We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their …
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48 results for “Yamabe Invariant”
The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.
problem Calculating λ-Yamabe invariants for specific compact manifolds with boundary.
method Generalizing Kobayashi's connected-sum inequality and applying it to specific manifolds.
result The paper proves that certain manifolds have the same λ-Yamabe invariants as the hemi-sphere.
Relative Yamabe Invariantmath.DG
We define a relative Yamabe invariant of a smooth manifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the classic Yamabe invariant. We develop approximation technique which leads to gluing theorems of two manifolds along their boundaries for the relative Yamab…
The Yamabe invariant is linked to static potentials and eigenvalues.
problem The relationship between Yamabe invariant and static potentials/eigenvalues.
method Analyzes the Yamabe invariant in the context of static potentials and eigenvalues of the Laplacian.
result The Yamabe invariant is closely tied to static potentials and the first eigenvalue of the Laplacian.
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
New proof of Yamabe invariant for RP^3 using harmonic functions.
problem Yamabe invariant of
method Using harmonic functions
result New proof of Yamabe invariant for
Yamabe invariants of certain non-Kähler surfaces are zero.
problem Determining the sign of Yamabe invariants for non-Kähler surfaces.
method Analyzing Inoue surfaces and Kodaira surfaces, their blowups, and applying Seiberg-Witten theory.
result Yamabe invariants of Inoue surfaces and their blowups are all zero.
Solutions to the Einstein constraint equations with a small TT-tensor and vanishing Yamabe invariantmath.AP
Existence proof for Einstein equations with small TT-tensor and vanishing Yamabe invariant.
problem Existence of metrics with vanishing Yamabe invariant and small TT-tensor.
method Existence proof for Einstein conformal constraint equations assuming small TT-tensor and vanishing Yamabe invariant.
result Existence result for Einstein equations under specified conditions.
We show that the S^1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S^1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S^1-actio…
We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular, given a compact smooth manifold M which does not admit metrics of positive scalar curvature, we prove that the Yamabe invariant of M is an upper bound for the Yamabe invariant of any ma…
Optimal pinching results on Einstein manifolds with positive Yamabe invariant.
problem Understanding the rigidity of Einstein manifolds with positive Yamabe invariant.
method Optimal pinching results and bounds on scalar curvature and Weyl tensor norms.
result Improved bounds on the Yamabe invariant and scalar curvature.
The paper classifies invariant gradient -Yamabe solitons in pseudo-Euclidean spaces.
problem Characterizing invariant gradient -Yamabe solitons in pseudo-Euclidean spaces.
method Characterization through the action of an -dimensional translation group and classification of rotational invariant solutions.
result Infinitely many explicit examples of geodesically complete steady gradient -Yamabe solitons are constructed.
We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…
In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called lambda-bar. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals + i…
Improved upper bound for equivariant Yamabe invariant in 3D.
problem Upper bound for -equivariant Yamabe invariant in 3-manifolds.
method Used topological assumptions to show an upper bound.
result Improved Hebey-Vaugon conjecture in dimension 3.
We consider the Yamabe invariant of a compact orbifold with finitely many singular points. We prove a fundamental inequality for the estimate of the invariant from above, which also includes a criterion for the non-positivity of it. Moreover, we give a sufficient condition for the equality in the inequality. In order t…
Innovates contact structure invariant, non-decreasing under operations.
problem Contact structures and CR structures on manifolds.
method Introduced invariant σ_c, proved non-decreasing under handle attaching and connected sum.
result Lower bound on σ_c in a specific case.
The study shows that certain conformal classes on S^7 cannot be conformal infinities of Poincaré-Einstein metrics.
problem Existence of conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics.
method Proved the existence of infinitely many positive conformal classes on S^7 that cannot be conformal infinities of Poincaré-Einstein metrics on the ball B^8.
result Proved a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior.
The second Yamabe invariantmath.DG
Let be a compact Riemannian manifold of dimension . We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.
New proof shows inequality without restrictions.
problem Sharp inequality relating Yamabe invariants on Poincare-Einstein manifolds.
method New proof without restrictions.
result Sharp inequality holds without restrictions.
New approach linking CR Yamabe invariant to Sasaki structures.
problem Existence of constant transversal scalar curvature Sasaki structures.
method Drawing on CR Yamabe problem ideas, establishing link between invariant, Sasaki structures, and K-stability.
result CR Yamabe invariant value determines K-semistability of Sasaki manifolds.
We compute the Yamabe invariant for a class of symplectic 4-manifolds of general type obtained by taking the rational blowdown of Kahler surfaces. In particular, for any point on the half-Noether line we exhibit a simply connected minimal symplectic manifold for which we compute the Yamabe invariant.
