Study existence of conformal metrics with specific curvature properties on compact manifolds.
problem Existence of conformal metrics with constant scalar curvature and boundary mean curvature.
method Proving existence through specific cases and sequences of metrics.
result Existence of conformal metrics in various cases, including positive Yamabe constant.
No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given i…
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 …
Study Minkowski formula for conformal Killing-Yano 2-forms in constant curvature spacetimes.
problem Derive Minkowski formula for conformal Killing-Yano 2-forms.
method Analyze spacetime Alexandrov theorem with a free boundary.
result Established spacetime Alexandrov theorem with a free boundary.
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.
Study finds multiple conformal metrics with constant Q-curvature.
problem Finding complete conformal metrics with constant Q-curvature.
method Analyzing manifolds of dimension ≥5, including spheres and complex projective spaces.
result Infinitely many branches of metrics with constant Q-curvature, bifurcating from Berger metrics.
Study negative scalar curvature metrics with positive boundary mean curvature.
problem Bounding conformal metrics with specific curvature properties.
method Analyzing Riemannian manifolds with boundary conditions.
result A priori boundedness of metrics in specific cases.
Let (V,g) and (W,h) be compact Riemannian manifolds of dimension at least 3. We derive a lower bound for the conformal Yamabe constant of the product manifold (V x W, g+h) in terms of the conformal Yamabe constants of (V,g) and (W,h).
Study on minimal two-spheres in complex hyperquadric, proving non-congruence of constant curvature spheres.
problem Classifying minimal two-spheres of constant curvature in complex hyperquadric.
method Construction of non-homogeneous constant curved minimal two-spheres and classification theorem.
result Minimal two-spheres of constant curvature in Q4 are not congruent. Killing-Yano and conformal Killing-Yano superalgebras are rigid in constant curvature manifolds.
problem Understanding the rigidity of Killing-Yano and conformal Killing-Yano superalgebras in constant curvature manifolds.
method Defining Z-gradations and filtrations, showing trivial second cohomology groups, and proving non-deformability. result Killing-Yano and conformal Killing-Yano superalgebras are rigid and correspond to geometric invariants of constant curvature manifolds.
Constructed static vacuum metrics in 5D with negative cosmological constant.
problem Finding static vacuum solutions in 5D with negative cosmological constant.
method Numerically constructed two families of metrics with squashed conformal infinity.
result Constructed metrics with squashed conformal infinity conformal to any squashed 3D sphere.
Study constant Q-curvature metrics on conic 4-manifolds.
problem Find metrics with constant Q-curvature on conic 4-manifolds.
method Analyze related differential equations in the given conformal class.
result Solve constant Q-curvature problem on conic 4-manifolds.
Classifies metrics with specific curvature properties on a ball.
problem Classifying conformal metrics with constant σk curvature and constant boundary mean curvature. method Uses the Obata-Escobar argument to classify metrics on the upper hemisphere.
result Extends a result of Escobar for k=1 to include positive and negative cones. Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.
problem Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal Kähler manifolds
method Prove the conjecture using curvature identities and properties of Kähler metrics
result Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler
Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.
problem Computing Cheeger constants for conformally compact asymptotically constant mean curvature submanifolds.
method Analyzes conformally compact asymptotically constant mean curvature submanifolds in asymptotically hyperbolic spaces.
result Identifies conditions for Cheeger constant equality and vanishing mean curvature.
The study preserves upper bounds of total scalar curvature in conformal classes.
problem Preserving upper bounds of total scalar curvature in conformal classes.
method Analyzing Yamabe constant and scalar curvature conditions.
result The upper bound condition of total scalar curvature is preserved in a conformal class.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
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…
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 non-singular spacetimes found with negative cosmological constant.
problem Finding non-singular spacetimes with a negative cosmological constant.
method Constructing infinite-dimensional families of solutions to complex equations.
result Infinite-dimensional families of non-singular stationary space-times with negative cosmological constant.
The conformally covariant split system generates non-constant mean curvature vacuum initial data.
problem Creating non-constant mean curvature vacuum initial data for the Einstein equations.
method Proved existence of solutions to the conformally covariant split system on compact 3-manifolds using the implicit function theorem.
result The conformally covariant split system provides non-constant mean curvature vacuum initial data for the Einstein equations.
Compact metrics found with specific curvature properties on 3D surfaces.
problem Finding compact metrics with constant curvature on 3D surfaces.
method Blow-up analysis of Yamabe equation with critical Sobolev exponents.
result Proved the compactness of conformal metrics with constant scalar curvature and boundary mean curvature.
In this paper we first use the result in [12] to remove the assumption of the L2 boundedness of Weyl curvature in the gap theorem in [9] and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature est…
The study of scalar curvatures on almost Hermitian manifolds and existence of conformally constant Chern scalar curvature metrics.
problem Existence of almost Kähler metrics with conformally constant Chern scalar curvature.
method Integrability theorems and adaptation of methods from the Chern-Yamabe problem to the non-integrable case.
result The problem is solved for ruled manifolds and a complementary case.
