Yamabe-type Problem Research Articles

Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric data

Let (M,g,σ) be an m-dimensional closed spin manifold, with a fixed Riemannian metric g and a fixed spin structure σ; let S(M) be the spinor bundle over M. The spinorial Yamabe-type problems address the solvability of the following equationDgψ=f(x)|ψ|g2m−1ψ,ψ:M→S(M),x∈M where Dg is the associated Dirac operator and f:M→R is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when m=2, a solution corresponds to a conformal isometric immersion of the universal covering M˜ into R3 with prescribed mean curvature f; meanwhile, for general dimensions and f≡constant≠0, a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant.The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold (Sm,g) where the solution set of the spinorial Yamabe-type problem is not compact: 1). the geometric potential f is constant (say f≡1) with the background metric g being a Ck perturbation of the canonical round metric gSm, which is not conformally flat somewhere on Sm; 2). f is a perturbation from constant and is of class C2, while the background metric g≡gSm.

The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus

Abstract In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset Ω \Omega of the CR manifold S 3 {{\mathbb{S}}}^{3} bounded by the Clifford torus Σ \Sigma is discussed. The Yamabe-type problem of finding a contact form on Ω \Omega which has zero Tanaka-Webster scalar curvature and for which Σ \Sigma has a constant p p -mean curvature is also discussed.

Solutions of spinorial Yamabe-type problems on 𝑆^{𝑚}: Perturbations and applications

This paper is part of a program to establish the existence theory for the conformally invariant Dirac equation \[ D g ψ = f ( x ) | ψ | g 2 m − 1 ψ D_g\psi =f(x)|\psi |_g^{\frac 2{m-1}}\psi \] on a closed spin manifold ( M , g ) (M,g) of dimension m ≥ 2 m\geq 2 with a fixed spin structure, where f : M → R f:M\to \mathbb {R} is a given function. The study on such nonlinear equation is motivated by its important applications in Spin Geometry: when m = 2 m=2 , a solution corresponds to an isometric immersion of the universal covering M ~ \widetilde M into R 3 \mathbb {R}^3 with prescribed mean curvature f f ; meanwhile, for general dimensions and f ≡ c o n s t a n t f\equiv constant , a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. Comparing with the existing issues, the aim of this paper is to establish multiple existence results in a new geometric context, which have not been considered in the previous literature. Precisely, in order to examine the dependence of solutions of the aforementioned nonlinear Dirac equations on geometrical data, concrete analyses are made for two specific models on the manifold ( S m , g ) (S^m,g) : the geometric potential f f is a perturbation from constant with g = g S m g=g_{S^m} being the canonical round metric; and f ≡ 1 f\equiv 1 with the metric g g being a perturbation of g S m g_{S^m} that is not conformally flat somewhere on S m S^m . The proof is variational: solutions of these problems are found as critical points of their corresponding energy functionals. The emphasis is that the solutions are always degenerate: they appear as critical manifolds of positive dimension. This is very different from most situations in elliptic PDEs and classical critical point theory. As corollaries of the existence results, multiple distinct embedded spheres in R 3 \mathbb {R}^3 with a common mean curvature are constructed, and furthermore, a strict inequality estimate for the Bär-Hijazi-Lott invariant on S m S^m , m ≥ 4 m\geq 4 , is derived, which is the first result of this kind in the non-locally-conformally-flat setting.

Improved higher-order Sobolev inequalities on CR sphere

We improve higher-order CR Sobolev inequalities on S2n+1 under the vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain the existence of minimizers in C2k(N) of higher-order CR Yamabe-type problems when Yk(N)<Yk(Hn).

Yamabe problem in the presence of singular Riemannian Foliations

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

Kähler Finsler metrics and conformal deformations

The conformai properties of complex Finsler metrics are studied. We first give a characterization of a compact complex Finsler manifold to be globally conformai Kahler. By considering the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes, we then study the variational properties of Kahler Finsler metrics. By studying the spectral properties of two average metrics, the stabilities of critical Kähler Finsler metrics are verified. Finally, a Yamabe type problem for mean holomorphic Ricci curvature is considered, and a partial existence result is obtained.

A generalization of Aubin's result for a Yamabe-type problem on smooth metric measure spaces

The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin, and Schoen. In particular, Aubin solved the case when the Riemannian manifold is compact, non-locally conformally flat and with dimension equal and greater than 6. In 2015, Case considered a Yamabe-type problem in the setting of smooth measure space in manifolds and for a parameter m, which generalize the original Yamabe problem when m=0. Also, Case solved this problem when the parameter m is a natural number. In the context of Yamabe-type problem we generalize Aubin's result for non-locally conformally flat manifolds, with dimension equal and greater than 6 and small parameter m.