New scalars measure failure of CC metrics to solve singular Yamabe problem.
problem Measuring failure of CC metrics to solve singular Yamabe problem.
method Introducing conformally invariant scalar curvature quantities along conformal infinity.
result CC boundary curvature scalars compute canonical expansion coefficients for singular Yamabe metrics.
We show a surgery formula for the relative Yamabe invariant and give applications to the study of concordance classes of metrics.
One invariant solution found for a specific Ricci soliton equation.
problem Investigating invariant solutions for a specific Ricci soliton equation.
method Analyzing the Yamabe equation on a Kähler metric invariant by action.
result Proves the existence of exactly one -invariant solution up to homothecies.
A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
problem Defining and analyzing a mass-type invariant for smooth metric measure spaces
method Defining a mass-type quantity and showing its geometric invariance properties
result The mass-type quantity has a close relation with the fractional Yamabe problem and the relevant Green's function
The Yamabe Invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using spin^c Dirac o…
We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal…
Study on -equivariant second Yamabe constant for manifolds.
problem Existence of -invariant nodal solutions of the Yamabe equation.
method Definition and study of -equivariant second Yamabe constant.
result Obtained results on the existence of -invariant nodal solutions.
Paper extends Kodaira dimension's role in Yamabe invariant for most complex surfaces.
problem Determining the sign of Yamabe invariant for compact complex surfaces.
method Analyzing Kodaira dimension and using simplified proof techniques.
result Pattern of Yamabe invariant sign depends on Kodaira dimension for most surfaces.
The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is non-negative for all closed simply connected manifolds of dimension . We extend this to show that Yamabe invariant is non-negative for a…
In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities chang…
Multiplicity of Solutions and Degenerate Solutions of Yamabe-Type Equations on Projective Spacesmath.DG
Study on solutions of Yamabe-type equations on projective spaces.
problem Existence and multiplicity of solutions of Yamabe-type equations on projective spaces.
method Investigation of solutions invariant under cohomogeneity one actions of U(n) and Sp(n).
result Existence of degenerate solutions on projective spaces.
We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of struct…
Yamabe Classification and Prescribed Scalar Curvature in the Asymptotically Euclidean Settingmath.DG
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…
In this paper we define the bi-orthogonal sectional curvature and we present two modified Yamabe invariants for compact 4-dimensional manifolds. In particular we obtained a relationship between one of these invariants and a Hopf conjecture.
Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are a…
Study of -almost Yamabe solitons in perfect fluid spacetimes.
problem Analyzing -almost Yamabe solitons in perfect fluid spacetimes.
method Examined perfect fluid spacetimes and -almost Yamabe solitons using Einstein field equations.
result Characterized properties of -almost Yamabe solitons in perfect fluid spacetimes.
In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescal…
Generalized stability theorem for compact manifolds with boundary.
problem Stability of manifolds with boundary.
method Equivariant μ-bubbles technique.
result Yamabe invariant of compact manifolds with boundary is positive if and only if the invariant of the manifold times S^1 is positive.
The Yamabe invariant is an invariant of a closed smooth manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold T^m\times B where T^m$ is the m-dimensional torus, and B is a closed spin manifold of nonzero \hat{A}-genus has zero Yamabe invariant. We generaliz…
We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there …
Study proves area estimates for stable capillary hypersurfaces with nonpositive Yamabe invariant.
problem Estimating areas of stable capillary hypersurfaces with nonpositive Yamabe invariant.
method Proves area estimates using stable capillary hypersurfaces in Riemannian manifolds.
result Local rigidity result for embedded, -energy-minimizing hypersurfaces.
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.
Explains peculiarities of 4D scalar curvature via Yamabe invariant.
problem Interplay between scalar curvature and differential topology in 4-manifolds.
method Careful discussion of Yamabe invariant (sigma constant).
result Proves new results and identifies open problems.
Study CR Yamabe constant and CR structures on manifolds.
problem Understanding CR Yamabe constant and its role in CR geometry.
method Developed integral formulae and constructed families of CR structures.
result Found an infinite family of CR structures with varying CR Yamabe constants.
Note proves a mathematical invariant can be close to a sphere's.
problem Fourth order conformal invariant on circle-sphere product.
method Generalizes Schoen's result about classical Yamabe invariant.
result Fourth order conformal invariant can be arbitrarily close to sphere's.
We make use of -structures and technology developed by Paternain - Petean to compute minimal entropy, minimal volume, and Yamabe invariant of symplectic 4-manifolds, as well as to study their collapse with sectional curvature bounded from below. À la Gompf, we show that these invariants vanish on symplecti…