Study Cheeger constant and Yamabe type for ALH manifolds.
problem Understanding the Cheeger constant and Yamabe type of ALH manifolds.
method Analyzes the Cheeger constant and Yamabe type of asymptotically locally hyperbolic manifolds with conformal compactification.
result Establishes a relationship between the Cheeger constant and the Yamabe type of the conformal infinity.
The study finds multiple solutions for constant Q-curvature metrics.
problem Finding multiple metrics with constant Q-curvature.
method Proving subcritical equations have at least Cat(M) positive solutions.
result Proves existence of at least Cat(M) metrics with constant Q-curvature.
We consider the conformal class of the Riemannian product g0+g, where g0 is the constant curvature metric on Sm and g is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect…
Study on curvature conditions for non-conformally flat spheres using quasiconformal maps and Ricci flow.
problem Curvature conditions for non-conformally flat spheres.
method Construct quasiconformal maps and apply Ricci flow.
result Controlled bilipschitz constant between metrics.
The paper finds conditions for Yamabe solitons' metrics to be constant scalar curvature.
problem Finding conditions for Yamabe solitons' metrics to be constant scalar curvature.
method Using properties of conformal vector fields to find sufficient conditions.
result Sufficient conditions on soliton vector fields under which their metrics are of Yamabe metrics.
Researchers find unique metrics solving complex PDEs for constant scalar curvature.
problem Finding metrics with constant scalar curvature in complex manifolds.
method Proving existence and uniqueness of smooth functions f that solve a fourth-order nonlinear PDE related to the Calabi functional. result Critical metrics minimize the Calabi functional and have constant Chern scalar curvature.
Optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces are determined.
problem Determining optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces.
method Analyzing Riemannian surfaces with boundary, considering both given topology and conformal class, and proving inequalities relating conformal invariants and eigenvalues.
result New examples of topological disks realizing optimal constants and inequalities relating conformal invariants of Steklov eigenvalues on surfaces and disks are provided.
Researchers solve a geometry conjecture for a specific class of manifolds.
problem Compact Hermitian manifolds with constant curvature.
method Utilized techniques from Chen, Chen, Nie and Huang-Wan to solve the conjecture.
result Locally conformally Kähler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature were proven.
In this paper we study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending to these spaces a result of J. Simons on the minimality of cones in Euclidean space, and apply…
Researchers found infinite families of non-singular static spacetimes with negative cosmological constant.
problem Finding non-singular static spacetimes with negative cosmological constant.
method Constructing infinite-dimensional families of solutions to the Einstein-Maxwell equations.
result Infinite-dimensional families of non-singular static space times with negative cosmological constant.
Constructs metrics with constant scalar curvature and unbounded volumes on spheres.
problem Creating metrics with constant scalar curvature on spheres with unbounded volumes.
method Constructs a sequence of metrics conformal to a given metric with scalar curvature 1 and unbounded volumes.
result Constructs metrics with constant scalar curvature and unbounded volumes on spheres.
Paper studies metrics with constant Q-curvature near singular points.
problem Deriving properties of metrics with constant Q-curvature near singularities.
method Refined asymptotic expansion for metrics with constant Q-curvature and scalar curvature.
result Modelled results on similar metrics with scalar curvature, analyzing linearization about Delaunay metrics.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.
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.
The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.
problem Discrete conformal geometry of polyhedral surfaces.
method Establishing rigidity for hexagonal triangulations and estimating quasiconformal constants.
result Discrete conformal maps converge to Riemann mappings for Jordan domains.
Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on S2 with two conical singularities. We study conformal metrics on R^{2m} with constant Q-curvature and finite volume. When m=3 we show that there exists V* such that for any V\in [V*,\infty) there is a conformal metric g on R^{6} with Q_g = Q-curvature of S^6, and vol(g)=V. This is in sharp contrast with the four-dimensional case, treated by C-S. Lin. We…
Paper proves uniqueness of conformally compact Einstein metrics with specific conformal infinity.
problem Proving uniqueness of conformally compact Einstein metrics with homogeneous conformal infinity.
method Analyzing Berger metrics on S3 and using properties of Yamabe constant. result Uniqueness of conformally compact Einstein metric for specific conformal infinity.
The abstract discusses nonuniqueness results for specific Riemannian invariants.
problem Identifying conditions for nonhomothetic conformal rescalings with constant Riemannian invariants.
method Identifying sufficient conditions for finite and infinite geometrically distinct periodic conformal rescalings.
result Improves and establishes nonuniqueness results for various Riemannian invariants.
The paper defines and analyzes conformal trajectories in 3D space forms.
problem Understanding trajectories in curved 3D spaces.
method Defined conformal trajectories and studied their properties in R3, S3, and H3. result Conformal trajectories in S3 and H3 have constant curvature and torsion. For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.