Philippe Delanoë, Frédéric Robert
We prove in this article that the local image of each conformal $Q$-curvature operator of arbitrary order on the sphere admits no scalar constraint. However, we prove that identities of Kazdan--Warner type hold for its graph.
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In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Δ_{S^2} f>0$ at the saddle points.
We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures $K$ contained in naturally defined stable regions. We prove that in such stable regions, the map $u \rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also p…
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,…
On Fully Nonlinear Loewner-Nirenberg Problem of Ricci curvature
本文研究了Ricci曲率的完全非线性Loewner-Nirenberg问题，证明了在欧氏空间一般区域上特定条件下存在具有预定第k初等对称函数的负Ricci曲率光滑完备共形度量，并将其推广至更一般方程。
We consider the fractional Nirenberg problem on the standard sphere $\mathbb{S}^n$ with $n\geq 4$. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature satisfying assumptions of Bahri-Coron type.
In this paper, the fractional order curvature equation $(-Δ)^γu = (1 + \varepsilon K(x))u^{\frac{N + 2γ}{N - 2γ}}$ in $\mathbb{R}^N$ is considered. Assuming $K(x)$ has two critical points satisfying certain local conditions, we prove the existence of two-peak solutions.
We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partialΩ$ is a smooth bounded hypersurface (of codimension one). When $\partialΩ$ contains a compact smooth submanifold $Σ$ of higher codimension with $\partialΩ\setminusΣ$ being compact, we also give a `sharp' condition for the di…
We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure $n$-space (i.e., the Euclidean metric $n$-space).
We prove the concavity of $p$-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on $\mathbb{R}^n$ or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp $L^p$-Sobolev inequality and $L^p$-Gagliardo-Nirenberg inequalities on $\mathbb{R}^n$. Moreover, two improvement of $L^p$-Gagliardo-Nirenberg inequalities are derived.
In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli-Kohn-Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.
We consider the problem of prescribing the $σ_k$-curvature on the standard sphere $\mathbb{S}^n$ with $n \geq 3$. We prove existence and compactness theorems when $k \geq n/2$. This extends an earlier result of Chang, Han and Yang for $n = 4$ and $k = 2$.
Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on $§^n$ for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory inte…
Admissibility, Topology, and the $σ_k$-Loewner--Nirenberg problem
研究$σ_k$-Loewner--Nirenberg问题中可容许性条件的拓扑障碍，通过例子说明存在拓扑限制。
Non-uniqueness and symmetries for the Nirenberg problem using computer assistance
利用计算机辅助方法证明了Nirenberg问题中特定曲率下的解存在性，并确定了解的对称群，其中一种情况与给定曲率的对称性不同。
A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary
研究在紧致流形边界上，通过给定非递减旋转对称平均曲率和固定共形类，证明里奇流解在时间趋于无穷时局部一致收敛到完备双曲度量，且曲率保持小于-1。
On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting
本文利用Morse理论和计数指标方法，在无对称性假设下证明标准球面上Nirenberg问题在扰动背景下存在任意多个共形度量解。
In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if a complete $n$-dimensional Finsler manifold of nonnegative $n$-Ricci curvature satisfies the Gagliardo-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. The other one is that we give an alternative proof to Mao's main result in [23] for s…
For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in $\mathbb{R}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a one-directional analog of this result: if every pair of points $(x',a), (x',b)\in M$ with $a<b$ has ordered mean curvature $H(x',b)\leq H(x',a)$, then $M$ is symmetric about some hyperplane $x_{n+1}=c$ under some additional conditions. Their proof was done by the moving plane method and some variations of the Hopf Lemma. We obtain the symmetry of $M$ under s…
In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed besides the combination of interior and boundary blow ups, there are non simple blow up points for subcritical sol…
We propose a new approach to the question of prescribing Gaussian curvature on the 2-sphere with at least three conical singularities and angles less than $2π$, the main result being sufficient conditions for a positive function of class at least $C^2$ to be the Gaussian curvature of such a conformal conical metric on the round sphere. Our methods particularly differ from the variational approach in that they don't rely on the Moser-Trudinger inequality. Along the way, we also prove a general precompactness theorem for compact Riemann surfaces with at least three conical singularities and angl…
A Machine Learning Approach to the Nirenberg Problem
本文提出Nirenberg神经网络，通过机器学习方法解决高斯曲率指定问题，实现低损失训练并区分可实现与不可实现曲率。
In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial \mathbb{S}^n_+$. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g_0} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \e…
We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,γ}_{\rm loc}$ for any $γ> \frac{1}{k}$. Optimal regularity for solutions to the $σ_k$-Yamabe problem on annuli with finite constant boundary values is also established.
