Khalil El Mehdi
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.
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We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetrization for higher order variational problems.
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.
We investigate fourth order Paneitz equations of critical growth in the case of $n$-dimensional closed conformally flat manifolds, $n \ge 5$. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the $Q$-curvature. We obtain sharp asymptotics for arbitrary bounded energy sequences of solutions of our equations from which we derive stability and compactness properties. In doing so we establish the criticality of the geometric equation with respect to the trace of its second order te…
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.
In this paper we prescribe a fourth order conformal invariant 9the Paneitz Curvature) on five and six spheres. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results.
Let $(M^4,g)$ be a closed Riemannian manifold of dimension four. We investigate the properties of metrics which are critical points of the eigenvalues of the Paneitz operator when considered as functionals on the space of Riemannian metrics with fixed volume. We prove that critical metrics of the aforementioned functional restricted to conformal classes are associated with a higher-order analog of harmonic maps (known as extrinsic conformal-harmonic maps) into round spheres. This extends to four-manifolds well-known results on closed surfaces relating metrics maximizing laplacian eigenvalues i…
In this paper, we establish that: Suppose a closed Riemannian manifold $(M^n,g_0)$ of dimension $\geq 8$ is not locally conformally flat, then the Paneitz-Sobolev constant of $M^n$ has the property that $q(g_0)<q(S^n)$. The analogy of this result was obtained by T. Aubin in 1976 and had been used to solve the Yamabe problem on closed manifolds. As an application, the above result can be used to recover the sequential convergence of the nonlocal Q-curvature flow on closed manifolds recently introduced by Gursky-Malchiodi.
In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q-curvature. We show also C^{\infty}-regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls.
Global bifurcation for Paneitz type equations and constant Q-curvature metrics
本文研究闭黎曼流形上Paneitz型方程，利用等参函数将其约化为四阶常微分方程，证明非常数解仅具非退化临界点，并通过全局分歧理论证明正解多重性，进而得到共形常Q-曲率度量数量趋于无穷的结论。
We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are universally bounded. If the $Q$-curvature is $1$, assuming that Paneitz operator's kernel is trivial and its Green function is positive, we establish universal energy bounds on manifolds which are either locally conformally flat (LCF) or of dimension $\le 9$. Moreover, assuming in addition that a positive mass type theorem holds for the Paneitz operator, we prove compact…
Global Convergence of the Gursky-Malchiodi $Q$-curvature Flow
本文证明了Gursky-Malchiodi Q-曲率流在任意初始能量下的全局收敛性，通过建立非局部Lojasiewicz-Simon不等式并构造测试气泡来解决Q-曲率和Paneitz算子的复杂性。
We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant condition for positivity of the Paneitz operator, which allows us to give many new examples of manifolds admitting metrics with constant $Q$-curvature.
A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in $\bbR^n$, or $\bbS^n$, $n\in \bbN$, which are distributed according to the configurational microcanonical measure $δ(E-H)$, or rather some regularization thereof, where H is the configurational Hamiltonian and E the configurational energy. When $N\to\infty$ with non-extensive energy scaling $E=N^2 \vareps$, the particle positions become i.i.d. according to a self-consistent Boltzmann distribution, respectively a superposition of such distributions. The self-con…
Multiplicity results for constant Q-curvature conformal metrics
本文证明闭流形上次临界Paneitz-Branson方程至少存在Cat(M)个正解，进而得出正爱因斯坦流形乘积共形类中常数Q-曲率度量的多重性。
Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-negative and the Webster curvature is positive. Our lower bound for the non-zero eigenvalues is sharp and is attained on S^3. A consequence of our lower bound is that all compact CR 3-manifolds with non-negative CR Paneitz operator and positive CR Yamabe constant are embeddable. N…
Complete Spectrum and Sharp Local Stability for the Critical Exponential Biharmonic Choquard Equation in \(\mathbb R^{4}\)
本文研究\u211d⁴中共形不变的指数双调和Choquard方程，完成其共形气泡处线性化算子的完整谱分解，得到尖锐局部稳定性结论，还证明有限质量解为显式平移-伸缩气泡。
In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the $Q$-curvature of the metric is a constant. Existence of solutions is obtained through the combination of variational methods, second order Sobolev inequalities, and the $W^{2,2}$ blow-up theory developed by Hebey and Robert. Positivity of the solutions is obtained from a novel argument proven here for the first time that is rooted in the conformal covariance property of the Paneitz-Bra…
Classification of solutions to the $Q$-flat and constant $T$-curvature equation on the half-space and ball
本文通过严格定义双调和 Poisson 核并给出显式表示，对半空间和球上 $Q$-平坦及常 $T$-曲率方程的非负解建立了分类定理。
Given a compact and connected four dimensional smooth Riemannian manifold $(M,g_0)$ with $k_P := \int_M Q_{g_0} dV_{g_0} <0$ and a smooth non-constant function $f_0$ with $\max_{p\in M}f_0(p)=0$, all of whose maximum points are non-degenerate, we assume that the Paneitz operator is nonnegative and with kernel consisting of constants. Then, we are able to prove that for sufficiently small $λ>0$ there are at least two distinct conformal metrics $g_λ=e^{2u_λ}g_0$ and $g^λ=e^{2u^λ}g_0$ of $Q$-curvature $Q_{g_λ}=Q_{g^λ}=f_0+λ$. Moreover, by means of the "monotonicity trick", we obtain crucial estim…
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.
