We consider the semi-Riemannian Yamabe type equations of the form
\[
-\square u + λu = μ\vert u\vert^{p-1}u\quad\text{ on }M
\] where $M$ is either the semi-Euclidean space or the pseudosphere of dimension $m\geq 3$, $\square$ is the semi-Riemannian Laplacian in $M$, $λ\geq0$, $μ\in\mathbb{R}\smallsetminus\{0\}$ and $p>1$. Using semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a generalized Emden-Fowler ODE of the form \[ w''+q(r)w'+λw = μ\vert w\vert^{p-1}w\quad\text{ on } I, \] where $I\subset\mathbb{R}$ is $[0,\infty)$ 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.
From fractional Lane-Emden-Serrin equation -- existence, multiplicity and local behaviors via classical ODE -- to fractional Yamabe metrics with singularity of "maximal" dimension
We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $σ_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.
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…
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 $(M\times N , g + \ve^2 h )$, for $\ve >0$ small. For example, if $M$ is a closed Riemann surface of genus ${\bf g}$ and $(N,h) = (S^2 , g_0)$ is the round 2-sphere,…
In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds for dimension $n\ne 2$. As one application, we solve a generalized Yamabe problem on locally conforamlly flat manifolds via a new designed energy functional and a new variational approach. Even for the classic Yamabe problem on locally conformally flat manifolds, our approach provides a new and relatively simpler solution.
In this paper, we study the existence of complete Yamabe metric with zero scalar curvature on an n-dimensional complete Riemannian manifold $(M,g_0)$, $n\geq 3$. Under suitable conditions about the initial metric, we show that there is a global fine solution to the Yamabe flow. The interesting point here is that we have no curvature assumption about the initial metric. We show that on an n-dimensional complete Riemannian manifold $(M,g_0)$ with non-negative Ricci curvature, $n\geq 3$, the Yamabe flow enjoys the local $L^1$-stability property from the view-point of the porous media equation. Co…
We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-Δ_g u + λu = λu^q$, with $λ>0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1 \geq d_2 $ we show that for any $q<\frac{ n-d_2+2 }{n - d_2 -2}$ the number of positive solutions of the equation $-Δ_g u + λu = λu^q$ tends to $\infty$ as $λ\rightarrow +\infty$. We apply this result to prove multiplicity results for positive solutions of critical and supercritical equations. In particular we prove multiplicity results …
We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.
This work is devoted to the analysis of the Yamabe problem on Spin manifolds and some applications to CMC immersions. Despite the efforts of many authors, very little is known on the existence of Yamabe metrics on general Spin manifolds. Motivated to bubbling phenomena for the Riemannian problem and recent multiplicity results in this setting, we investigate special spinorial Yamabe metrics on product manifolds developing a bubbling analysis which has independent interest in the present setting.
Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -Δ_{g}u+au=0 & \text{ on }M \\ \partial_νu+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $ν$ 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_…
Sign-changing solutions to the Yamabe problem on a spherical cap
Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）；Dipartimento di Matematica e Fisica, Università della Campania “Luigi Vanvitelli”（坎帕尼亚路易吉凡维泰利大学数学与物理系）；Dipartimento SBAI, Sapienza Università di Roma（罗马智慧大学SBAI系）
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.
In this paper, we analyze the asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe type equations on an asymptotically hyperbolic Riemannian manifold. We prove that Palais-Smale sequences can be decomposed into the solution of the limit equation plus a finite number of bubbles, which are the rescaling of the fundamental solutions to the fractional Yamabe equation on Euclidean space. We also verify the non-interfering fact for multi-bubbles.
In this paper, we set up a new Yamabe type flow on a compact Riemannian manifold $(M,g)$ of dimension $n\geq 3$. Let $ψ(x)$ be any smooth function on $M$. Let $p=\frac{n+2}{n-2}$ and $c_n=\frac{4(n-1)}{n-2}$. We study the Yamabe-type flow $u=u(t)$ satisfying {u_t}=u^{1-p}(c_nΔu-ψ(x)u)+r(t)u, \ \ in \ M\times (0,T),\ T>0 with r(t)=\int_M(c_n|\nabla u|^2+ψ(x)u^2)dv/ \int_Mu^{p+1}, which preserves the $L^{p+1}(M)$-norm and we can show that for any initial metric $u_0>0$, the flow exists globally. We also show that in some cases, the global solution converges to a smooth solution to the equation c…
Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -Δ_{g_0}u+λu=λ | u\ | ^{p-1}u\qquad\text{ on }\mathbb{S}^n \] with $λ>0$ and $1<p$, into a singular ODE in $[0,π]$ of the form $w'' + \frac{h(r)}{\sin r} w' + \fracλ{\ell^2}\ (| w|^{p-1}w - w\ )=0$, where $h$ is an strictly decreasing function having exactly one zero in this interval and $\ell$ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a …
T. Riviere proved an energy quantization for Yang-Mills fields defined on n-dimensional Riemannian manifolds, when $n$ is larger than the critical dimension 4. More precisely, he proved that the defect measure of a weakly converging sequence of Yang-Mills fields is quantized, provided the $W^{2,1}$ norm of their curvature is uniformly bounded. In the present paper, we prove a similar quantization phenomenon for the Yamabe problem in a bounded domain $Ω$ of $R^n$.
For a smooth, compact Riemannian manifold (M,g) of dimension $N \geg 3$, we are interested in the critical equation $$Δ_g u+(N-2/4(N-1) S_g+εh)u=u^{N+2/N-2} in M, u>0 in M,$$ where Δ_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), $h\in C^{0,α}(M)$, and $ε$ is a small parameter.
We 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 non degenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of P.D.E.'s on the submanifold with a singular term of attractive or repulsive type is established.
