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As a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup G of the isometry group, find a G-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we introduce the equivariant CR Yamabe problem and prove some related results. [ABSTRACT FROM AUTHOR]

Let (M,∂M,g)\left(M,\partial M,g) be a compact Riemannian manifold with boundary. As a generalization of the Yamabe invariant with boundary Y(M,∂M,g)Y\left(M,\partial M,g), we define the kth Yamabe invariant with boundary Yk(M,∂M,g){Y}_{k}\left(M,\partial M,g). We prove some of its properties and study when it can be attained by the generalized metric. We also prove a version of conformal Schwarz lemma on (M,∂M,g)\left(M,\partial M,g) by using the Yamabe flow with boundary.

Yamabe soliton is a self-similar solution to the Yamabe flow on manifolds without boundary. In this paper, we define and study the Yamabe soliton with boundary and conformal mean curvature soliton, which are natural generalizations of the Yamabe soliton. We study these solitons from equation point of view. We also study their two-dimensional analog: the Gauss curvature soliton with boundary and geodesic curvature soliton. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of conformal metrics with constant holomorphic d-scalar curvature and the prescribed holomorphic d-scalar curvature problem on closed, connected almost Hermitian manifolds of dimension n ⩾ 6. In addition, we obtain an application and a variational formula for the associated conformal invariant. [ABSTRACT FROM AUTHOR]

We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider Yamabe problem for a smooth manifold with a W 1 , p metric. The curvature is considered as a distribution since our metric is not twice differentiable. We prove that for any metric g which is W 1 , p on M such that the Yamabe constant of (M, g) is less than that of the standard sphere, there exists a W 1 , p metric on M which is conformal to g and has constant distributional scalar curvature. [ABSTRACT FROM AUTHOR]

The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m), the weighted Yamabe problem consists on finding another smooth metric measure space conformal to (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ+μe−ϕ∕m\lambda +\mu {e}^{-\phi /m} for some constants μ\mu and λ\lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m). On the other hand, we prove a version of the conformal Schwarz lemma on (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m). All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a large positive maximum.

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n ≥ 7 and the trace-free part of the second fundamental form is nonzero everywhere on the boundary. [ABSTRACT FROM AUTHOR]

Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature. [ABSTRACT FROM AUTHOR]

In this paper, we study the smooth Yamabe invariant with boundary, which is defined to be the supremum of all the Yamabe constants on a manifold with boundary. We also study the problem of prescribing curvatures on manifolds with boundary under the condition of unit volume. [ABSTRACT FROM AUTHOR]

We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a large positive maximum. [ABSTRACT FROM AUTHOR]

We prove that a minimizer of the Yamabe functional does not exist for a sphere S n of dimension n ≥ 3 , endowed with a standard edge-cone spherical metric of cone angle greater than or equal to 4 π , along a great circle of codimension two. When the cone angle along the singularity is smaller than 2 π , the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set. [ABSTRACT FROM AUTHOR]

Let (M ,) be a compact, connected, strictly pseudo-convex CR manifold. In this paper, we give some properties of the CR Yamabe Operator L . We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions. [ABSTRACT FROM AUTHOR]

We provide explicit formulae for the first eigenvalue of the Laplace–Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric. As consequences, we prove that homogeneous metrics on CROSSes are isospectral if and only if they are isometric, and also discuss their stability (or lack thereof) as solutions to the Yamabe problem. [ABSTRACT FROM AUTHOR]

Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that ∂ M has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above. [ABSTRACT FROM AUTHOR]

Let (M n , g) be a closed Riemannian manifold of dimension n ≥ 3. Assume [ g ] is a conformal class for which the Conformal Laplacian L g has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of L g over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility. [ABSTRACT FROM AUTHOR]