The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $σ_k$ curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding of the analytic behavior of solutions of the PDE. However, the singular behavior of these solutions, which is important in describing many important questions in conformal geometry, is little understood. This note classifies all possible radial solutions, in particular, the \emph{singular} solutions of the $σ_k$ Yamabe equation, which describes conformal metrics whose $σ_k$ curvature equals a constant. Although the analysis involved is of elementary nature, these results should provide useful guidance in studying the behavior of singular solutions in the general situation.
We study positive solutions of the Yamabe equation with isolated singularity and prove the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.
We study asymptotic behaviors of positive solutions to the Yamabe equation and the $σ$k-Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck, and a work by Korevaar, Mazzeo, Pacard, and Schoen, on the Yamabe equation, and a work by Han, Li, and Teixeira on the $σ_k$-Yamabe equation. The study is based on a combination of classification of global singular solutions and an analysis of linearized operators at these global singular solutions. Such linearized equations are u…
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.
We study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not conformally flat. We prove that, in dimension $6$, any solution is asymptotically close to a Fowler solution, which is an extension of the same result for lower dimensions by F.C. Marques in 2008.
We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity $(-Δ)^γw = c_{n, γ} w^{\frac{n+2γ}{n-2γ}}, w>0 \ \mbox{in} \ \mathbb{R}^n \backslash \{0\}$ We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.
In this paper we construct singular solutions to the critical Dirac equation on spheres. More precisely, first we construct solutions admitting two points singularities that we call Delaunay-type solutions because of their similarities with the Delaunay solutions constructed for the singular Yamabe problem in \cite{MP1 , Schoen1989}. Then we construct another kind of singular solutions admitting a great circle as a singular set. These solutions are the building blocks for singular solutions on a general Spin manifold.
$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equa…
Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder-type estimates for the heat operator on certain Hölder spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting.
This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov-Safonov …
We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d>(n-2)/2. Our main theorem is inspired by the classical results by Aviles-McOwen and Loewner-Nirenberg …
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 works of Loewner and Nirenberg, and Schoen and Yau concerned with the locally conformally flat manifolds. The goal of this work is to apply ideas from the nonlinear potential theory to the problem. They allow, in particular, to solve the problem in the case of the constant negative scalar curvature.
We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted $L^\infty$ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an \emph{Ansatz} for the Yamabe problem with a prescribed singular set of maximal dimension $(n-2)/2$, for which, using the same machinery,…
We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on t…
We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.
Our primary purpose is to study a class of strongly coupled nonlinear elliptic systems with critical growth in a compact Riemannian manifold with constant scalar curvature. Using a gluing technique and perturbation arguments, we show the existence of singular solutions asymptotic to a Fowler-type solution near the isolated singularity.
We prove the existence of a homogeneous singular solution of the critical equation $$-Δu = u^{\frac{Q+2}{Q-2}}$$ on the Heisenberg group $H^n$, where $Q$ is the \textit{homogeneous dimension}. In order to do this, we introduce a suitable concept of normal curvature for hypersurfaces. Furthermore we study the bifurcation of non-homogeneous solutions from the homogeneous one.
We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding version of the Yamabe problem on cylindrical manifolds. We affirmatively solve this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almos…
Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,π]$, of the form $w" + (h(r)/\sin r)w'+λ(\vert w\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with exactly one zero on $[0,π]$ and $λ>0$ is a constant. The natural boundary conditions in order to obtain smooth solutions are $w'(0)=0$ and $w'(π)=0$. We show that for any positive integer $k$ there exists a solution with exactly $k$-zeroes yielding solutions to the Yamabe equatio…
Let $Ω$ be a domain on the unit $n$-sphere $ \mathbb S^n$ and $\mathring{g}$ the standard metric of $\mathbb S^n$, $n\ge 3$. We show that there exists a conformal metric $g$ with vanishing scalar curvature $R(g)=0$ such that $(Ω, g)$ is complete if and only if the Bessel capacity $\mathcal C_{α, q}(\mathbb S^n\setminus Ω)=0$, where $α=1+\frac2n$ and $q=\frac n2$. Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf-Rinow theorem for the divergent curves.
Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty,T)$ for some $T\in\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as $|x|\to\infty$. We also prove that both the $3$-parameters ancient solution and the $4$-parameters ancient solution are singular limit solution of the $5$-parameters ancient solutions.
We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.
We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a r…
From fractional Lane-Emden-Serrin equation -- existence, multiplicity and local behaviors via classical ODE -- to fractional Yamabe metrics with singularity of "maximal" dimension
Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for Poincaré-Einstein metrics. The results are formulated in terms of Weyl conformal invariants arising from the ambient metric of Fefferman-Graham. As applications we obtain "Green function" characterizations of locally conformally flat mani…
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global …
We study a behavior of the conformal Laplacian operator $Ł_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator $Ł_g$ on such manifolds. We describe the asymptotic of a general solution of the equation $Ł_g u = Q u^α$ with $1\leq α\leq \frac{n+2}{n-2}$ near each singular point. In particular, we derive the asymptotic of the Yamabe metric near such singularity.