The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
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 $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $φ$ solution of the following Yamabe type equation Δφ+ hφ= \tilde h φ^{\frac{n+2}{n-2}} where $h\in L^p(M)$, $p>n/2$ and $\tilde h\in \mathbb R$. We give the regularity of $φ$ with respect to the value of $p$. Finally, we consider the results in geometry when $g$ is a singular Riemannian metric and $h=\frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar curvature of $g$.
On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(α,β)$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(α,β)$-Ricci-Yamabe flow by estimating its curvatures.
Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold of dimension $N \ge 3$ is known to exist for all time $t$ and converges to a solution to the Yamabe problem as $t \to \infty$. We prove that if a suitable perturbation, which may be smooth and arbitrarily small, is imposed on the Yamabe flow on any given Riemannian manifold $M$ of dimension $N \ge 5$, the resulting flow may blow up at multiple points on $M$ in the infinite time. Our proof is constructive, and indeed we construct such a flow by using solutions of the Yamabe problem on the un…
Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\nabla u)+bu=c|u|^{2^{\ast}-2}u\quad on\ M$$ where $a,b,c\in C^{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ is coercive, and $2^{\ast}=\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_{g}$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $$Δ_{g}u+\frac{m-2}{4(m-1} R_{g}u=κu^{2^{\a…
Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $ξ_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the Yamabe problem $$(P_ε)\qquad-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g) .$$ We prove that for any $k\in \mathbb N$, there exists $ε_k>0$ such that for all $ε\in (0, ε_k)$ the problem $(P_ε)$ has a symmetric solution $u_ε,$ which looks like the superposition of $k$ positive bubbles centered at the point $ξ_0$ as $ε\to 0$. In particular, $ξ_0$ is a {\em to…
Let (V,g) and (W,h) be compact Riemannian manifolds of dimension at least 3. We derive a lower bound for the conformal Yamabe constant of the product manifold (V x W, g+h) in terms of the conformal Yamabe constants of (V,g) and (W,h).
On any closed Riemannian manifold of dimension $n\geq 3$, we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the conformal class. Generically, this distance is controlled quadratically by the Yamabe energy deficit. Finally, we produce an example for which this quadratic estimate is false.
Let $G=(V,E)$ be a locally finite graph, $Ω\subset V$ be a bounded domain, $Δ$ be the usual graph Laplacian, and $λ_1(Ω)$ be the first eigenvalue of $-Δ$ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if $α<λ_1(Ω)$, then for any $p>2$, there exists a positive solution to $-Δu-αu=|u|^{p-2}u$ in $Ω^\circ$, $u=0$ on $\partialΩ$, where $Ω^\circ$ and $\partialΩ$ denote the interior and the boundary of $Ω$ respectively. Also we consider similar problems involving the $p$-Laplacian and poly-Laplacian by the same method. Such p…
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 3$. In this paper, we give various properties of the eigenvalues of the Yamabe operator $L_g$. In particular, we show how the second eigenvalue of $L_g$ is related to the existence of nodal solutions of the equation $L_g u = ε| u|^{N-2} u$, where $ε= +1, 0,$ or -1.
Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show that, when the dimension is greater than or equal to 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.
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 $σ_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-homoth…
In this paper,we prove the following Myers-type theorem: if $(M^n,g)$, $n\geq 3$, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition $Rc\geq εRg>0$, where $ε>0$ is an uniform constant, then $M^n$ must be 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 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 a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates would imply the compactness of the set of solutions of the Yamabe equation on such manifolds.
We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if $n=3$ all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions $4\leq n\leq 7$. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a lo…
Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n \geq 52.
We consider the problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$ dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature $K$ and boundary mean curvature $H$ of arbitrary sign which are non-constant and $\mathfrak D_n=\sqrt{n(n-1)}{|K|}^{-1/2}>1$ at some point of the boundary. It is known that this problem admits a positive mountain pass solution if $n=3$, while no existence results are known for $n\geq 4$. We will consider a perturbation of the geometric problem and show the existence of a posit…
In this paper we produce families of Riemannian metrics with positive constant $σ_k$-curvature equal to $2^{-k} {n \choose k}$ by performing the connected sum of two given compact {\em non degenerate} $n$--dimensional solutions $(M_1,g_1)$ and $(M_2,g_2)$ of the (positive) $σ_k$-Yamabe problem, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.
We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds which are bounded and L^p for p=2n/(n-2) are also L^2. This L^p-L^2-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in a previous article of the authors. As an application we see that the smooth Yamabe invariant of any 2-connected compact 7-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions at least 11.
Let (M,g) be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar's work. Moreover, we reduce the remaining cases to the positive mass theorem.
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…
Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k=1,2,..n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k=1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions to the k-Yamabe problem was recently proved by Gursky and Viaclovsky for k>n/2. In this paper we prove the existence of solutions for the remaining cases k <n/2, k=n/2, assuming t…
In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M,g) is compact. Established in the locally conformally flat case by Schoen [43,44] and for n\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally false for n\geq 25 as shown by Brendle [8] and Brendle-Marques [9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1) Scal_g, Scal_g being the Scalar curvature of (M,g). We show that…
We consider a closed cohomogeneity one Riemannian manifold $(M^n,g) $ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.
In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$, when $g$ has not necessarily the usual regularity (it can be $C^1$). To solve this problem, we start the study of the Yamabe type equations. We show that all the known properties in the smooth case are still valid. Under some assumptions, we prove the existence and uniqueness of solutions. The second part is dedicated to the Hebey-Vaugon conjecture, stated in their paper about th…
Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that the boundary 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.
Let M,g a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M 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.
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 …
