Surgery and the Yamabe Invariant

We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular: for a compact connected manifold M with no metric of positive scalar curvature, we prove that the Yamabe invariant of any manifold obtained by performing surgery on spheres of codimension greater than 2 on M is not smaller than the invariant of M. Submitted: August 1998.     

Yamabe flow on manifolds with edges

Let be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder‐type estimates for the heat operator on certain Hölder spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting.     

Further Results on the Fractional Yamabe Problem: The Umbilic Case

We prove some existence results for the fractional Yamabe problem in the case that the boundary manifold is umbilic, thus covering some of the cases not considered by González and Qing. These are inspired by the work of Coda-Marques on the boundary Yamabe problem but, in addition, a careful understanding of the behavior at infinity for asymptotically hyperbolic metrics is required.     

Weak and smooth solutions for a fractional Yamabe flow: The case of general compact and locally conformally flat manifolds

As a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the manifold is locally conformally flat with positive Yamabe invariant, we also prove that the flow is smooth and converges to a constant fractional curvature metric. We provide different proofs using extension properties introduced by Chang and González (2011) for the conformally covariant fractional order operators.     

Yamabe metrics on cylindrical manifolds

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 version of the Yamabe problem on cylindrical manifolds. We find a positive solution to this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almost cuspsingularities. We describe the supremum case, i.e., when the cylindrical Yamabe constant is equal to the Yamabe invariant of the sphere. We prove that in this case such a cylindrical manifold coincides conformally with the standard sphere punctured at a finite number of points. In the course of studying the supremum case, we establish a Positive Mass Theorem for specific asymptotically flat manifolds with two almost conical singularities. As a by-product, we revisit known results on surgery and the Yamabe invariant. Submitted: Submitted: August 2001. Revision: January 2003 RID="*" ID="*"Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 14540072.     

First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

The Yamabe problem on quaternionic contact manifolds

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ ⁿ ). Mathematics Subject Classification (2000) 53C17, 53D10, 35J70     

The second Yamabe invariant

Let (M,g) be a compact Riemannian manifold of dimension n?3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.     

Positive mass theorem for the Yamabe problem on spin manifolds

Let (M, g) be a compact connected spin manifold of dimension n ≥ 3 whose Yamabe invariant is positive. We assume that (M, g) is locally conformally flat or that n ∈ {3, 4, 5}. According to a positive mass theorem by Schoen and Yau the constant term in the asymptotic development of the Green’s function of the conformal Laplacian is positive if (M, g) is not conformally equivalent to the sphere. The proof was simplified by Witten with the help of spinors. In our article we will give a proof which is even considerably shorter. Our proof is a modification of Witten’s argument, but no analysis on asymptotically flat spaces is needed.Received: March 2004 Revised: June 2004 Accepted: June 2004     

Q‐Curvature on a Class of Manifolds with Dimension at Least 5

For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q‐curvature, and dimension at least 5, we prove the existence of a conformal metric with constant Q‐curvature. Our approach is based on the study of an extremal problem for a new functional involving the Paneitz operator.© 2016 Wiley Periodicals, Inc.     

An Estimate on Riemannian Manifolds of Dimension 4

We give an estimate of type sup × inf on Riemannian manifold of dimension 4 for a Yamabe type equation.     

Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

Aubin’s Lemma says that, if the Yamabe constant of a closed conformal manifold (M, C) is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. We generalize this lemma to the one for the Yamabe constant of any (M _∞, C _∞) of its infinite conformal coverings, provided that π ₁(M) has a descending chain of finite index subgroups tending to π ₁(M _∞). Moreover, if the covering M _∞ is normal, the limit of the Yamabe constants of the finite conformal coverings (associated to the descending chain) is equal to that of (M _∞, C _∞). For the proof of this, we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities.     

Conformal paracontact curvature and the local flatness theorem

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.     

The Weyl functional near the Yamabe invariant

For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant Y_C(M) and the L^n/2-norm W_C(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y_C(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant W_C(M) is arbitrarily large. We study the image of the mapYW:C→(Y_C(M), W_C(M))∈R ² near the line {(Y(M), w)|w∈R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.     

Sur quelques problèmes de courbure scalaire

In this article we prove, among other things, some results about two problems which are the subject of announces these last decades: (1) the compactness of the set of the solutions of the Yamabe equation on a compact Riemannian manifold, (2) a generalization of a result of the author which is necessary to solve the Yamabe problem, when 2ω?n−6.     

The second CR Yamabe invariant

Let be a closed, connected, strictly pseudoconvex CR manifold with dimension . We define the second CR Yamabe invariant in terms of the second eigenvalue of the Yamabe operator and the volume of M over the pseudo‐convex pseudo‐hermitian structures conformal to θ. Then we study when it is attained and classify the CR‐sphere by its second CR Yamabe invariant. This work is motivated by the work of B. Ammann and E. Humbert 1 on the Riemannian context.     

Best Sobolev Constants and Manifolds with Positive Scalar Curvature Metrics

We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N ⁿ ,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M ^m , the Yamabe invariantof M ^m × N ⁿ is no less than K times the invariant ofS ^{n + m} . We will find some estimates for the constant K in the case N =S ⁿ .     

Evolution of Yamabe constant under Ricci flow

In this note under a crucial technical assumption, we derive a formula for the derivative of Yamabe constant , where g(t) is a solution of Ricci flow on closed manifold. We also give a simple application. Mathematics Subject Classifications (2000): 53C21 and 53C44     

“A priori” estimates, uniqueness and existence of positive solutions of Yamabe type equations on complete manifolds

We study the asymptotic behaviour of non-negative solutions of Yamabe type equations on a complete Riemannian manifold. Then we provide a comparison result, based on a form of the weak maximum principle at infinity, which together with the “a priori” estimates previously obtained, yields uniqueness under very general Ricci assumptions. The paper ends with an existence result and an application to the non-compact Yamabe problem.     

Positive solutions of $\Delta u+u^p = 0$ whose singular set is a manifold with boundary

The aim of this paper is to prove the existence of weak solutions to the equation , with , which are positive in a domain and which are singular along a k-dimensional submanifold with smooth boundary. Here, the exponent p is required to lie in the interval , where is the dimension of the singular set. In the particular case where and , solutions correspond to solutions of the singular Yamabe problem. Received: 7 October 2001 / Accepted: 7 March 2002 / Published online: 6 August 2002