The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.
problem Yang-Mills theory on manifolds with positive Yamabe constant.
method Extending Gursky-Kelleher-Streets results to complete manifolds.
result Equality in gap theorem described in terms of basic instanton.
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.
Study of scattering on singular Yamabe spaces using asymptotically hyperbolic manifolds.
problem Understanding conformal geometry of compact manifolds with boundary.
method Application of scattering theory to singular Yamabe metrics.
result Definition of extrinsic GJMS operators and Q-curvatures on boundary.
Solves Yamabe problem for 3D metrics of Sobolev class W2,q.
problem Yamabe problem on closed 3-manifolds for Sobolev metrics.
method Developed elliptic theory for conformal Laplacian on rough metrics.
result Existence, regularity, and blow-up analysis for Green function.
Explains conformal symmetry with examples in geometry and analysis.
problem None explicitly stated; focuses on introduction.
method Introduction based on examples of Yamabe operator and its applications.
result Illustrates conformal symmetry in geometry and analysis.
We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non homothetic solutions of the Yamabe problem. Using local rigi…
In this paper, we study multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.
Study degenerate solutions on product of spheres using bifurcation theory.
problem Existence of degenerate solutions on product of spheres.
method Bifurcation theory, isoparametric functions, Gegenbauer polynomials.
result Existence of degenerate solutions that depend on both factors.
Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N)\ge min (σ(M),Λ) for Λ>0. We derive explicit lower bounds for Λin dimensions where previous methods failed, namely for (n,k)\in {(4,1),(5,1),…
Study on positive solutions of Yamabe-type equation on spheres.
problem Existence and multiplicity of conformal metrics with constant scalar curvature.
method Reduction to an ODE and application of bifurcation theory.
result Existence of degenerate solutions and multiplicity results.
Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.
problem Existence of Yamabe metrics on conical 4-manifolds with singular points.
method Min-max scheme adapted to singular setting, leveraging recent positive mass theorems.
result Existence of Yamabe metrics on conical 4-manifolds with finitely-many singular points.
The paper examines Euclidean domains with nearly maximal Yamabe quotients.
problem Understanding domains with nearly maximal Yamabe quotients in Euclidean space.
method Analyzes the properties of domains in R3 with nearly maximal Yamabe quotients, proving conditions for equality and comparing to quasi-conformal maps. result Domains with nearly maximal Yamabe quotients are diffeomorphic to balls and are close to a ball in a metric space sense.
Study on scalar curvature in wedge spaces with existence and obstruction results.
problem Existence and obstructions of scalar curvature in wedge spaces.
method Utilized established tools for wedge spaces including Yamabe, elliptic, and index theories.
result Provided existence and obstruction results for scalar curvature under suitable positivity assumptions.
Solves geometric problems using fully nonlinear equations and Morse theory.
problem Geometric problems, specifically Loewner-Nirenberg and Yamabe problems.
method Investigates structure of fully nonlinear equations and applies Morse theory techniques.
result Constructs admissible metrics under weak conditions and demonstrates topological obstructions.
New existence results for curvature problem on balls with specific conditions.
problem Existence of solutions for a prescribed mean curvature problem on a ball.
method Combining critical points at infinity approach with Morse theory.
result New existence results for higher dimensional case n≥5 under pinching conditions. Let (Mm,gM) be a closed, connected manifold with positive scalar curvature and (Tk,g) some flat k-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique f:M→R>0 such that the warped product M×fTk has constant scalar curvature and unit volume…
The paper solves spinorial Yamabe-type problems on spheres, with applications in geometry.
problem Existence of solutions to a conformally invariant Dirac equation on spin manifolds.
method Perturbation techniques to establish the existence of solutions.
result Established existence of solutions for the conformally invariant Dirac equation on Sm. Let (M,g) be any closed Riemannianan manifold and (N,h) be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product (M×N,g+δh) has at least Cat(M)+1 solutions for δ small enough, where Cat(M) denotes the Lusternik-Schnirelmann-categ…
By using the gluing formula of the Seiberg-Witten invariant, we compute the Yamabe invariant Y(X) of 4-manifolds X obtained by performing surgeries along points, circles or tori on compact Kaehler surfaces. For instance, if M is a compact Kaehler surface of nonnegative Kodaira dimension, and N is a smooth closed orient…
This paper is devoted to the study of the constraint equations of the Lovelock gravity theories. In the case of an empty, compact, conformally flat, time-symmetric, and space-like manifold, we show that the Hamiltonian constraint equation becomes a generalisation of the σk-Yamabe problem. That is to say, the prescri…
We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.
Study CR Yamabe constant, flow, and soliton on CR manifolds.
problem Analyzing CR Yamabe constant, flow, and soliton on CR manifolds.
method Using maximum principles, proving uniqueness for CR Yamabe flow, and studying properties of CR Yamabe soliton.
result Uniqueness theorem for CR Yamabe flow and properties of CR Yamabe soliton.
Study on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
The paper classifies a type of solitons in Euclidean spaces.
problem Classifying generalized Yamabe solitons on hypersurfaces.
method Completely classified solitons arising from the position vector field.
result Classification of generalized Yamabe solitons on hypersurfaces in Euclidean spaces.
