On a fully nonlinear Yamabe problem

Riemannian流形上完全非线性Yamabe问题的边界估计

Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds

本文研究带边Riemannian流形上完全非线性Yamabe方程的Dirichlet边值问题，在假设下解存在的前提下导出先验边界二阶导数估计并证明光滑解的存在性，且该估计对插值方程族保持一致。

关于具有常边界平均曲率的完全非线性Yamabe问题. I

On the fully nonlinear Yamabe problem with constant boundary mean curvature. I

本文研究了完全非线性Yamabe问题，证明了共形不变二阶椭圆方程在半欧氏空间中的最优Liouville型定理。

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 $σ_k$-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…

Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow

Let 0<m<(n-2)/n, n>2, $α=(2β+ρ)/(1-m)$ and $β>mρ/(n-2-mn)$ for some constant $ρ>0$. Suppose v is a radially symmetric symmetric solution of $\frac{n-1}{m}Δv^m+αv+βx\cdot\nabla v=0$, v>0, in $R^n$. When m=(n-2)/(n+2), the metric $g=v^{4/(n+2)}dx^2$ corresponds to a locally conformally flat Yamabe shrinking gradient soliton with positive sectional curvature. We prove that the solution $v$ of the above nonlinear elliptic equation has the exact decay rate $\lim_{r\to\infty}r^2v(r)^{1-m}=\frac{2(n-1)(n(1-m)-2)}{(1-m)(α(1-m)-2β)}$.

Existence and uniqueness of Green's functions to nonlinear Yamabe problems

For a given finite subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theore…

Connected sum construction for $σ_k$-Yamabe metrics

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.

Some properties of the Yamabe soliton and the related nonlinear elliptic equation

We will prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation $\frac{n-1}{m}Δv^m+αv+βx\cdot\nabla u=0$ in $R^n$ when $n\ge 3$, $0<m\le\frac{n-2}{n}$, $α<0$ and $β\le 0$. Let $n\ge 3$ and $g=v^{\frac{4}{n+2}}dx^2$ be a metric on $\R^n$ where $v$ is a radially symmetric solution of the above elliptic equation in $R^n$ with $m=\frac{n-2}{n+2}$, $α=\frac{2β+ρ}{1-m}$ and $ρ\in R$. For $n\ge 3$, $m=\frac{n-2}{n+2}$, we will prove that $\lim_{r\to\infty}r^2v^{1-m}(r)=\frac{(n-1)(n-2)}ρ$ if $β>\fracρ{n-2}>0$, the scalar curvature $R(r)\toρ$ as $r\to\infty$ …

Thinness for Scalar-Negative Singular Yamabe Metrics

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.

Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.

The Boundary Yamabe Problem, II: General Constant Mean Curvature Case

This article uses the iterative schemes and perturbation methods to completely solve the Han-Li conjecture, i.e. the general boundary Yamabe problem with prescribed constant scalar curvature and constant mean curvature on compact manifolds $ (M, \partial M, g) $ with non-empty smooth boundary, $ \dim M \geqslant 3 $. It is equivalent to show the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n - 2} Δ_{g} u + R_{g} u = λu^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u}{\partial ν} + \frac{n-2}{2} h_{g} u = \frac{n-2}{2} ζu^{\frac{n}{n - 2}} $ on $ \partial M $ with some…

An analytic construction of singular solutions related to a critical Yamabe problem

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,…

The k-Yamabe problem on CR manifolds

We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem was proposed and partially solved by Jerison and Lee for CR manifolds non-locally CR-equivalent to the CR sphere. For k > 1, the problem can be translated in terms of the study of a fully nonlinear equation of type complex k-Hessian. We provide some partial answers related to the …

对偶Yamabe流及相关积分流

A dual Yamabe flow and related integral flows

研究黎曼流上涉及Riesz势的非线性积分流，通过引入对偶Q曲率分析HLS临界区域的集中紧性现象，并在积分核匹配Green函数时将其视为对偶Yamabe流，证明了其在单位球及局部共形平坦流形上的收敛性。

Boundary value problems for some fully nonlinear elliptic equations

We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.

