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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.

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]

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]

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]

The classical Yamabe problem in Riemannian geometry states that every conformal class contains a metric with constant scalar curvature. In Finsler geometry, the C-convexity is needed in general. In this paper, we study the strong C-convexity of Randers metrics, and provide a result on the Yamabe problem for the metrics of Randers type.

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]

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the dimension of the manifold is n ≥ 8 and that the Weyl tensor W g is not vanishing on ∂ M. [ABSTRACT FROM AUTHOR]

In this work, we study the nonlinear problem on compact d-dimensional (d ≥ 3) Riemannian manifolds with respect to absence of boundary. The existence of one non-trivial weak solution is established, and its application to solve Emden-Fowler equations which contain infinity nonlinear terms. We also introduce an example to illustrate the results obtained, which can be applied to many of the problems resulting in astrophysics, conformal Riemannian geometry, gas combustion, isothermal stationary gas sphere and in the theory of thermionic emission. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the formula of scalar curvature for Kropina metrics. Then we prove that a Kropina metric of isotropic scalar curvature must be of isotropic S-curvature. We also focus on Yamabe problem in Kropina metrics and give a negative answer to the Yamabe problem on Kropina metrics with isotropic S-curvature. It turns out that C-reducible metrics do not satisfy Yamabe problem. [ABSTRACT FROM AUTHOR]

We construct singular solutions to the Yamabe equation using a reduction of the problem in an equivariant setting. This provides a non-trivial geometric example for which the analysis is simpler than in Mazzeo-Pacard program. Our construction provides also a non-trivial example of a weak solution to the Yamabe problem involving an equation with (smooth) coefficients. [ABSTRACT FROM AUTHOR]

In order to study the Yamabe changing-sign problem, Bahri and Xu proposed a conjecture which is a universal inequality for p points in ℝm. They have verified the conjecture for p ≤ 3. In this paper, we first simplify this conjecture by giving two sufficient and necessary conditions inductively. Then we prove the conjecture for the basic case m = 1 with arbitrary p. In addition, for the cases when p = 4, 5 and m ≥ 2, we manage to reduce them to the basic case m = 1 and thus prove them as well. [ABSTRACT FROM AUTHOR]

In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in C 1 . The full conjecture from the conformal perspective is also discussed as a possible approach to resolving the conjecture. [ABSTRACT FROM AUTHOR]

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential. [ABSTRACT FROM AUTHOR]

We construct a new family of entire solutions to the Yamabe equation − Δ u = n (n − 2) 4 | u | 4 n − 2 u in D 1 , 2 (R n). If n = 3 our solutions have maximal rank, being the first example in odd dimension. Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in S 3 (1). [ABSTRACT FROM AUTHOR]

Given any closed Riemannian manifold (M, g) we use the Lyapunov–Schmidt finite-dimensional reduction method and the classical Morse and Lusternick–Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on (M, g). If (N, h) is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product (M × N , g + ε 2 h) , for ε > 0 small. For example, if M is a closed Riemann surface of genus g and (N , h) = (S 2 , g 0) is the round 2-sphere, we prove that for ε > 0 small enough and a generic metric g on M, the Yamabe equation on (M × S 2 , g + ε 2 g 0) has at least 2 + 2 g solutions. [ABSTRACT FROM AUTHOR]

Let a compact Riemannian -dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of there are scalar-flat metrics that have as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term with a negative smooth function the set of solutions of Yamabe problem is still a compact set. [ABSTRACT FROM AUTHOR]

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to- Neumann operators of uniformly degenerate elliptic boundary value problems observed by Chang and González, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf-type maximum principle together with interplay between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extends the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem. © 2013 Mathematical Sciences Publishers.

In this paper, we define the second generalized Yamabe invariant on manifolds with boundary. We prove some of its properties and study when the invariant is attained by some metric. In another direction, by using the conformal mean curvature flow, we prove a version of the conformal Schwarz lemma for manifolds with boundary. [ABSTRACT FROM AUTHOR]

Let (M, g) be a closed Riemannian manifold of dimension n ≥ 3 and x 0 ∈ M be an isolated local minimum of the scalar curvature s g of g. For any positive integer k we prove that for ϵ > 0 small enough the subcritical Yamabe equation - ϵ 2 Δ u + (1 + c N ϵ 2 s g) u = u q has a positive k-peaks solution which concentrate around x 0 , assuming that a constant β is non-zero. In the equation c N = N - 2 4 (N - 1) for an integer N > n and q = N + 2 N - 2 . The constant β depends on n and N, and can be easily computed numerically, being negative in all cases considered. This provides solutions to the Yamabe equation on Riemannian products (M × X , g + ϵ 2 h) , where (X, h) is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.

