Ali Hyder, Angela Pistoia, Yannick Sire
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.
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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 the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time $T$ or as $t \to +\infty$.
Compactness in Dimension Five and Equivariant Noncompactness for the CR Yamabe Problem
本文研究了五维CR Yamabe方程的紧性和非紧性现象，通过建立先验估计和构造非标准CR结构，证明了在等价设定下的非紧性。
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…
Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant
本文证明了一类含不完全锥边奇点的流形上，具有正 Yamabe 常数的归一化 Yamabe 流的长时存在性，通过公理化方法和抛物型 Moser 迭代获得解与数量曲率的统界。
Convergence of the Yamabe flow on singular spaces with positive Yamabe constant
本文研究具有正Yamabe常数的分层奇异空间上规范化Yamabe流的收敛性，在低能量条件下证明收敛，并建立集中-紧性二分法及分析Viaclovsky不收敛例子。
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…
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…
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 …
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 $g$.
Multiplicity of singular solutions to the fractional Yamabe problem on spheres
本文利用分岔技术证明球面上分数阶 Yamabe 问题存在多个奇异正解，揭示了相应完备共形度量的非唯一性。
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 …
Isolated singularities of solutions to the Yamabe equation in higher dimensions I
本文在黎曼流形上完成了 Yamabe 方程正解孤立奇点的分类（3≤n≤29），证明奇点要么可去，要么渐近于 Fowler 解，并通过 n≥30 的反例确立范围最优性。
Yamabe problem in the presence of singular Riemannian Foliations
本文利用变分方法和奇异黎曼叶状结构的对称性，证明了 Yamabe 型问题存在无穷多个变号解及一个最小能量正解，并给出了新的 Sobolev 嵌入定理和对称临界性原理。
Thinness for Scalar-Negative Singular Yamabe Metrics
本文讨论球面上区域上的标准度量到具有常数量曲率的完备度量的共形变形。允许这种变形的域的描述问题起源于Loewner和Nirenberg以及Schoen和Yau关于局部共形平坦流形的工作。这项工作的目标是应用思想的非线性位势论的问题。他们允许，特别是，解决问题的情况下，恒定的负数量曲率。
Min-max theory and Yamabe metrics on conical four-manifolds
该研究首次在奇异情形下采用极小极大方案，结合锥形正质量定理分析共形爆破质量发散规律，证明了带有限$\u2124_2$群锥形点的四维流形上Yamabe度量的存在性。
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,…
Classification of singular radial solutions to the $σ_k$ Yamabe equation on annular domains
关于黎曼度量的Weyl-Schouten曲率张量的$k$阶初等对称函数，即所谓的$σ_k$曲率的研究，近年来在共形几何中取得了许多丰硕的成果。在这些研究中，变形保形因子被认为是一个完全非线性椭圆型偏微分方程的解决方案。近年来，在对偏微分方程解的解析行为的理解方面取得了重要进展。然而，这些解决方案的奇异性，这是很重要的，在描述共形几何中的许多重要问题，是很少了解。本文对所有可能的径向解进行了分类，特别是$σ_k$ Yamabe方程的\emph{singular}解，该方程描述了$σ_k$曲率等于常数的共形度量。虽然所涉及的分析是初等性质的，但这些结果对研究一般情况下的奇异解的行为具有指导意义。
Properties of hypersurface singular sets of solutions to the $σ_k$-Yamabe equation in the negative cone
研究 $σ_k$-Yamabe 方程在负锥中解的超曲面奇异性性质，证明解沿超曲面满足特定偏微分方程，并探讨 $k=2$ 时超曲面的极小性及解的展开问题。
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.
A Parametrization of Solutions to the Yamabe Equation with Isolated Singularity
本文受 Caffarelli 等人工作启发，证明了与调和函数对应的形式无穷级数可产生带不可去孤立奇点的 Yamabe 方程实际解，并提供了构造该类方程在孤立奇点及无穷远处所有解的通用方法。
Yamabe metrics on conical manifolds
本文在具有锥点和Einstein型锥链的奇异流形上证明了一般类型Yamabe度量的存在性，通过向标准气泡添加隐式低阶修正项并利用对称二张量场分解与Fourier分析推导Yamabe商的渐近展开，得到了Aubin经典结果的奇异对应版本。
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.
