Related papers: A surgery formula for the second Yamabe invariant

We estimate from below the isoperimetric profile of $S^2 \times \re^2$ and use this information to obtain lower bounds for the Yamabe constant of $S^2 \times \re^2$. This provides a lower bound for the Yamabe invariants of products $S^2…

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…

In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the $Q$-curvature of the metric…

Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order.…

If $G$ is a compact Lie group endowed with a left invariant metric $g$, then $G$ acts via pullback by isometries on each eigenspace of the associated Laplace operator $\Delta_g$. We establish algebraic criteria for the existence of left…

We complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of $\RP^3$, by showing that such manifolds are either $S^3$ or finite connected sums $# m(S^2 \times…

We study the H^n-Yamabe constants of Riemannian products (H^n \times M^m, g_h^n +g), where (M,g) is a compact Riemannian manifold of constant scalar curvature and g_h^n is the hyperbolic metric on H^n. Numerical calculations can be carried…

Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated with $\mathcal{T}$ and…

By using the Yamabe flow, we prove that if $(M^n,g)$, $n\geq3$, is an $n$-dimensional locally conformally flat complete Riemannian manifold $Rc\geq \epsilon Rg>0$, where $\epsilon>0$ is a uniformly constant, then $M^n$ must be compact. Our…

The Schouten tensor \ $A$ \ of a Riemannian manifold \ $(M,g)$ provides important scalar curvature invariants $\sigma_k$, that are the symmetric functions on the eigenvalues of $A$, where, in particular, $\sigma_1$ \ coincides with the…

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.…

Let $(M,g,\sigma)$ be a compact Riemmannian surface equipped with a spin structure $\sigma$. For any metric $\tilde{g}$ on $M$, we denote by $\mu\_1(\tilde{g})$ (resp. $\lambda\_1(\tilde{g})$) the first positive eigenvalue of the Laplacian…

Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be…

It is shown in the paper "Variational Properties of the Gauss-Bonnet Curvatures" of M.L. Labbi, that metrics with constant 2k-Gauss-Bonnet curvature on a closed n-dimensional manifold, 1<2k<n, are critical points for a certain Hilbert type…

We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $\lambda$ of the Dirac operator satisfies the inequality $\lambda^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$.…

We obtain existence results for the $Q$-curvature equation of order $2k$ on a closed Riemannian manifold of dimension $n\ge 2k+1$, where $k\ge1$ is an integer. We obtain these results under the assumptions that the Yamabe invariant of order…

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the…

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $\Delta_g$ on…

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2…

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $\delta(n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb{S}^{n+1}$,…