We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas, and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [11] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.
We consider the sphere $\Sph^{2n+1}$ equipped with its standard CR structure. In this paper we construct explicit contact forms on $\Sph^{2n+1}\setminus \Sph^{2k+1}$, which are conformal to the standard one and whose related Webster metrics have constant Webster curvature; in particular the curvature is positive if $2k< n-2$. As main applications, we provide two perturbative results. In the first one we prove the existence of infinitely many contact structures on $\Sph^{2n+1}\setminus τ(\Sph^{1})$ conformal to the standard one and having constant Webster curvature, where $τ(\Sph^{1})$ is a sma…
An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of $M$ into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of or…
In this article, we set up the continuous maximal regularity theory for a class of linear differential operators on manifolds with singularities. These operators exhibit degenerate or singular behaviors while approaching the singular ends. Particular examples of such operators include differential operators defined on domains, which degenerate fast enough toward the boundary. Applications of the theory established herein are shown to the Yamabe flow, the porous medium equation, the parabolic $p$-Laplacian equation and the thin film equation. Some comments about the boundary blow-up problem, an…
We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular solutions are rotationally invariant and weakly positive. More precisely, they are the product of a fourth order spherical solution by a unit vector with nonnegative coordinates. If the origin is a non-removable singularity, we show that the solutions are radially symmetric and stro…
We construct positive solutions to the equation $$-Δ_{\mathbf{H}^n} u = u^{\frac{Q+2}{Q-2}}$$ on the Heisenberg group, singular in the origin, similar to the Fowler solutions of the Yamabe equations on $\mathbf{R}^n$. These satisfy the homogeneity property $u\circδ_T=T^{-\frac{Q-2}{2}}u$ for some $T$ large enough, where $Q=2n+2$ and $δ_T$ is the natural dilation in $\mathbf{H}^n$. We use the Lyapunov-Schmidt method applied to a family of approximate solutions built by periodization from the global regular solution classified by Jerison and Lee.
We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of isolated singularities of some semilinear elliptic equations, such as the Yamabe equation, Hardy-Hénon equation etc.
This work investigates a nonlocal sinh-Gordon equation with a singularly perturbed parameter in a ball. Under the Robin boundary condition, the solution asymptotically forms a quite steep boundary layer in a thin annular region, and rapidly becomes a flat curve outside this region. {Focusing more particularly on the structure of the thin annular layer in this region, the pointwise asymptotic expansion involving the domain-size is evaluated more sharply, where the domain-size exactly appears in the second term of the asymptotic expansion.} It should be stressed that the standard argument of mat…
In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class $[g_0]$ of the Riemannian product $S^1\times S^{n-1}$ : a circle of length $T$ crossed with the (n-1)-dimensional standard sphere. Such metrics have a harmonic Riemannian curvature and a non parallel Ricci tensor, except for the cylindric one. Thus, it appears a natural link between them and the Derdzinski metrics which are warped product and classify a family of Riemannian manifolds. The…
Let $A=\{x\in \R^{2m} : 0< a< |x| <b\}$ be an annulus. Consider the following singularly perturbed elliptic problem on $A$ \begin{equation} \begin{array}{lll} -\eps^2{\De u} + |x|^ηu = |x|^ηu^p, &\mbox{\qquad in} A \notag u>0 &\mbox{\qquad in} A u = 0 &\mbox{\qquad on} \partial A \end{array} %\label{a1} \end{equation} $1<p<2^*-1$. We shall prove the existence of a positive solution $u_\eps$ which concentrates on two different orthogonal spheres of dimension $(m-1)$ as $\eps\to 0$. We achieve this by studying a reduced problem on an annular domain in $\R^{m+1}$ and analyzing the profile of a tw…
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 theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $Δ_g$ denote…
For any $n\ge 3$, $0<m\le (n-2)/n$, and constants $η>0$, $β>0$, $α$, satisfying $α\leβ(n-2)/m$, we prove the existence of radially symmetric solution of $\frac{n-1}{m}Δv^m+αv +βx\cdot\nabla v=0$, $v>0$, in $\R^n$, $v(0)=η$, without using the phase plane method. When $0<m<(n-2)/n$, $n\ge 3$, and $α=2β/(1-m)>0$, we prove that the radially symmetric solution $v$ of the above elliptic equation satisfies $\lim_{|x|\to\infty}\frac{|x|^2v(x)^{1-m}}{\log |x|} =\frac{2(n-1)(n-2-nm)}{β(1-m)}$. In particular when $m=\frac{n-2}{n+2}$, $n\ge 3$, and $α=2β/(1-m)>0$, the metric $g_{ij}=v^{\frac{4}{n+2}}dx^2$…