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 paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \end{equation*} w…
Hardy type inequalities on manifolds with nonnegative Ricci curvature
本文研究非负 Ricci 曲率且具欧氏体积增长的高维流形，证明了三类 Hardy 型不等式。
We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for Gagliardo-Nirenberg-Sobolev inequalities $\|w\r|_q \leq C\|\nabla w\|_2^θ\|w\|_p^{1-θ}$. These invariants are constructed via a minimization procedure for the weighted scalar curvature functional in the conformal class of a smooth metric measure space. We then describe critical points which are also critical points for variations in the metric or the measure. When the me…
Regularity of viscosity solutions of the $σ_k$-Loewner-Nirenberg problem
研究 $σ_k$-Loewner-Nirenberg 问题粘性解的正则性，证明其在边界附近的光滑性与 Hölder 连续性，并给出边界不变量刻画光滑性，且多分支边界时解不可微。
Smooth solutions to the Christoffel problem in $\mathbb{H}^{n+1}$
本文通过证明满秩定理建立双曲空间中 Christoffel 问题光滑解的存在性，并据此推出 $\mathbb{S}^n$ 上给定标量曲率的 Nirenberg-Kazdan-Warner 问题解的存在性。
Given a positive function $u\in W^{1,n}$, we define its John-Nirenberg radius at point $x$ to be the supreme of the radius such that $\int_{B_t}|\nabla\log u|^n<ε_0^n$ when $n>2$, and $\int_{B_t}|\nabla u|^2<ε_0^2$ when $n=2$. We will show that for a collapsing sequence in a fixed conformal class under some curvature conditions, the radius is bounded below by a positive constant. As applications, we will study the convergence of a conformal metric sequence on a $4$-manifold with bounded $\|K\|_{W^{1,2}}$, and prove a generalized Hélein's Convergence Theorem.
We study the problem of prescribing $σ_k$-curvature for a conformal metric on the standard sphere $\mathbb{S}^n$ with $2 \leq k < n/2$ and $n \geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $β\in [2,n)$. We prove among other things the following statements. (1) When $β>n-2k$, the solution set is compact, has a nonzero total de…
In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savaré and Toscani \cite{ST} on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein …
A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of the open unit disk there is even an indestructible Blaschke product $B$ such that the critical sets of $f$ and $B$ coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighte…
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 …
Let $(M; g)$ be a smooth compact Riemiannian manifold without boundary and $g_{k}$ be a metric conformal to $g$. Suppose $vol(M; g_{k})+||R_{k}||_{L^{p}(M;g_{k})} < C$, where $R_{k}$ is the scalar curvature and $p > \frac{n}{2}$. We will use the 3-circle theorem and the John-Nirenberg inequality to study the bubble tree convergence of $g_{k}$.
Answering a question by M. Struwe (Vietnam J. Math. 2020) related to the blow-up behaviour in the Nirenberg problem, we show that the prescribed $Q$-curvature equation $$Δ^2 u=(1-|x|^p)e^{4u}\text{ in }\mathbb{R}^4,\quad Λ:=\int_{\mathbb{R}^4}(1-|x|^p)e^{4u}dx<\infty$$ has normal solutions (namely solutions which can be written in integral form, and hence satisfy $Δu(x) =O(|x|^{-2})$ as $|x|\to \infty$) if and only if $p\in (0,4)$ and $$\left(1+\frac{p}{4}\right)8π^2\le Λ<16π^2.$$ We also prove existence and non-existence results for the positive curvature case, namely for $Δ^2 u=(1+|x|^p)e^{4…
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.
We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck [4] and Ivochkina, Trundinger and Lin [19] to more general curvature functions and less convex domains.
