Yilong Ni, Meijun Zhu
This is the second paper of our series of papers on one dimensional conformal metric flows. In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in math.AP/0611254. We prove the global existence and convergence of the one dimensional Yamabe and affine flows. Furthermore, we obtain exponential convergence of the metrics under these flows.
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In this paper we prove that given a smoothly conformally compact metric there is a short-time solution to the Ricci flow that remains smoothly conformally compact. We adapt recent results of Schnürer, Schulze and Simon to prove a stability result for conformally compact Einstein metrics sufficiently close to the hyperbolic metric.
We define two conformal structures on $S^1$ which give rise to a different view of the affine curvature flow and a new curvature flow, the ``$Q$-curvature flow". The steady state of these flows are studied. More specifically, we prove four sharp inequalities, which state the existences of the corresponding extremal metrics.
We study the boundary behaviors of a complete conformal metric which solves the $σ_k$-Ricci problem on the interior of a manifold with boundary. We establish asymptotic expansions and also $C^1$ and $C^2$ estimates for this metric multiplied by the square of the distance in a small neighborhood of the boundary.
In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the $n$-sphere, with $n\geq 5$. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existene of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.
Parabolic frequency monotonicity on the conformal Ricci flow
本文研究了在维数不小于3且标量曲率恒定的闭黎曼流形上，共形Ricci流下抛物频率泛函的单调性，通过应用Bakry-Emery Ricci曲率界证明了该性质并给出了相关推论。
Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author.
We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. \begin{equation*}\begin{cases}-Δu=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_νu+2=2h(z)e^\frac u2&\hbox{on}\;\partial\mathbb{D}^2,\end{cases} \end{equation*} where $K,h$ are the prescribed curvatures. We construct a family of conformal metrics with curvatures $K_\varepsilon,h_\varepsilon$ converging to $K,h$ respectively as $\varepsilon$ goes to $0$, which blows up at one boundary point under some generic assumptions.
This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the $C^2$-norm and the total Leray-Schauder degree of all solutions is equal to -1. Then we deduce from this compactness result the existence of at least one solution to our problem.
This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.
In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive scalar curvature is compact in the $C^\infty$ topology.
Regularized $n$-Conformal heat flow and global smoothness
本文提出结合正则化 $n$-调和映射流与共形度量演化的正则化 $n$-共形热流，并证明其不会形成有限时间奇点，从而获得全局光滑性。
In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in \cite{GL} \cite{Lee} \cite{Bi} of conformally compact Einstein metrics with conform…
In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential convergence of metrics for the 1-Q and 4-Q flows are obtained.
On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.
We prove that in many cases the existence of an extremal metric for some Laplace eigenvalue in a conformal class allows to find extremal metrics in conformal classes close by. As a consequence and as part of the arguments we obtain perturbed harmonic maps with constant density.
We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the "link" of the singular set. Within this class of "conic metrics," we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar …
Uniqueness of conformal metrics with constant Q-curvature on closed Einstein manifolds
本文在具有正数量曲率且不与标准球面共形微分同胚的闭黎曼流形上，证明了具有4阶常Q-曲率的共形度量只能是常数倍的原度量，从而建立了该类度量的唯一性。
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+\infty$.
Conformal metrics with finite total Q-curvature revisited
本文研究有限总Q-曲率共形度量，引入共形质量并在不假设完备性下给出正规性充要条件，进而得到体积比较定理和正质量型定理。
We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $σ\in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.
In this short note, we prove that conformal classes which are small perturbations of a product conformal class on a product with a standard sphere admit a metric extremal for some Laplace eigenvalue. As part of the arguments we obtain perturbed harmonic maps with constant density.
Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various geometric conditions on $X$ and $M$. We also obtain the existence results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$, $γ= 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to be smooth on $M$.
In this paper we pursue the work initiated in \cite{Bahuaud, BahuaudGicquaud}: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to -1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tenso…
A new conformal heat flow of harmonic maps
本文提出调和映射的新共形热流，通过映射与共形因子的耦合演化推迟奇点形成，并证明除有限多个点外光滑的Struwe型全局弱解存在。
Let (M,g) be a compact Riemannian manifold with boundary. This paper addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some remaining cases of this problem.
We study conformal metrics on R^{2m} with constant Q-curvature and finite volume. When m=3 we show that there exists V* such that for any V\in [V*,\infty) there is a conformal metric g on R^{6} with Q_g = Q-curvature of S^6, and vol(g)=V. This is in sharp contrast with the four-dimensional case, treated by C-S. Lin. We also prove that when $m$ is odd and greater than 1, there is a constant V_m>\vol (S^{2m}) such that for every V\in (0,V_m] there is a conformal metric g on R^{2m} with Q_g = Q-curvature of S^{2m}, vol(g)=V. This extends a result of A. Chang and W-X. Chen. When m is even we prove…
In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are of Yamabe metrics.
A priori estimates for negative constant scalar curvature conformal metrics with positive constant boundary mean curvature
针对带边界紧致黎曼流形上内部负常数数量曲率、边界正常数平均曲率的共形度量集合，在Yamabe共形不变量为正的条件下，证明其在三维及任意≥3维局部共形平坦且边界脐点的情形下先验有界。
We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.
We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane domain there exist universal conformal metrics with prescribed constant curvature.
We show that existing decay results for scalar fields on the Schwarzschild metric are sufficient to obtain a conformal scattering theory. Then we re-interpret this as an analytic scattering theory defined in terms of wave operators, with an explicit comparison dynamics associated with the principal null geodesic congruences. The case of the Kerr metric is also discussed.
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