Related papers: Quantitative Stability for Yamabe minimizers on ma…

The Yamabe Invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive…

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 Gonzalez and Qing. These are inspired by the work of Coda-Marques on the…

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the…

In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincar\"{e} exponent less than…

For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a…

We derive lower bounds on the scalar curvature of complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that the corresponding potential functions have at most quadratic growth in…

We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension…

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar…

We study an optimal boundary control problem associated to the boundary obstacle problem for the couple conformal Laplacian and conformal Robin operator on n-dimensional compact Riemannian manifolds with boundary and with n\geq 3. When the…

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure.…

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…

In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean…

We prove a local minimizing property for strictly stable free-boundary minimal hypersurfaces in the relative current setting. Let $\Sigma^n$ be a compact, two-sided, properly embedded free-boundary minimal hypersurface in a compact…

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main…

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The…

Let $(X^{n},g_+) $ $(n\geq 3)$ be a Poincar\'{e}-Einstein manifold which is $C^{3,\alpha}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=\rho^2g_+)$ via some…

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…

We present in this note a lower bound for the Calabi functional in a given K\"ahler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.

This paper deals with the conformal deformation of the standard metric in a domain on the sphere to a complete metric with the constant scalar curvature. The problem of description of domains allowing such deformation originates in the…