We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a new hypersurface conformal invariant which generalises to higher dimensions the important Willmore invariant of embedded surfaces. We call this the obstruc…
We introduce two flow approaches to the Loewner--Nirenberg problem on comapct Riemannian manifolds $(M^n,g)$ with boundary and establish the convergence of the corresponding Cauchy--Dirichlet problems to the solution of the Loewner--Nirenberg problem. In particular, when the initial data $u_0\in C^{4,α}(M)$ is a solution or a strict subsolution to the equation $(1.1)$, the convergence holds for both the direct flow $(1.3)-(1.5)$ and the Yamabe flow $(1.10)$. Moreover, when the background metric satisfies $R_g\geq0$, the convergence holds for any positive initial data $u_0\in C^{2,α}(M)$ for th…
For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $Ω: = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $Δu + f(u) = 0 $ in $Ω$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\partial Ω$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \cite{bcn2}. In 1971, Serrin \cite{serrin} proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy t…
The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates
本文完整分类了上半空间中完全非线性 Loewner-Nirenberg 问题的正粘性解，证明解的唯一性或单参数族结构取决于锥参数 μ_Γ^+ 是否大于 1，并据此构造了局部边界 C^0 估计的反例。
Nonlocal Ordered Mean Curvature with Integrable Kernel
本文引入非局部有序曲率概念，利用 Alexandrov 移动平面方法的推广，在核函数可积且紧支撑的非局部设定下，证明了有界光滑表面在特定方向平均曲率有序时必关于垂直超平面对称，扩展了经典微分设定下的对称性猜想。
We investigate the geometry and topology of extremal domains in a manifold with negative sectional curvature. An extremal domain is a domain that supports a positive solution to an overdetermined elliptic problem (OEP for short). We consider two types of OEPs. First, we study narrow properties of such domains in a Hadamard manifold and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity and how the domain behaves at infinity. This shows interesting relations with the Singular Yamabe Problem. Later, we focus on extremal domains i…
Sharp $L^p$-uncertainty principles on Finsler measure spaces
研究在芬斯勒测度空间上 $1 < p < n$ 的 $L^p$ -不确定性原理，通过曲率界等条件证明相关原理并刻画其尖锐性，推广了芬斯勒和黎曼情形下的相关结果。
Conformally Invariant Dirac Equation with Non-Local Nonlinearity
本文研究受4维共形爱因斯坦-狄拉克问题启发的共形不变狄拉克方程，通过分析能量泛函的性质、刻画基态解、证明Aubin型不等式及研究线性扰动，确保了问题解的存在性并得到相关结果。
In this paper we prove a local curvature estimate for the $κ$-LYZ flow over Kähler manifolds introduced in [FGP1] and [LYZ]. In particular, we generalize the long time existence of the flow.
By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci curvature.
Local and global blow-downs of transport twistor space
本文研究传输扭量空间的退化复结构，通过构造全纯 blow-down 映射，为常曲率度量提供整体解、任意度量提供局部解，并证明了退化情形的 Newlander--Nirenberg 定理。
We develop a local version of Huisken-Stampacchia iteration, using it to obtain local versions of a host of important sharp curvature pinching estimates for mean curvature flow. The local estimates we obtain do not depend on the quality of noncollapsing of the solution and the method adopted applies in a host of other settings.
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。
Assuming local uniform bounds on the metric for a solution of the Chern-Ricci flow, we establish local Calabi and curvature estimates using the maximum principle.
The main purpose of this paper is to determine the admissible forms of the sectional curvature operator on a three-dimensional locally homogeneous Lorentzian manifolds.
Local classification of Kähler metrics with constant holomorphic sectional curvature
该研究借助全纯函数1-节丛的几何特性，完成了常全纯截面曲率Kähler度量的局部分类证明，推进了复微分几何相关研究。
In this paper we consider the local $L^p$ estimate of Riemannian curvature for the Ricci-harmonic flow or List's flow introduced by List \cite{List2005} on complete noncompact manifolds. As an application, under the assumption that the flow exists on a finite time interval $[0,T)$ and the Ricci curvature is uniformly bounded, we prove that the $L^p$ norm of Riemannian curvature is bounded, and then, applying the De Giorgi-Nash-Moser iteration method, obtain the local boundedness of Riemannian curvature and consequently the flow can be continuously extended past $T$.
The Skew Mean Curvature Flow(SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general dimensional SMCF in Euclidean spaces.
It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4,5,6,7 and 8. It is shown that any n-dimensional $(4\leq n \leq 8)$ locally conformal flat Riemannian manifold with constant principal Ricci curvatures is a Riemannian locally symmetric space.
We extend our previous definition of quasi-local mass to 2-spheres whose Gauss curvature is negative and prove its positivity.
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