We study the polyharmonic problem $Δ^m u = \pm e^u$ in ${\mathbb R}^{2m}$, with $m \geq 2$. In particular, we prove that {\sl for any} $V > 0$, there exist radial solutions of $Δ^m u = -e^u$ such that $$\int_{{\mathbb R}^{2m}} e^u dx = V.$$ It implies that for $m$ odd, given arbitrary volume $V > 0$, there exist conformal metrics $g$ on ${\mathbb R}^{2m}$ with positive constant $Q$-curvature and vol$(g) =V$. This answers some open questions in Martinazzi's work.
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.
Using the Lyapunov-Schmidt reduction method, we describe how to use annular domains to construct (scalar curvature) functions on S^n, (n > 5), so that each one of them enables the conformal scalar curvature equation to have a blowing-up sequence of positive solutions. The prescribed scalar curvature function is shown to have C^{n - 1, β} smoothness.
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.
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.
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 case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular with negative scalar curvature and boundary mean curvature of arbitrary sign, which to our knowledge has not been treated in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that bubbling may occur…
Conformal Metrics on the unit Ball with Constant $Q$-Curvature, Constant $T$-Curvature, and Minimal Boundary
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
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.
By using the Lyapunov-Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on S^n (n greater or equal to 3) when the prescribed function (after being projected to R^n) has two close critical points, which have the same value (positive), equal "flatness" (twin, flatness < n - 2), and exhibit maximal behavior in certain directions (pseudo-peaks). The proof relies on a balance between the two main contributions to the reduced functional - one from the critical points and the other from the interaction of the two bubbles.
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.
We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Δ)^m u = K(x) u^{\frac{n+2m}{n-2m}} ~~~~~~ in ~ Ω\backslash Λ$$ near their singular set $Λ$, where $Ω\subset \mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $Ω$, $Λ$ is a closed subset of $\mathbb{R}^n$, $2 \leq m < n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $Λ$, we prove that $u(x) \leq C [{dist}(x, Λ)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Λ$ is less than $(n-2m)/2$.
We give sufficient conditions on a function invariant under the action of an isometry group to be Branson's Q-curvature of a metric in a given conformal class, using the conformal GJMS operators.
We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a conformal deformation of the metric. We derive some existence results using a variational approach, either by minimization of the Euler-Lagrange energy or via min-max methods. One of the main tools in our approach is a blow-up analysis of solutions, which in the present setting can have diverging volume. To our knowledge, this is the first time in which such an aspect is treated. Key ingredients in our arguments are: a blow-up analysis around a sequence of points different fr…
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 $+\infty$.
The article contains a brief description on the study of conformal scalar curvature equations, and discusses selected topics and questions concerning the equations in open spaces.
A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure o…
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 $M$.
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.
We review recent compactness and non-compactness results for the Yamabe equation. We also discuss the asymptotic behavior of the parabolic Yamabe flow.
We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal invariant $Y(M,\partial M)<0$. Given any negative smooth functions $f$ in $M$ and $h$ on $\partial M$, there exists a unique conformal metric of $g_0$ such that its scalar curvature equals $f$ and mean curvature curvature equals $h$. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In…
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…
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 …
In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary there exists a metric conformal to g with constant T-curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth order nonlinear elliptic boundary value problem (BVP) with boundary conditions given by a third-order pseudodifferential operator, and homogeneous Neumann one. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from below, we l…
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 study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension $4$, and an existence theorem which holds in dimensions $n \geq 4$. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the $\rm{U}(2)$-invariant Leray-Schauder degree for a family of negative-mass orbifolds found by LeBrun.
We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with constant $Q$-curvature, but without constant scalar curvature, are found to bifurcate from Berger metrics on spheres and complex projective spaces. These provide examples of nonisometric metrics with the same constant negative $Q$-curvature in a conformal class with negative Yamabe invariant, echoing the absence of a Maximum Principle. We also discover infini…
We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results.
We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.