Let $(M, g)$ be a compact Riemannian manifold of dimension $N \geq 5$ and $Q_g$ be its $Q$ curvature. The prescribed $Q$ curvature problem is concerned with finding metric of constant $Q$ curvature in the conformal class of $g$. This amounts to finding a positive solution to \[ P_g (u)= c u^{\frac{N+4}{N-4}}, u>0 {on} M\] where $P_g$ is the Paneitz operator. We show that for dimensions $N \geq 25$, the set of all positive solutions to the prescribed $Q$ curvature problem is {\em non-compact}.
We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We show that this problem is equivalent to the study of the Composite Membrane Problem, a free boundary problem studied earlier by the author and his collaborators in all dimensions. Thus complete answers, existence of the limit metric, regularity of the minimizing eigenfunction and va…
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.
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.
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…
Remarks on constructing biharmonic and conformal biharmonic maps to spheres
本文提出一种几何算法，用于构造双调和与共形双调和映射到球面，并通过实例展示算法在不同域下的灵活性及方法的有效性。
In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. Our results extend the classifications of Escobar in \cite{E0} when $n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.
Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions
该研究证明3至10维中接近球面圆度量的共形类里，共形拉普拉斯第二共形特征值无极小值子，与11维及以上的存在性结果形成对比，还推广了低维Yamabe方程变号解能量下界的结论。
We will show that in the conformal class of the standard metric $g_{S^n}$ on $S^n$, the scaling invariant functional $(μ_g(S^n))^{\frac{2m-n}{n}}\int_{S^n}Q_{2m,g}dμ_g$ maximizes at $g_{S^n}$ when $n$ is odd and $m=\frac{n+1}{2}$ or $\frac{n+3}{2}$. For $n$ odd and $m\geq\frac{n+5}{2}$, $g_{S^n}$ is not stable and the functional has no local maximizer. Here $Q_{2m,g}$ is the $2m$th order $Q $-curvature.
The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces
该论文研究球面上的共形对数拉普拉斯算子，建立其与ℝ^N中对数拉普拉斯算子的联系，引入对应希尔伯特空间，衔接Yamabe型问题分类结果并推广至弱情形。
Conformally invariant equations with negative critical exponents on the three dimensional hyperbolic space
研究三维双曲空间上四阶方程正解的增长率对称性，证明正解必须径向严格递减。
We study some conformally invariant integral equations using the method of moving spheres.
In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric analysis. For instance, it underlies positive mass theorems associated to the Paneitz operator, playing a similar role in the positive $Q$-curvature conformal prescription problem as the Schoen-Yau positive energy theorem does for the Yamabe problem…
The regularized determinant of the Paneitz operator arises in quantum gravity (see Connes 1994, IV.4.$γ$). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in Branson (1996). A similar formula holds for Cheeger's half-torsion, which plays a role in self-dual field theory (see Juhl, 2009), and is defined in terms of regularized determinants of the Hodge laplacian on $p$-forms ($p < n/2$). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal…
Reverse conformally invariant Sobolev inequalities on the sphere
针对n维球面上阶数2s>n的算子对应的共形不变Sobolev不等式最佳常数优化问题，将已有结论推广至s为非整数的情形，解决了所有n≥2维度下不等式的成立性问题。
Let M be a compact Riemannian manifold and let $μ$,d be the associated measure and distance on M. Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M -> M pushing forward $μ$ to a measure $ν$: each S factors uniquely a.e. into the composition S = T \circ U, where U : M -> M is volume preserving and T : M -> M is the optimal map transporting $μ$ to $ν$ with respect to the cost function d^2/2. In this article we study the polar factorization of conformal and projective maps of the sphere S^n. For conformal maps, which m…
The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities
本文研究了单位球面上的共形分数-对数拉普拉斯算子的分析与谱性质，推导了Yamabe型方程，并应用于尖锐Sobolev型不等式，恢复了尖锐对数Sobolev不等式，揭示了分数-对数类比的失效，并建立了新的尖锐分数-对数不等式。
Given $(M,g)$ a smooth compact Riemannian manifold without boundary of dimension $n\geq 3$, we consider the first conformal eigenvalue which is by definition the supremum of the first eigenvalue of the Laplacian among all metrics conformal to $g$ of volume 1. We prove that it is always greater than $nω_n^{\frac{2}{n}}$, the value it takes in the conformal class of the round sphere, except if $(M,g)$ is conformally diffeomorphic to the standard sphere.
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