arXiv research

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

New proof shows inequality without restrictions.

The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.

Inequality found for a specific equation on 5D manifolds.

In this paper, we study the CR Yamabe flow with zero CR Yamabe invariant. We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

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…

Study on gradient h-almost Yamabe solitons with scalar curvature estimation.

The paper extends inequalities to closed Riemannian manifolds.

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 …

Sharp inequality for compactifying Poincaré-Einstein manifolds.

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $ν$-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family …

The stability of the Yamabe invariant of $S^3$ is discussed.

A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the $L^2$ Folland-Stein embedding theorem is determined.

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds $(M, g_0)$ of dimension $n\ge 3$. For $n/2 <k<n$, we prove a sharp Harnack inequality for admissible metrics when $(M,g_0)$ is not conformally equivalent to the unit sphere $S^n$ and that the set of …

In this note we prove the existence of infinitely many positive conformal classes on $S^7$ which cannot be the conformal infinity of a Poincaré-Einstein metric on the ball $B^8$. We also prove a sharp inequality between the Yamabe invariant of the conformal infinity and the Yamabe invariant of the interior (after a sui…

An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.

Optimal pinching results on Einstein manifolds with positive Yamabe invariant.

The paper studies inequalities for fractional GJMS operators on conformal infinity.

For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is …

We prove a positive mass theorem for some noncompact spin manifolds that are asymptotic to products of hyperbolic space with a compact manifold. As conclusion we show the Yamabe inequality for some noncompact manifolds which are important to understand the behaviour of Yamabe invariants under surgeries.

On a compact stratified space (X, g) there exists a metric of constant scalar curvature in the conformal class of g, if the scalar curvature satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the local Yamabe constant , another conformal invariant introduced in the recent work…

We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf fo…

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.

We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequali…

In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.

The paper solves a Yamabe problem involving the quotient of Q-curvature and scalar curvature.

Researchers prove a stability result for a 3-sphere inequality, extending previous work.

We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hija…

Paper proves a spinor inequality for magnetic fields on spin manifolds.

Paper simplifies and proves Bahri-Xu conjecture for various cases.

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the…

The paper improves CR Sobolev inequalities and classifies minimizers.

We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then we show that the…

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Yamabe invaria…

Global convergence proved for Gursky-Malchiodi $Q$-curvature flow in dimensions $n \geq 5$.

The paper proves inequalities for zero mode equations on spin manifolds.

In this paper we provide a sharp characterization of the smooth four-dimensional sphere. The assumptions of the theorem are conformally invariant, and can be reduced to an L^2 inequality of the Weyl tensor and positivity of the Yamabe invariant.

The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.

Simplified Obata-Vétois argument for Einstein manifolds with nonnegative scalar curvature.

Paper introduces new fractional Dirac operator and Q-curvature.

We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n/2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^{n}$. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result f…

In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities between the Green's functions of the conformal Laplacian operator and the Paneitz…

We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical $CR$ three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.

We prove that if a closed unit volume Riemannian manifold, $(M^n, g)$, has Ricci curvature bounded from below by r>0 then the Yamabe constant of the conformal class of $g$ is at least $n.r$. This inequality has already been proved by S. Ilias (Constantes explicites pour les inegalites de Sobolev sur les varietes rieman…

We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetriza…