Zindine Djadli, Andrea Malchiodi, Mohameden Ould Ahmedou
We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.
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This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.
We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.
In this work we prove the existence and uniqueness of Killing graphs with prescribed mean curvature considering functions which are not necessarily constant along the flow lines of the Killing vector field.
Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three
本文通过积分方法，证明了三维空间中预定数量曲率方程超曲面的内部曲率估计，关键发现是满足该方程的图函数对应的拉格朗日子流形具有有界平均曲率。
This paper focuses on the problem of prescribing mean curvature on the unit ball. Assume that $f$, which is allowed to change sign, satisfies Morse index counting or certain kind of symmetry condition. By using a negative gradient flow method, we then prove that $f$ can be realized as the boundary mean curvature of some conformal metric.
Prescribing curvatures on surfaces with conical singularities and corners
本文研究了带锥奇点和角点的带边界紧致曲面上的高斯曲率与测地曲率预定问题，通过将新变分公式推广至奇异情形求得解，是该奇异曲面预定曲率问题的首个结果。
In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree.
Prescribing negative curvature with cusps and conical singularities on compact surface
本研究针对紧曲面，证明了一类仅在有限个带指定角度的尖点或锥形奇点处例外、曲率由负函数指定的共形度量的存在性与唯一性。
We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $σ_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate for these equations obtained under a weakened condition.
In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a priori estimates combined with the perturbation result of Part I allow us to obtain some existence results. On the six dimensional sphere we prove the existence of at least one solution when an associated index is different from zero.
Prescribing almost constant curvatures on manifolds with boundary
本文研究标准球面上预定数量曲率与边界平均曲率的边界情形，利用微扰方法和 Han-Li ansatz 证明了当预定曲率接近常数时共形度量的存在性。
Prescribing positive curvature with conical singularities on $\mathbb S^2$
本文研究了$\mathbb S^2$上具有锥奇点的共形度量的正曲率规定问题，通过证明收敛定理并分析 bubble trees 和面积恒等式，给出了 prescribing data 在开区域内不存在的判据。
Prescribing curvatures in the disk via conformal changes of the metric: the case of negative Gaussian curvature
本文通过共形变形在对称圆盘上规定负高斯曲率及边界测地曲率，采用变分方法结合单调性技巧、爆破分析和Morse指标证明存在性，并给出不存在性结果。
Prescribing $σ_k$ curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function $K$ to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the $σ_2$ curvature equation with the given $K$; and rule out the possibility of blowing up solutions when $K$ satisfies a non-degeneracy condition. We also prove uniform a priori estimates for solutions to a family of $σ_2$ curvature equations deforming $K$ to a positive constant under the same non-degene…
We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with at least two boundary components. This seems to be the first result in this setting. Moreover, we allow to have conical singularities with both positive and negative orders, that is cone angles both less and grater than $2π$.
Prescribing nearly constant curvatures on balls
针对 Kazdan-Warner 问题的边界情形，研究通过共形度量变换在圆盘及高维球上预设近常的内、外曲率，并利用 Ljapunov-Schmidt 方法证明负内曲率、正边界曲率条件下的新存在性结果。
In this paper we study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove the existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality…
Prescribing Chern scalar curvatures on noncompact manifolds
针对非紧非 Kähler 流形上的预定 Chern 标量曲率问题，论文借助新的解析条件建立 Kazdan-Warner 型方程的存在性结果，并据此给出 Ni 经典多重性定理的新证明。
Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset…
A direct approach to prescribing scalar curvature on bundles
本文直接利用Kazdan--Warner结果和变分方法，研究纤维丛全空间上标量曲率函数的可实现性，并应用于Calabi--Yau流形上的丛。
The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(Σ,β)$ be a closed Riemann surface with a divisor $β$, and $K_λ=K+λ$, where $K:Σ\rightarrow\mathbb{R}$ is a Hölder continuous function satisfying $\max_ΣK= 0$, $K\not\equiv 0$, and $λ\in\mathbb{R}$. If the Euler characteristic $χ(Σ,β)$ is negative, then by a variational method, it is proved that there exists a constant $λ^\ast>0$ such that for any $λ\leq 0$, there is a unique conformal metric with the Gaussian curvature $K_λ$; for any $λ$, $0<λ<λ^\ast$, there are at least two conformal…
Prescribing $Q$-curvature on even-dimensional manifolds with conical singularities
该研究针对带锥形奇点的$2m$维闭流形的指定$Q$-曲率问题，通过爆破分析与极小极大变分论证，得到超临界情形下的存在性与多重性结果，是$m>1$时非球面超临界锥形流形的首个相关存在性结论。
Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on $M$ (resp. metric on $M$ with geodesic boundary). In order to provide analogous results for this problem with $n\geq 3,$ we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on $\partial M$ (resp. on $M$) is a mean curvature of a …
For $p\in (1,2]$ and a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ let $μ_p(\barΩ,\cdot)$ be the $p$-capacitary curvature measure (generated by the closure $\barΩ$ of $Ω$) on the unit circle $\mathbb S^1$. This paper shows that such a problem of prescribing $μ_p$ on a planar convex domain: "Given a finite, nonnegative, Borel measure $μ$ on $\mathbb S^1$, find a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ such that $dμ_p(\barΩ,\cdot)=dμ(\cdot)$" is solvable if and only if $μ$ has centroid at the origin and its support $\mathrm{supp}(μ)$ does not comprise any pair of …
In this note we show that any real exact G-invariant (1,1)-form is the Ricci form of a Kaehler metric on the complexification of an irreducible compact symmetric space G/K.
We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $\mathbb{S}^{d_1+1}\times \mathbb{S}^{d_2}$, where $d_i \geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show that $T$ can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.
Prescribing scalar curvatures: non compactness versus critical points at infinity
研究通过例子展示在黎曼流形上，标量曲率的规范问题中，非紧性与无穷远处临界点的转变现象。
Almost prescribing scalar curvature by mixed convex integration
本文提出混合凸积分方法，用于处理一类半线性二阶偏微分关系，并借此重新证明 Lohkamp 关于标量曲率的结果。
The mixed scalar curvature is one of the simplest curvature invariants of a foliated Riemannian manifold. We explore the problem of prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibred manifold, we reduce the problem to solution of a leafwise elliptic equation, which has three stable solutions -- only one of them corresponds to the case of a foliated Riemannian manifold.
The study of radial prescribing scalar curvature of X.Xu and P.C.Yang [2] in 1993 showed a nonexistence result on $S^2$. Later in 1995, W.Chen and C.Li [2] generalized the nonexistence result to higher dimensions. G.Bianchi and E.Egnell [1] suggested that there may exist some non-negative smooth radial function which cannot be scalar curvature in the standard conformal class of $RP^n.$ However, in our paper, we prove that all smooth radial non-negative smooth functions which are positive on the pole could be prescribing scalar curvatures. We consider the quotient $\frac{v_λ}{V_λ}$ rather than …
An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of $M$ into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of or…
Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial $α$-curvature for vertex scaling of piecewise linear metrics on surfaces is a discretization of Gaussian curvature on surfaces. In this paper, we investigate the prescribing combinatorial $α$-curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory for piecewise linear metrics on surfaces and variational principles with constraints, we prove some Kazdan-Warner type theorems for prescribing combina…
The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M=G/K is studied. We focus on the metrics at which the Ricci curvature map is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.
In this paper, we study the problem of conformally deforming a metric on a $3$-dimensional manifold $M^3$ such that its $k$-curvature equals to a prescribed function, where the $k$-curvature is defined by the $k$-th elementary symmetric function of the eigenvalues of the Einstein tensor, $1\le k\le 3$. We prove the solvability of the problem and the compactness of the solution sets on manifolds when $k=2$ and $3$, provided the conformal class admits a negative $k$-admissible metric with respect to the Einstein tensor.