An ODE reduction method for the semi-Riemannian Yamabe problem on space forms

We consider the semi-Riemannian Yamabe type equations of the form −□u+λu=μ|u|p−1uonMwhere M is either the semi-Euclidean space or the pseudosphere of dimension m≥3, □ is the semi-Riemannian Laplacian in M, λ≥0, μ∈R∖︀{0} and p>1. Using semi-Riemannian isoparametric functions on M, we reduce the PDE into a generalized Emden–Fowler ODE of the form w′′+q(r)w′+λw=μ|w|p−1wonI,where I⊂R is [0,∞) or [0,π], q(r) blows-up at 0 and w is subject to the natural initial conditions w′(0)=0 in the first case and w′(0)=w′(π)=0 in the second. We prove the existence of blowing-up and globally defined solutions to this problem, both positive and sign-changing, inducing solutions to the semi-Riemannian Yamabe type problem with the same qualitative properties, with level and critical sets described in terms of semi-Riemannian isoparametric hypersurfaces and focal varieties. In particular, we prove the existence of sign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in the pseudosphere having a prescribed number of nodal domains.

A generalization of the Escobar–Riemann mapping-type problem to smooth metric measure spaces

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization of Sobolev Trace Inequality deduced by Bolley at. al. This trace inequality allows us to introduce an Escobar quotient and its infimum. This infimum we call the Escobar weighted constant. The Escobar-Riemann mapping type problem for smooth metric measure spaces in manifolds with boundary consists of finding a function which attains the Escobar weighted constant. Furthemore, we resolve the later problem when Escobar weighted constant is negative. Finally, we get an Aubin type inequality connecting the weighted Escobar constant for compact smooth metric measure space and the optimal constant for the trace inequality due to Bolley at. al.

Compactness for a class of Yamabe-type problems on manifolds with boundary

In this paper we are interested in studying the set of positive solutions to the following Yamabe–type problem(⁎){−Δgu+a(x)u=0,inM,∂u∂ηg+b(x)u=(n−2)unn−2,on∂M, where (Mn,g) is a compact Riemannian n−manifold, n≥3, with boundary, Δg is the Laplace–Beltrami operator for g, ηg 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−24(n−1)Rg(ξ) and Φ(ξ¯)=b(ξ¯)−n−22hg(ξ¯), where Rg denotes the scalar curvature of the metric g and hg is the mean curvature of ∂M, we prove some results of compactness for the set of positive solutions to the problem (⁎).

On the existence of extremals for the weighted Yamabe problem on compact manifolds

In [3], J.S. Case considered a Yamabe type problem in the setting of smooth measure space for a Riemannian manifold and a parameter m, which generalize the original Yamabe problem when m=0 and Perelman's ν-entropy when m=∞. In the context of Euclidean space this generalization consists to find the functions that satisfy the sharp Gagliardo, Nirenberg, Sobolev inequalities. We point out that J.S. Case also solved this problem when the parameter m is natural number. In this work, by using the Schoen's argument [12], we prove the result of J.S. Case for a special class of compact n-dimensional smooth metric measure space when the parameter m belongs to I=⋃i=0∞[i,14(4−3n+n2+16n+16)+i).

Gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces and applications

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.

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $\sigma_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem which allows us to construct examples of compact Riemannian manifolds $(X,g)$ for which this problem admits multiple non-homothetic solutions in the case when $2k<\dim X$. Our examples are all such that the boundary with its induced metric is a Riemannian product of a round sphere with an Einstein manifold.

Gradient estimates for a nonlinear parabolic equation and Liouville theorems

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.

A Conformally Invariant Gap Theorem in Yang–Mills Theory

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary

We consider the Yamabe-type problem \begin{document}$\begin{equation*} \begin{cases} Δ_g u+fu=0 in M\\ \frac{\partial u}{\partial ν}+hu=u^{\frac{n}{n-2}} on \partial M \end{cases}\end{equation*} $ \end{document} when \begin{document} $(M,g)$ \end{document} is the standard half sphere of dimensions \begin{document} $n≥ 3$ \end{document} . We establish existence results of positive blowing-up solutions with unbounded energy to this problem for all dimensions \begin{document} $n≥ 3$ \end{document} .

Renormalized Volume

We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.

Bubbling phenomena in calculus of variations

This paper is a survey on bubbling phenomena occurring in some geometric problems. We present here a few problems from conformal geometry, gauge theory and contact geometry and we give the main ideas of the proofs and important results. We focus in particular on the Yamabe type problems and the Weinstein conjecture, where A. Bahri made a huge contribution by introducing new methods in variational theory.

Concentration on minimal submanifolds for a Yamabe-type problem

ABSTRACTWe construct solutions to a Yamabe-type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a nondegenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of PDE's on the submanifold with a singular term of attractive or repulsive type is established.

On Yamabe-type problems on Riemannian manifolds with boundary

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $\nu$ is the outward pointing unit normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$ is the boundary mean curvature.