Solutions of the singular Yamabe problem near singular boundaries
本文研究了奇异 Yamabe 问题在奇异边界附近的解行为，证明了局部正解在切锥中被正解近似，并扩展了相关研究成果。
Yamabe metrics on cylindrical manifolds
我们研究一类特殊的开流形。在黎曼流形的范畴中，这些流形是具有圆柱端点的完备流形。我们给这样的流形上的共形几何，包括一个适当的概念的圆柱Yamabe常数/不变的自然设置。这导致了相应版本的Yamabe问题的圆柱形流形。我们肯定地解决这个Yamabe问题：我们证明了最小化度量的存在性，并分析了它们在无穷大附近的奇异性。这些奇点是非常特殊的类型：几乎圆锥奇点或几乎尖点奇点。我们描述的上确界的情况下，即当圆柱Yamabe常数等于Yamabe不变的球。我们证明，在这种情况下，这样一个圆柱形的流形符合共形的标准领域穿刺在有限数量的点。在研究上确界情形的过程中，我们建立了具有两个几乎锥奇点的渐近平坦流形的正质量定理。作为一个副产品，我们重新审视手术和Yamabe不变量的已知结果。关键词：带圆柱端点的流形，Yamabe常数/不变量，Yamabe问题，圆锥度量奇点，尖点度量奇点，正质量定理，外科手术和Yamabe不变量。
Qualitative Properties of Singular Solutions to the Fractional Yamabe Problem
本文研究分数阶Yamabe问题中具有孤立奇点的解，通过Emden Fowler型变换将其化为非局部一维问题，并证明这类解的Morse指标为无穷大。
We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there is phrased in terms of a strict inequality of the global Yamabe invariant with a `local Yamabe invariant', which captures information about the local singular structure. All of this is generalized here to the setting of Dirichlet spaces which admit a Sobolev inequality and satisf…
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…
Classification of fractional, singular Yamabe metrics on a twice punctured sphere I
本文研究了双点 punctured 球面上分数 Q-曲率度量的分类，证明当 s 接近但小于 1 时，所有完整、conformally flat 常 Q-曲率度量均为 Delaunay 度量，并给出了 conformal 因子的 sharp 估计。
Isolated Singularities of Solutions to the Yamabe Equation with boundary in Dimension 3 and 4
研究三维和四维带边界的 Yamabe 方程正解在孤立奇点附近的渐近行为，建立了精确上界与不可去情形下的匹配下界及可去性条件，证明不可去奇点解渐近柱对称，扩展相关理论。
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.
Asymptotic behavior of complete conformal metric near singular boundary
本文研究负常数量曲率奇异 Yamabe 问题在奇异边界附近的解渐近行为，对非共形平坦背景度量建立最优估计，并证明解可由奇异点处切锥中的解逼近。
Asymptotic behavior of solutions to the Yamabe equation with an asymptotically flat metric
该研究证明了满足平坦阶数条件的渐近平坦流形上Yamabe方程正解的无穷远渐近二择一性，通过孤立奇点附近渐近分析推广了多类前人结果。
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.
Asymptotic approach to singular solutions for the CR Yamabe equation
研究海森堡群中福兰德 - 斯坦 - 索伯列夫嵌入次临界逼近最优函数紧致性损失，防边界爆破时中心对称最大化器集中于中心，获远离极点爆破率与渐近轮廓，建立近临界索伯列夫问题对应渐近。
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
本文为分数阶 Lane-Emden-Serrin 方程找到决定奇异渐近的一维局部 ODE，并构造沿 (n−2s)/2 维子流形奇异的完备 s-分数阶 Yamabe 度量。
The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces
该论文研究球面上的共形对数拉普拉斯算子，建立其与ℝ^N中对数拉普拉斯算子的联系，引入对应希尔伯特空间，衔接Yamabe型问题分类结果并推广至弱情形。
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.
Super-Liouville equation with a spinorial Yamabe type term
本文研究带有旋量Yamabe型项的超Liouville方程，为爆破序列建立精细定性性质，证明旋量与函数部分的能量恒等式并计算局部质量，揭示了由非线性项导致的两种奇性新现象。
Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation
本文研究Einstein流形上预定Q-曲率方程的最优分割问题，利用余一维群作用精确描述解域，证明对称解存在性，并给出Paneitz-Branson算子及Yamabe问题变号解的多重性结果。
We study local behavior of positive solutions to the fractional Yamabe equation with a singular set of fractional capacity zero.
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 …
In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2+1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from $\mathbb{R}^2$ into $S^2$.
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 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.
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