In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Δu=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.
This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results …
On a manifold $(\mathbb{R}^n, e^{2u} |dx|^2)$, we say $u$ is normal if the $Q$-curvature equation that $u$ satisfies $(-Δ)^{\frac{n}{2}} u = Q_g e^{nu}$ can be written as the integral form $u(x)=\frac{1}{c_n}\int_{\mathbb R^n}\log\frac{|y|}{|x-y|}Q_g(y)e^{nu(y)}dy+C$. In this paper, we show that the integrability assumption on the negative part of the scalar curvature implies the metric is normal. As an application, we prove a bi-Lipschitz equivalence theorem for conformally flat metrics.
A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan-Chern-Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in the complex vector s…
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy …
Let $h$ be a complete metric of Gaussian curvature $K_0$ on a punctured Riemann surface of genus $g \geq 1$ (or the sphere with at least three punctures). Given a smooth negative function $K$ with $K=K_0$ in neighbourhoods of the punctures we prove that there exists a metric conformal to $h$ which attains this function as its Gaussian curvature for the punctured Riemann surface. We do so by minimizing an appropriate functional using elementary analysis.
We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.
We define a generalization of convex functions, which we call $δ$-convex functions, and show they must satisfy interior Hölder and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear equations in conformal geometry with isolated singularities, in the case of non-negative Ricci curvature. We prove that such solutions either extend to a Hölder continuous function across the singularity, or else have the same singular behavior as the fundamental solution of the conformal Laplacian. We also obtain various removable singularity theorems for these equa…
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.
In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test function construction, we are able to seattle the most cases left by Escobar's and Marques's works. Moreover, we reduce the remaining case to the positive mass theorem. In this proof, we use the method developed in previous works by Brendle and by Brendle and the author.
We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with the (conformal Laplacian power) GJMS operators, a suitable selection of the boundary operators yields formally self-adjoint elliptic conformal boundary problems. Working on a conformal manifol…
In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities between the Green's functions of the conformal Laplacian operator and the Paneitz operator.
In this article we study the nonlocal equation \begin{align} (-Δ)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb{R}^n$}, \quad\int_{\mathbb{R}^n}e^{nu}dx<\infty, \notag \end{align} which arises in the conformal geometry. Inspired by the previous work of C. S. Lin and L. Martinazzi in even dimension and T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong in dimension three we classify all solutions to the above equation in terms of their behavior at infinity.
We give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric $ g $ for both closed manifolds and compact manifolds with boundary, including the interesting cases $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, in dimensions $ n \geqslant 3 $, provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of $ \mathbb{S}^{n} $, we show that, on one hand, any smooth function that is a positive consta…
We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fixed conformally congruent area. On the way to establish this result we prove that tori which are critical points of the area for perturbations within a given conformal class and which are degenerate points of the conformal class mapping - i.e. isothermic - are either minimal surf…
In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, $Q$-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaffian curvature. For a class of even-dimensional complete LCF manifolds with integrable $Q$% -curvature, we establish a Gauss-Bonnet-Chern inequality. Second, a finiteness theorem for certain classe…
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least $ 3 $. The results are classified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a "Trichotomy Theorem" in terms of both scalar and mean curvature functions, which is a full extension of the "Trichotomy Theorem" given by Kazdan and Warner. We also discuss prescribing Gauss and geodesic curvature problems on compact R…
In this paper, we bend a closed Riemannian manifold in the conformal class, through solving a fully nonlinear equation. As a result, we prove that each metric of quasi-negative Ricci curvature is conformal to a metric with negative Ricci curvature.
In this paper, we consider the problem of existence and multiplicity of conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$ with positive scalar curvature. We prove new exitence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri-Coron and Chang-Gursky-Yang Euler-Hopf type criterium. Our argument gives estimates on the Morse index of the solutions and has the advantage to extend known existence results. Moreover it provides, for generic $K$ {\it Morse Inequalities at Infinity}, which give a lower bound on the numb…
