lorentzian spaces.
problem Defining consistent conformal transformations in spaces of low regularity.
method Introducing conformal transformations in metric and Lorentzian spaces, focusing on Lorentzian pre-length spaces.
result Established a consistent notion of conformal length and proved its properties.
Length metrics can be closely approximated by conformally flat metrics.
problem Approximating length metrics with conformally flat metrics.
method Uniform approximation of length metrics by conformally flat Riemannian metrics.
result Any length metric on \(\mathbb{R}^d\) can be uniformly approximated by conformally flat Riemannian metrics.
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.