A class of fully nonlinear equations arising in conformal geometry

In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $σ_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for solutions to these equations and establish an existence result associated to them.

带有边界的杨-贝尔问题的变号解

Sign-changing solutions to the Yamabe problem on manifolds with boundary

研究了带有边界的人 manifold 上杨-贝尔问题变号解的存在性，通过变分方法和合适的共形不变量分析，证明了在特定条件下存在最小能量变号解。

The mixed Yamabe problem for harmonic foliations

The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a harmonic foliation, we derive the leafwise elliptic equation and explore the corresponding nonlinear heat type equation. We assume that the leaves are compact submanifolds and the manifold is fibered instead of being foliated, and use spectral parameters of certain Schrödinger ope…

The Boundary Yamabe Problem, I: Minimal Boundary Case

We apply iteration schemes and perturbation methods to provide a complete solution of the boundary Yamabe problem with minimal boundary scenario, or equivalently, the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n - 2} Δ_{g} u + S_{g} u = λu^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u}{\partial ν} + \frac{n-2}{2} h_{g} u = 0 $ on $ \partial M $. Thus $ g $ is conformal to to the metric $ \tilde{g} = u^{\frac{4}{n -2}} g $ of constant scalar curvature $ λ$ with minimal boundary. In contrast to the classical method of calculus of variations with assumptions on Weyl …

紧致与非紧致 Yamabe 问题的一种局部方法

A Local Method for Compact and Non-compact Yamabe Problems

本文提出一种新的局部方法，用于解决维数至少为4的紧致流形及逐点共形于紧致流形子集的非紧致完备流形上的 Yamabe 问题，尤其适用于 Yamabe 常数为正的困难情形，并将 Brezis-Nirenberg 非线性特征值问题推广到流形子集。

Variational status of a class of fully nonlinear curvature prescription problems

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.

近厄米几何中的指定Gauduchon（高东雄）标量曲率问题

The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

该研究针对近厄米流形的指定Gauduchon标量曲率问题，通过共形变化推导将其转化为带指数非线性的半线性偏微分方程，用上下解方法证明解的存在性与Gauduchon度相关常数的符号有关，负号时给出充要条件，还复现了三类相关Yamabe问题。

The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let $g_0$ be a smooth Riemannian metric on a closed manifold $M^n$ of dimension $n\geq 3$. We study the existence of a smooth metric $g$ conformal to $g_0$ whose Schouten tensor $A_g$ satisfies the differential inclusion $λ(g^{-1}A_g)\inΓ$ on $M^n$, where $Γ\subset\mathbb{R}^n$ is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric $g_1$ conformal to $g_0$ satisfying $λ(g_1^{-1}A_{g_1})\in\bar{Γ'}$ in the viscosity sense on $M^n$, to…

Local estimates for elliptic equations arising in conformal geometry

In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations associated with the problems.

The Gursky-Streets equations

Gursky-Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the $σ_2$-Yamabe problem. The geodesic equation of Gursky-Streets' metric is a fully nonlinear degenerate elliptic equation and Gursky-Streets have proved uniform $C^{0, 1}$ regularity for a perturbed equation. Gursky-Streets apply the results and parabolic smoothing of Guan-Wang flow to show that the solution of $σ_2$-Yamabe problem is unique. A key ingredient is the convexity of Chang-Yang's $\cF$-functional along the (smoot…

A fully nonlinear conformal flow on locally conformally flat manifolds

We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when $k \neq n/2$.

完全非线性方程的结构及其在完备共形度量预设问题中的应用

The structure of fully nonlinear equations and its applications to prescribed problems on complete conformal metrics

该研究探讨完全非线性方程的结构，引入Morse理论在弱条件下构造容许度量，求解两类几何问题的完全非线性版本，并通过拓扑障碍验证结构条件的最优性。

具有变号非线性的Yamabe型方程与预定曲率问题

Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

本文研究非紧黎曼流形上变号函数的预定标量曲率问题，通过标准与共形拉普拉斯算子的谱性质，在极低假设下证明了两个保证拟等距的新存在性结果，并推广至拟线性方程与Schrödinger型算子的次临界理论。

Moduli Spaces of Singular Yamabe Metrics

Complete, conformally flat metrics of constant positive scalar curvature on the complement of $k$ points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were constructed by R\. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally rea…

$S^m$ 上旋量 Yamabe 型问题的解：扰动与应用

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。

具有非光滑几何数据的旋量Yamabe型问题的非紧性结果

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。

A general Liouville type theorem for some conformally invariant fully nonlinear equations

We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.