Study properties of hyperbolic Yamabe solitons in submanifolds.
problem Characterize and understand hyperbolic Yamabe solitons in submanifolds.
method Define hyperbolic Yamabe flow, prove properties, and classify solitons.
result Prove that hyperbolic Yamabe solitons are pseudosymmetric or metallic shaped.
The Yamabe flow converges to a specific function on compactified manifolds.
problem Analyzing the Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant.
method Studied the Yamabe flow on asymptotically flat manifolds with Y≤0 and showed convergence after rescalings. result The Yamabe flow converges to the unique positive function solving the Yamabe problem on a compactification of the original manifold.
Study on η-Ricci-Yamabe solitons on Riemannian submersions.
problem Characterizing η-Ricci-Yamabe solitons on Riemannian submersions. method Analyzing conditions for η-Ricci-Yamabe solitons on submersions and deriving Laplacian equations. result Classification of fiber and target manifolds as η-Ricci-Yamabe solitons under various conditions. Paper examines properties of specific solutions to Yamabe flow.
problem Characterizing quotient almost Yamabe solitons.
method Investigates quotient almost Yamabe solitons and presents conditions for their rigidity.
result Sufficient conditions for quotient almost Yamabe solitons to be trivial or isometric with a sphere.
The differential geometry of Kenmotsu manifold is a valuable part of contact geometry with nice applications in other fields such as theoretical physics. In fact, its statistical counterpart, that is, Kenmotsu statistical manifold also has same importance as that of Kenmotsu manifold. Theoretical physicists have also b…
The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M. (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g. affect …
The study classifies h-almost Ricci-Yamabe solitons in various paracontact manifolds.
problem Classifying h-almost Ricci-Yamabe solitons in paracontact geometry.
method Characterization and classification of para-Kenmotsu, para-Sasakian, and para-cosymplectic manifolds.
result Characterizations and classifications of various paracontact manifolds.
We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t→−∞, to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments,…
We study multiplicity of constant scalar curvature metrics in products of a compact closed manifold and a compact manifold with boundary using equivariant bifurcation theory.
In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hy…
Paper proves uniqueness of Type II Yamabe metrics on manifolds.
problem Uniqueness of Type II Yamabe metrics on compact manifolds.
method Investigates sufficient conditions for metric uniqueness and proves corresponding theorems.
result Establishes sufficient condition for a metric to be the unique Type II Yamabe metric.
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.
The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of σ2 Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform C1,1 estima…
We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the Lq-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asym…
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.
The paper classifies 3D complete gradient Yamabe solitons.
problem Classifying nontrivial 3D complete gradient Yamabe solitons.
method Analyzing properties of Yamabe solitons to show rotationally symmetry.
result Nontrivial non-flat 3D complete steady gradient Yamabe solitons are rotationally symmetric.
Paper shows k-Yamabe solitons have constant curvature under certain conditions.
problem Understanding the properties of k-Yamabe solitons.
method Analyzing the curvature and gradient conditions for k-Yamabe solitons.
result Compact k-Yamabe solitons have constant σk-curvature under certain conditions. Study convergence of Yamabe flow on singular spaces with positive constant.
problem Analyzing convergence of Yamabe flow on singular spaces.
method Normalized Yamabe flow with positive Yamabe constant on pseudo-manifolds, including stratified spaces.
result Established convergence under low energy condition and investigated alternatives.
The study compares and finds Yamabe constants on warped products.
problem Comparing Yamabe constants on warped products.
method Using fiberwise spherical symmetrization.
result Existence of radially-symmetric Yamabe minimizers on product manifolds.
We study the Yamabe invariants of cylindrical manifolds and compact orbifolds with a finite number of singularities, by means of conformal geometry and the Atiyah-Patodi-Singer L2-index theory. For an n-orbifold M with singularities ΣΓ={(pˇ1,Γ1),...,(pˇs,Γs)} (where each group $Γ_j<O…
The paper studies Yamabe metrics and stability in Riemannian manifolds.
problem Existence of complete Yamabe metrics with zero scalar curvature.
method Yamabe flow and local L1-stability analysis. result Local L1-stability of the Yamabe flow on manifolds with non-negative Ricci curvature. Quantitative stability for nearly minimizing Yamabe metrics.
problem Understanding the stability of nearly minimizing metrics in Riemannian geometry.
method Proving quantitative closeness of nearly minimizing metrics to minimizing metrics in a specific sense.
result The distance between nearly minimizing metrics and minimizing metrics is controlled quadratically by the Yamabe energy deficit.
The study characterizes quasi Yamabe solitons with potential vector fields.
problem Characterizing quasi Yamabe solitons with specific properties.
method Analyzing potential vector fields and their norms in quasi Yamabe solitons.
result If the potential vector field has a finite global norm in a complete non-trivial, non-compact quasi Yamabe soliton with finite volume, the scalar curvature becomes constant and the soliton reduces to a Yamabe soliton.
Characterizes ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds.
problem Understanding ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds. method Analyzes the geometry of ∗-k-Ricci-Yamabe solitons and gradient solitons on Kenmotsu manifolds. result Characterizes the nature of ∗-k-Ricci-Yamabe solitons and gradient solitons.