In this paper we are interested in studying the set of positive solutions to the following Yamabe–type problem (⁎) { − Δ g u + a (x) u = 0 , in M , ∂ u ∂ η g + b (x) u = (n − 2) u n n − 2 , on ∂ M , where (M n , g) is a compact Riemannian n −manifold, n ≥ 3 , with boundary, Δ g is the Laplace–Beltrami operator for g , η g is the pointing unit outward normal to ∂ M , a ∈ C ∞ (M) and b ∈ C ∞ (∂ M). Assuming suitable hypotheses on the functions Ψ (ξ) = a (ξ) − n − 2 4 (n − 1) R g (ξ) and Φ (ξ ¯) = b (ξ ¯) − n − 2 2 h g (ξ ¯) , where R g denotes the scalar curvature of the metric g and h g is the mean curvature of ∂ M , we prove some results of compactness for the set of positive solutions to the problem (⁎). [ABSTRACT FROM AUTHOR]

Given an isoparametric function f on the n -dimensional sphere, we consider the space of functions w ∘ f to reduce the Yamabe equation on the round sphere into a singular ODE on w in the interval [ 0 , π ] , of the form w ″ + (h (r) / sin ⁡ r) w ′ + λ (| w | 4 / n − 2 w − w) = 0 and boundary conditions w ′ (0) = 0 = w ′ (π) , where h is a monotone function with exactly one zero on (0 , π) and λ > 0 is a constant. For any positive integer k we obtain a solution with exactly k -zeroes yielding solutions to the Yamabe equation with exactly k connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on the singularities, and then to solve the corresponding double shooting problem, matching the values of w and w ′ at the unique zero of h. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy. [ABSTRACT FROM AUTHOR]

Given an isoparametric function f on the n-dimensional round sphere, we consider functions of the form u=w∘fu=w∘f to reduce the semilinear elliptic problem −Δg0u + λu = λ|u|p−1u on Sn with λ > 0 and 1 < p, into a singular ODE in [0,π] of the form w" + h(r)/sinr w′ + λ/ℓ2(|w|p−1w − w) = 0, where h is an strictly decreasing function having exactly one zero in this interval and ℓ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to ff and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when p > n+2/n−2, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets. [ABSTRACT FROM AUTHOR]

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution's asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20]. [ABSTRACT FROM AUTHOR]

Given (M,g)${(M,g)}$, a smooth compact Riemannian n-manifold, we prove that for a generic Riemannian metric g the critical points of the function 𝒲g⁢(ξ):=|Weylg⁢(ξ)|g2${\mathcal{W}_{g}(\xi):=\lvert\mathrm{Weyl}_{g}(\xi)\rvert^{2}_{g}}$ with 𝒲g⁢(ξ)≠0${\mathcal{W}_{g}(\xi)\not=0}$ are nondegenerate.

In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equationΔfu+p(x)u+q(x)uα=0, on complete smooth metric measure spaces (MN,g,e−fdv) with ∞-Bakry-Émery Ricci tensor bounded from below, where α is an arbitrary real constant, p(x) and q(x) are smooth functions. As an application, Liouville-type theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on (MN,g,e−fdv) with nonpositive weighted scalar curvature.

Abstract We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n ≥ 11. [ABSTRACT FROM AUTHOR]

Abstract Through desingularization of Clifford torus, we prove the existence of a sequence of nondegenerate (in the sense of Duyckaerts–Kenig–Merle ([8])) nodal nonradial solutions to the critical Yamabe problem − Δ u = n (n − 2) 4 | u | 4 n − 2 u , u ∈ D 1 , 2 (R n). The case n = 4 is the first example in the literature of a solution with maximal rank N = 2 n + 1 + n (n − 1) 2. [ABSTRACT FROM AUTHOR]

Abstract We develop an efficient calculus for varying hypersurface embeddings based on variations of hypersurface defining functions. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compact four-manifolds. We give explicit formulæ for both the energy functional and the obstruction. Vanishing of the latter is a necessary condition for solving the vacuum cosmological Einstein equations in four spacetime dimensions with data prescribed on a conformal infinity, while the energy functional generalizes the scheme-independent contribution to entanglement entropy across surfaces to hypersurfaces. [ABSTRACT FROM AUTHOR]

Abstract Let (M m , g M) be a closed (compact, without boundary) connected manifold with positive scalar curvature and (F k , h) a closed connected manifold with constant scalar curvature (m ≥ 3 and k > 3). By a Theorem of Dobarro and Lami Dozo (1987), there are weights f : M → R + such that the warped product (M m × F k , g M + f 2 h) has constant scalar curvature. We construct paths of warped product metrics (M m × F k , g M + f ϵ 2 h) , ϵ ∈ (0 , ϵ 0) , ϵ 0 small, with constant scalar curvature, that exhibit multiplicity of solutions to the Yamabe equation. Moreover, in the case that (F k , h) has a flat metric we add the constraints of unit volume and fixed constant scalar curvature to the construction of paths of warped metrics (M m × F k , g M + f 2 h ϵ) , ϵ ∈ (0 , ϵ 0) , that exhibit multiplicity. We use techniques from bifurcation theory along with spectral theory for warped products. [ABSTRACT FROM AUTHOR]

Abstract Let (M , g) be any closed Riemannianan manifold and (N , h) be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product (M × N , g + δ h) has at least Cat (M) + 1 solutions for δ small enough, where Cat (M) denotes the Lusternik–Schnirelmann-category of M. The solutions obtained are functions of M and C a t (M) of them have energy arbitrarily close to the minimum. [ABSTRACT FROM AUTHOR]

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q -curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson's Q -curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds. [ABSTRACT FROM AUTHOR]