Tian Xu
Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods, we study analytic aspects of the associated energy functional, culminating in a blow-up analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
Spinorial Yamabe-type equations and the Bär-Hijazi-Lott invariant
本文研究了闭Riemannian旋流形上的旋量Yamabe型方程，为归一化解的特征值找到了正下界，并以此给出了正数量曲率旋流形中Bär-Hijazi-Lott不变量的显式下界。
On an Iterative Scheme for the Spinorial Yamabe Equation on Manifolds with Boundary
本文针对带边流形上的旋量 Yamabe 方程，提出迭代格式证明非齐次方程解的存在性，并利用 bootstrapping 方法将正则性提升至内部及边界上的 $C^\infty$。
Fixed-spinor radial solutions of a perturbed Euclidean spinorial Yamabe equation
针对受扰欧几里得旋量Yamabe方程，通过径向旋量空间的有限维约化，结合K在原点的行为与加权积分量，得到固定旋量径向解的存在及多重性准则。
Solutions of the equation of a spinorial Yamabe-type problem on manifolds of bounded geometry
考虑有界几何开流形上的旋量Yamabe型问题。目的是研究相应的Euler-Lagrange方程解的存在性。我们表明，在适当的假设下，这样的解决方案存在。作为应用，我们证明了一个解决方案的存在性意味着共形Hijazi不等式的基础自旋流形。
Let $(M_1,\textit{g}^{(1)})$, $(M_2,\textit{g}^{(2)})$ be closed Riemannian spin manifolds. We study the existence of solutions of the spinorial Yamabe problem on the product $M_1\times M_2$ equipped with a family of metrics $\varepsilon^{-2}\textit{g}^{(1)}\oplus\textit{g}^{(2)}$, $\varepsilon>0$. Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor $M_1$, and which exhibit a spike layer as $\varepsilon\to0$. Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on $(M_1,\…
In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescaled hyperbolic spaces. We give variational characterizations of these threshold constants, and our investigations lead to an explicit positive lower bound for the spinorial threshold constants.
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>10.
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>7 and that the Weyl tensor W(x) is not vanishing on the boundary of M.
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>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.
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…
We investigate the blow-up behavior of sequences of sign-changing solutions for the Yamabe equation on a Riemannian manifold $(M,g)$ of positive Yamabe type. For each dimension $n\ge11$, we describe the value of the minimal energy threshold at which blow-up occurs. In dimensions $11 \le n \le 24$, where the set of positive solutions is known to be compact, we show that the set of sign-changing solutions is not compact and that blow-up already occurs at the lowest possible energy level. We prove this result by constructing a smooth, non-locally conformally flat metric on space forms $\mathbb{S}…
A CR structure with blowing up solutions to the CR Yamabe problem
本研究通过形变三维球面标准CR结构构造出一类CR结构，并用Lyapunov-Schmidt方法证明其上CR Yamabe问题存在爆破解序列，突破了解集紧致性的常规认知。
Blow-up phenomena for the Yamabe equation II
设n为整数，使得\leq n \leq 51。我们在S^n上构造了一个光滑度量g，该度量g的共形类中的常数数量曲率度量集不是紧的.
On the Bär-Hijazi-Lott invariant for the Dirac operator and a spinorial proof of the Yamabe problem
本文证明了非局部共形平坦闭自旋流形上Dirac算子的Bär-Hijazi-Lott不变量严格小于球面情形，作为Aubin估计的旋量类比，解决了该设定下的Yamabe问题。
Blow-up phenomena for the Yamabe equation
设（M，g）是n维紧致黎曼流形\geq 3.紧性猜想断言g的共形类中的常数数量曲率度量集是紧的，除非（M，g）共形等价于圆球。在本文中，我们构造反例，这一猜想在维数n \geq 52。
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 la…
Compactness of solutions to the Yamabe problem. III
对于10维或11维非局部共形平坦紧致黎曼流形上Yamabe方程的爆破解序列，我们建立了爆破点附近渐近轮廓的精确估计，以及爆破点处Weyl张量及其共变导数的精确衰减估计.如果正质量定理在10维和11维中成立，这些估计将意味着Yamabe方程在这样的流形上的解集的紧性。
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.