Nonclassical Solutions of Fully Nonlinear Elliptic Equations

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

Further results on Liouville type theorems for some conformally invariant fully nonlinear equations

We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.

Conformal geometry and fully nonlinear equations

This article is a survey of results involving conformal deformation of Riemannian metrics and fully nonlinear equations.

The Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds

We derive the existence of $C^{1,1}$-solutions to the Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds under appropriate assumptions.

Modulus of Continuity Estimates for Fully Nonlinear Parabolic Equations

We prove that the moduli of continuity of viscosity solutions to fully nonlinear parabolic partial differential equations are viscosity subsolutions of suitable parabolic equations of one space variable. As applications, we obtain sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data, via comparison with one-dimensional solutions. This work extends multiple results of Andrews and Clutterbuck for quasilinear equations to fully nonlinear equations.

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

Interior regularity for strong solutions to a class of fully nonlinear elliptic equations

We obtain local pointwise second derivative estimates for $W^{2,p}$-strong solutions to a class of fully nonlinear elliptic equations on Euclidean domains, motivated by problems in conformal geometry.

Homogeneous Solutions of Fully Nonlinear Elliptic Equations in Four Dimensions

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.

Conformally invariant fully nonlinear elliptic equations and isolated singularities

We study solutions to conformally invariant equations with isolated singularties.

Degenerate conformally invariant fully nonlinear elliptic equations

We establish Liouville type theorems for degenerate conformally invariant equations.

Some fully nonlinear problems on manifolds with boundary of negative admissible curvature

In this paper, we consider a fully nonlinear problem on manifolds with boundaries of negative admissible curvatures. As a consequence, we conclude the existence of certain types of metrics on the general differential manifolds with boundaries.

On fully nonlinear CR invariant equations on the Heisenberg group

In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully nonlinear CR invariant equations defined on bounded domains of the Heisenberg group and a comparison principle for solutions of a family of second order fully nonlinear equations on a punctured ball.

On estimates for fully nonlinear elliptic equations with Neumann boundary conditions on Riemannian Manifolds

We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.

On a class of fully nonlinear elliptic equations on closed Hermitian manifolds

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive $C^\infty$ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.

Local Estimates for Some Fully Nonlinear Elliptic Equations

We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be applied to a large class of equations.

On estimates for fully nonlinear parabolic equations on Riemannian manifolds

We present some new ideas to derive {\em a priori} second order estiamtes for a wide class of fully nonlinear parabolic equations. Our methods, which produce new existence results for the initial-boundary value problems in $\bfR^n$, are powerful enough to work in general Riemannian manifolds.

Estimates and Existence Results for some Fully Nonlinear Elliptic Equations on Riemannian Manifolds

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.

The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds

We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and optimal) result in the Euclidean case. We introduce some new ideas and methods in deriving a priori estimates, which can be used to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds.

Hermitian流形上若干指定曲率问题的完全非线性椭圆方程

Fully nonlinear elliptic equations for some prescribed curvature problems on Hermitian manifolds

该研究利用爆破论证与部分一致椭圆性研究Hermitian流形上的完全非线性椭圆方程，将其应用于寻找指定Chern-Ricci曲率的共形Hermitian度量的几何问题，且假设因几何函数理论阻碍几乎达最优。

关于在共形几何中全非线性方程奇异解的粘合构造

A gluing construction of singular solutions for a fully non-linear equation in conformal geometry

本文通过研究$σ_2$-Yamabe方程的奇异解构造，展示了经典粘合方法在全非线性设定中的应用，利用共形性质保证线性化算子在加权空间中的良好映射性质。

On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: $L^\infty$ estimate

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.

A Monge-Ampere Type Fully Nonlinear Equation on Hermitian Manifolds

We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.