Related papers: On some manifolds with positive sigma invariants a…

We view all smooth metrics $g$ on a closed surface $\Sigma$ through their Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We…

In this note we generalize our previous result, stating that if $(M_1,g_1)$ and $(M_2,g_2)$ are compact Riemannian manifolds, then any Einstein metric on the product $M:=M_1\times M_2$ of the form $g=e^{2f_1}g_1+e^{2f_2}g_2$, with $f_1\in…

We identify the smooth metrics $\mc{M}(M)$ on a manifold $M^n$ with the smooth isometric embeddings $f_g: (M,g) \rightarrow (\mb{S}^{\tn}, \tg)$ into a standard sphere of large dimension $\tn=\tn(n)$, and their Palais isotopic deformations,…

We let (M^m, g) be a closed smooth Riemannian manifold (m >1) with positive scalar curvature S_g, and prove that the Yamabe constant of (M \times R^n,g+g_E) is achieved by a metric in the conformal class of (g+g_E), where g_E is the…

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $\sigma_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is…

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe…

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…

We identify all metrics on a closed $n$-manifold with their Nash isometric embeddings into a standard sphere of large, but fixed dimension, and use the Palais' isotopic extension theorem to identify their deformations with the isotopic…

We start by taking the analytical approach to discuss how the minimizer of Yamabe functional provides constant scalar curvature and its relationship with the Sobolev Space $W^{1,2}.$ Then, after demonstrating the importance of the sphere…

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $\lambda_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $\sigma_1(M,g,\rho)\mathcal…

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…

We derive the general formulas for a special configuration of the sequential warped product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a conformal metrics.…

Let a sequence of conformal Riemannian metrics $\{g_k=u_k^2g_0\}$ be isospectral to $g_0$ over a compact boundaryless smooth 4-dimension manifold $(M,g_0)$. We prove that the subsequence of conformal factors $\{u_k\}$ converges to $u$…

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…

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $\lambda (M,[g],\si)= \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $\lambda_1^+(\tilde{g})$ is the smallest…

Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^\infty)$ on a compact manifold $M^n$ ($n\ge 3$) with negative Yamabe invariant $\sigma(M)$. It is well-known that if $g$ is a smooth…

We study, on a weighted Riemannian manifold of Ric$_{N} \geq K > 0$ for $N < -1$, when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product $\mathbb{R}…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For a metric $g$ on $M$, we let $\la_2(g)$ be the second eigenvalue of the Yamabe operator $L_g:= \frac{4(n-1)}{n-2} \Delta_g + \scal_g$. Then, the second Yamabe invariant…

We examine here the space of conformally compact metrics $g$ on the interior of a compact manifold with boundary which have the property that the $k^{th}$ elementary symmetric function of the Schouten tensor $A_g$ is constant. When $k=1$…

Let $(M, g_0)$ be a closed 4-manifold with positive Yamabe invariant and with $L^2$-small Weyl curvature tensor. Let $g_1 \in [g_0]$ be any metric in the conformal class of $g_0$ whose scalar curvature is $L^2$-close to a constant. We prove…