Solutions of Spinorial Yamabe-type Problems on $S^m$: Perturbations and Applications
本文研究了闭旋量流形上共形不变 Dirac 方程的存在性理论，通过求解该非线性方程，揭示了其在 Spin Geometry 中等距浸入与不变量上界估计的重要应用。
Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds
设M是紧致流形，具有黎曼度量g和自旋结构\si .设$λ(M,[g],\si)= \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$，其中$λ_1^+(\tilde{g})$是Dirac算子D在度量$\tilde{g}$中的最小正特征值.先前的结果表明，$λ(M,[g],\si) \leq λ(\mS^n) =\frac{n}{2} \om_n^{1/n}$，其中\om_n代表标准n球的体积。本文研究了n维共形平坦流形\geq 2中D是可逆的这一问题.例如，我们表明，严格的不等式保持在$n\equiv 0,1,2\mod 4$维，如果一定的自同态不为零。由于它与广义相对论中的ADM质量有着紧密的关系，因此自同态被称为质量自同态。我们将严格不等式应用到自旋共形谱理论中，证明了最小的正Dirac本征值在g的扩大的体积-1-共形类内达到其下确界。
Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric data
本文研究旋量Yamabe型方程解集的非紧性，分别在球面度量受非共形平坦$C^k$扰动及几何势受$C^2$扰动两种模型中给出具体结果。
Conformal Fractional Dirac Operator and Spinorial Q-curvature
本文引入共形分数阶Dirac算子及其分数阶旋量Yamabe问题，通过Caffarelli-Silvestre型延拓建立能量不等式与加权Sobolev不等式，并进一步定义推广经典数量Q-曲率的旋量Q-曲率算子。
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。
Blow-up phenomena for the $σ_2$-Yamabe equation
研究针对\(n\geq27\)时\(\mathbb{S}^n\)上特定光滑度量，解决归一化\(\sigma_2\)-Yamabe方程相关问题。通过构造特定度量，克服线性化算子椭圆性可能丧失的困难，给出有限维约化中的四次轮廓确定端点维数\(n = 27\)。
We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the 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} -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_…
We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2n + 1. The problem of finding solutions with minimum energy has been resolved for all dimensions except dimension 5 (n = 2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach.
A priori estimates for the Yamabe problem in the non-locally conformally flat case
给定一个紧黎曼流形，具有正Yamabe商，不共形同构于标准球面，我们证明了Yamabe问题解的先验估计。我们把自己限制在小于或等于7的维度上，在那里正质量定理是正确的。我们还表明，当维数是大于或等于6，外尔张量消失的一点，解决方案的Yamabe方程爆破。
Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstraß representation, we study the problem of prescribing mean curvature for the immersion $S^2\to\mathbb{R}^3$.
This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.
A spinorial analogue of Aubin's inequality
设$(M,g,\si)$是维数为$\geq 2$的紧致黎曼自旋流形.对于与$g$共形的任何度量$\tilde g$，我们用$\tildeλ$表示$(M,\tilde g,\si)$上狄拉克算子的第一个正本征值。我们证明了$$\inf_{\tilde{g} \in [g]} \tildeλ\Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$这个不等式是Aubin不等式的一个旋量类似物，Aubin不等式是Yamabe问题解中的一个重要不等式。该不等式在$n \geq 3$和$n = 2$，$\ker D=\{0\}$的情况下是已知的。我们的证明也适用于其余情况$n=2$，$\ker D\neq \{0\}$。用同样的方法，我们还证明了黎曼曲面上的任何共形类都包含一个具有$2\tildeλ^2\leq \tildeμ$的度量，其中$\tildeμ$表示Laplace算子的第一正特征值.
Assume that $(X, g^+)$ is an asymptotically hyperbolic manifold, $(M, [\bar{h}])$ is its conformal infinity, $ρ$ is the geodesic boundary defining function associated to $\bar{h}$ and $\bar{g} = ρ^2 g^+$. For any $γ\in (0,1)$, we prove that the solution set of the $γ$-Yamabe problem on $M$ is compact in $C^2(M)$ provided that convergence of the scalar curvature $R[g^+]$ of $(X, g^+)$ to $-n(n+1)$ is sufficiently fast as $ρ$ tends to 0 and the second fundamental form on $M$ never vanishes. Since most of the arguments in blow-up analysis performed here is irrelevant to the geometric assumption i…
We compute the Yamabe invariant for a class of symplectic 4-manifolds of general type obtained by taking the rational blowdown of Kahler surfaces. In particular, for any point on the half-Noether line we exhibit a simply connected minimal symplectic manifold for which we compute the Yamabe invariant.
Conformal Green functions and Yamabe metrics of Sobolev regularity
本文在闭3-流形上对W^{2,q}（q>3）Sobolev类度量完整解决Yamabe问题，通过建立粗糙度量下共形Laplace算子的椭圆理论及其格林函数的存在性、正则性与爆破分析，主要结果在n≥3且q>n/2时成立并具独立意义。
The Yamabe problem for Q-curvature
本文证明了在一般条件下，在维数大于4的光滑闭黎曼流形上，存在一个度量在任意度量的共形类中，使得该度量的$Q$ -曲率为常数.解的存在性是通过变分方法，二阶Sobolev不等式，和$W^{2,2}$ blow-up理论的Hebey和罗伯特的组合。正的解决方案是从一个新的论点证明这里的第一次，是植根于保形协方差属性的Paneitz-Branson运营商和积极的半定性的二阶导数的$C^2$功能在局部极小值。
Blow-up sets of Ricci curvatures of complete conformal metrics
本文研究完备共形度量在边界接近低维极限集时 Ricci 曲率的爆破现象，依据 Yamabe 不变量刻画爆破集，并证明所得结果的最优性。
We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $S^1$ inside $S^m$, $m\geq 5$, that are conformal to the round (incomplete) metric and "periodic" in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of $S^m \setminus S^1\cong S^{m-2}\times H^2$.
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 …
A Spinorial Heat Flow Framework for Geometric Degeneration on $3$-Manifolds
本文提出了一种基于旋量场驱动的 3 维流形几何退化分析框架，揭示了旋量节点行为与诱导度量退化之间的关系，并探讨了退化抛物理论中的开放问题。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。