Related papers: Conformal invariants for the zero mode equation

We extend the following result of Cochran ``A closed $m$-quasi Einstein manifold ($M,g,X$) with $m \ne -2$ has constant scalar curvature if and only if $X$ is Killing" covering the missing accidental case $m=-2$ and generalize it showing…

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems…

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…

The Yamabe invariant is an invariant of a closed smooth manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold T^m\times B where T^m$ is the m-dimensional torus, and B is a…

We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an…

We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear…

For a compact connected manifold M of dimension n greater than 3 and with no metric of positive scalar curvature, we prove that the Yamabe invariant is unchanged under surgery on spheres of dimension different from 1, n-2 and n-1. We use…

Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^{\alpha}\)-H\"older topologie on the space of…

IIn this article, we study the instanton equation on the cylinder over a closed manifold $X$ which admits non-zero smooth $3$-form $P$ and $4$-form $Q$. Our results are (1) if $X$ is a \textbf{good} manifold, i.e., $P,Q$ satisfying…

Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984.…

The purpose of this article is to study gradient Yamabe soliton on warped product manifolds. First, we prove triviality results in the case of noncompact base with limited warping function, and for compact base. In order to provide…

Let $(M^{n},g_{0})$ be a $n=3,4,5$ dimensional, closed Riemannian manifold of positive Yamabe invariant. For a smooth function $K>0$ on $M$ we consider a scalar curvature flow, that tends to prescribe $K$ as the scalar curvature of a metric…

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the $n$-sphere involving an operator of order $2s> n$. In this case the Sobolev exponent is negative. Our results…

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…

We prove that a compact, connected, and oriented 4-dimensional gradient $m$-quasi-Einstein manifold with $m\in [1, \infty]$ which is additionally a spin manifold must satisfy the Hitchin-Thorpe Inequality. We show further that the…

In this paper, we show that if a compact $n$-dimensional vacuum static space $(M^n, g, f)$ admits a non-trivial closed conformal vector field $V$, then $(M, g)$ is isometric to a standard sphere ${\Bbb S}^n(c)$. We also prove that if a pair…

Let us fix a conformal class $[g_0]$ and a spin structure $\sigma$ on a compact manifold $M$. For any $g\in [g_0]$, let $\lambda^+_1(g)$ be the smallest positive eigenvalue of the Dirac operator $D$ on $(M,g,\sigma)$. In a previous paper we…

We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$ Y(S^4, [g])…

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…

In this paper, we investigate local rigidity properties related to Gagliardo-Nirenberg constants and unweighted Yamabe-type constants. Let $V$ be an open bounded subset of an $n$-dimensional Riemannian manifold $(M,g)$ whose…