Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.
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48 results for “metrics of constant curvature”
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
Study on Hermitian metrics and curvature properties of complex manifolds.
problem Analyzing curvature properties of Hermitian metrics on complex manifolds.
method Derivation of formulae and proofs for Chern-Ricci curvatures and holomorphic sectional curvatures.
result Examples of metrics with specific curvature properties.
New Finsler metrics with constant flag curvature defined using Weyl-type curvature tensor.
problem Characterizing Finsler metrics with constant flag curvature.
method Defining a Weyl-type curvature tensor and constructing projectively related Finsler metrics.
result Construction of new families of Finsler metrics with constant flag curvature.
Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.
problem Creating Kähler metrics with constant scalar curvature on complex manifolds.
method Blowing up the manifold at points and constructing metrics with Poincaré-type singularities.
result Existence of constant scalar curvature Kähler metrics with Poincaré-type singularities.
A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics wit…
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.
Study spherically symmetric Finsler metrics with specific curvature properties.
problem Characterize Finsler metrics with scalar and constant flag curvature.
method Analyze spherically symmetric metrics on symmetric spaces with given curvature properties.
result Provide families of Finsler metrics with scalar and constant flag curvature.
New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl curvature.
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.
Classifies metrics with specific curvature properties on a ball.
problem Classifying conformal metrics with constant 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 to include positive and negative cones.
The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.
problem Existence of constant scalar curvature Kähler metrics with cone singularities.
method Log -polystability and -uniform log -stability are established.
result Uniform log -stability is achieved for normal varieties.
Paper builds singular metrics with constant Q-curvature.
problem Constructing metrics with constant Q-curvature on manifolds.
method Utilizes tools from recent years to build weak solutions.
result First construction of singular metrics with positive Q-curvature.
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.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
Classifies Kähler metrics with constant holomorphic curvature.
problem Classifying Kähler metrics with constant holomorphic sectional curvature.
method Exploiting the geometry of the bundle of 1-jets of holomorphic functions.
result Local classification of Kähler metrics with constant holomorphic sectional curvature.
Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
problem Characterize compact quasi-Einstein metrics with constant scalar curvature.
method Connection to Sasakian geometry and circle bundles over Einstein metrics.
result Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension . We prove the existence of such conformal metrics in the cases of or the manifold is spin and some other remai…
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.
This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already car…
Complete constant positive scalar curvature metrics on S^n - {p_1, ..., p_k} admit a definite asymptotic structure; i.e. the metric is asymptotic to a specific S^{n-1}-invariant metric near the puncture points. This allows one to glue together two such metrics near their puncture points, provided the asymptotic structu…
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.
We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) -curvature on compact and noncompact manifolds of dimension . Infinitely many branches of metrics with constant -curvature, but without constant scalar curvature, are found to bifur…
Constant Scalar Curvature Kähler Metrics on Resolutions of an Orbifold Singularity of Depth 1math.DG
New metrics found on orbifold resolutions with specific curvature.
problem Finding metrics with constant scalar curvature on orbifolds.
method Constructing metrics on resolutions of orbifolds with type I singularities.
result Constant scalar curvature Kähler metrics constructed on resolutions.
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 with two conical singularities.
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.
The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a manifold of dimension n \geq 3 is either Riemannian or locally Minkowskian.
In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a techn…
Concrete example of 2-sphere with constant curvature 1 and closed geodesics.
problem Constructing a 2-sphere with constant curvature 1 and closed geodesics.
method Constructing a concrete example of a Finsler metric on the 2-sphere.
result A 2-sphere with constant Gauss curvature 1 and all geodesics closed.
This paper solves Hilbert's fourth problem for constant curvature metrics.
problem Classifying metric geometries with shortest straight lines in constant curvature settings.
method Analyzing Finsler manifolds with constant flag curvature, deriving distance formulas, and proving global geometry theorems.
result Complete characterization of global geometry for constant flag curvature metrics.
A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…
Estimates for metrics with constant Chern scalar curvature on complex manifolds.
problem Finding metrics with constant Chern scalar curvature on complex manifolds.
method Proving a priori estimates conditional on an upper bound on entropy.
result Extending a recent result by Chen-Cheng in the Kähler setting.
We investigate projective spherically symmetric Finsler metrics with constant flag curvature in and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk are found to have constant negative flag curvature.
Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.
problem Existence of constant scalar curvature Kähler metrics on polarized manifolds.
method Direct proof using microscopic stability thresholds and conditions on the limit.
result Existence of a unique constant scalar curvature Kähler metric under specific conditions.
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
Paper proves algebraic condition for Finsler metrics with constant curvature.
problem Characterizing Finsler metrics with constant flag curvature.
method Proves algebraic identity for curvature of induced nonlinear connection.
result Algebraic characterisation for Finsler metrics of constant flag curvature.
The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…
We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and K"ahler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant…
The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.
problem Investigating the Funk-Finsler metric in spaces of constant curvature.
method Explicitly computed -curvature, Riemann curvature, Ricci curvature, and flag curvature.
result The -curvature and flag curvature of the Funk-Finsler metric in hyperbolic, spherical, and Euclidean spaces are bounded.
We consider the conformal class of the Riemannian product , where is the constant curvature metric on and 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…
In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of ; namely, the boundary complexes of cyclic polyt…
The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…
We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We …
This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/
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.
The paper explores properties of Finsler manifolds with specific curvature conditions.
problem Investigating Finsler metrics with various curvature conditions.
method Analyzing non-Riemannian -metrics and compact Finsler manifolds with specific curvature properties.
result Compact Finsler manifolds with relatively non-negative stretch curvature are Landsberg metrics.
Paper finds conditions for non-Einstein relative Yamabe metrics.
problem Finding relative Yamabe metrics with positive scalar curvature.
method Sufficient condition for positive constant scalar curvature metrics on manifolds with boundary.
result Examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant sc…