Prescribing scalar curvatures: on the negative Yamabe case
研究在负Yamabe不变量的闭合黎曼流形上规定 conformally 标量曲率的问题，证明当K严格负时总可解，并探讨K变号时的解的存在性和唯一性。
Prescribing Morse scalar curvatures: critical points at infinity
研究通过亚临界近似或负伪梯度流方法，证明在非退化条件下，两种方法在零弱极限下的等价性，包括有限能量亚临界爆破解、负型临界点及函数的负拉普拉斯临界点。
Prescribing Morse scalar curvatures: incompatibility of non existence
研究在正Yamabe不变量闭 manifold 上，通过 conformal 变换是否能将给定的Morse函数作为标量曲率的问题，揭示不存在性与能量有界解之间的关系。
We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
Prescribing scalar curvatures: loss of minimizability
本文研究了在 Yamabe 不变数为负的闭合流形上，标量曲率规定问题在可微性丧失时的行为，并构造了一个函数 $K$ 使得鞍点解仍存在。
The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condition on the measure for this problem to have a solution. In this paper, we address Alexandrov's problem for convex bodies in the hyperbolic space H m+1. After defining the Gauss curvature measure of an arbitrary hyperbolic convex body, we completely solve Alexandrov's problem in t…
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…
This paper presents results on the extent to which mean curvature data can be used to determine a surface in space or its shape. The emphasis is on Bonnet's problem: classify and study the surface immersions in $\R^3$ whose shape is not uniquely determined by the first fundamental form and the mean curvature function. The properties of immersions with umbilics and global rigidity results for closed surfaces are presented in the first part of this paper. The second part of the paper outlines an existence theory for conformal immersions based on Dirac spinors along with its immediate application…
We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-…
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.
In this paper we consider the problem of prescribing the $\bar{Q}'$-curvature on three dimensional Pseudo-Einstein CR manifolds. We study the gradient flow generated by the related functional and we will prove its convergence to a limit function under suitable assumptions.
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.
Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_σ(v)= Kv^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $\mathbb{S}^n$ for all $σ\in (0,n/2)$, where $P_σ$ is the intertwining operator of order $2σ$. Finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures. When $σ=1$, it is the prescribing scalar curvature problem or the Nirenberg problem,…
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling…
In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges to a metric with the prescribed function $f$ as its scalar curvature.
In this article, we first show that given a smooth function $ S $ either on closed manifolds $ (M, g) $ or compact manifolds $ (\bar{M}, g) $ with non-empty boundary, both for dimensions at least $ 3 $, the condition $ S \equiv 0 $, or $ S $ changes sign and $ \int_{M} S \dvol < 0 $ (with zero mean curvature if the boundary is not empty), is both the necessary and sufficient condition for prescribing scalar curvature problems within conformal class $ [g] $, provided that the first eigenvalue of the conformal Laplacian is zero. We then extend the same necessary and sufficient condition, in term…
We consider the problem of prescribing the $Q_{\ gamma}$ curvature on $\mathbb{S}^n$. Using a perturbation method, we obtain existence results for curvatures close to a positive constant.
We study conformal deformation problems on manifolds with boundary which include prescribing $σ_k\equiv0$ in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear operators as a key technical tool.
Prescribing the mean curvature of an achronal hypersurface as a measure: the case of 3D spacetimes
本文研究了在三维时空中的非时序超曲面的均曲率规定问题，证明了其存在性和正则性定理，并指出在更高维数中结论可能失效，同时探讨了其在 Born-Infeld 静电学中的应用。
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.
We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature of a Riemannian metric.
This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.
We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold $(M^n, g)$ endowed with the complementary orthogonal distributions $D_1$ and $D_2$. We provide conditions for symmetric $(0,2)$-tensors $T$ of a simple form (defined on $M$) to admit metrics $\tilde g$, conformal to $g$, that solve the partial Ricci equations. The solutions are given explicitly. Using above solutions, we also give examples to the problem of prescribing the mixed scalar curvature related to $D_i$. In aim to find "optimally placed" distributions, we calculate the variations of…
In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem.
The prescribed cross curvature problem on the three-sphere
本文研究三维球面上预定正交叉曲率的问题，通过给出多个存在性结果和一个非唯一性例子，否定了Hamilton的相关猜想。
We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy- Riemann spheres under Beta-flatness condition. To give a lower bound for the number of solutions, we use Bahri methods based on the theory of critical points at infinity and a Poincare-Hopf type formula.
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