Nanbo Chen, Xiaochun Liu
In this paper, we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator. With the help of the subcritical approach from variational method, we obtain the non-existence, existence, and multiplicity results under some given assumptions.
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For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.
We prove rigidity for the Lichnerowicz-type eigenvalue estimate for the Kohn Laplacian on strictly pseudoconvex three-manifolds with nonnegative CR Paneitz operator and positive Webster curvature.
We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.
We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.
Gradient estimates and Liouville theorems for Lichnerowicz-type equation on Riemannian manifolds
本文研究完备黎曼流形上 Lichnerowicz 椭圆方程正解的梯度估计，采用 Nash-Moser 迭代技术获得精细估计，导出刘维尔型定理，并应用于 Einstein-标量场方程的 Cheng-Yau 型估计。
We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds $M$ of finite volume. Sharp conditions ensuring $L^q(M)$ and $L^\infty (M)$ bounds for eigenfunctions are exhibited in terms of either the isoperimetric function or the isocapacitary function of $M$.
We study viscosity solutions to degenerate and singular elliptic equations of $p$-Laplacian type on Riemannian manifolds. The Krylov-Safonov type Harnack inequality for the $p$-Laplacian operators with $1<p<\infty$ is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear $p$-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.
The $p$-Laplacian overdetermined problem on Riemannian manifolds
该文研究非负Ricci曲率完备非紧黎曼流形上的$p$-拉普拉斯超定问题，证明弱解正则性、得到积分恒等式，并应用其证明该超定问题，推导出Heintze-Karcher不等式等经典结果。
We obtain a criterion for the existence of solutions of the problem $$ Δ_p u = 0 \quad \mbox{in } M \setminus \partial M, \quad \left. u \right|_{ \partial M } = h, $$ with the bounded Dirichlet integral, where $M$ is an oriented complete Riemannian manifold with boundary and $h \in W_{p, loc}^1 (M)$, $p > 1$.
Following \cite{B2}, we introduce a notion of para-products associated to a semi-group. We do not use Fourier transform arguments and the background manifold is doubling, endowed with a sub-laplacian structure. Our main result is a paralinearization theorem in a non-euclidean framework, with an application to the propagation of regularity for some nonlinear PDEs.
In this paper, we investigate the existence and uniqueness of a non-trivial solution for a class of nonlocal equations involving the fractional $p$-Laplacian operator defined on compact Riemannian manifold, namely, \begin{eqnarray}\label{k1} \begin{gathered} \left\{\begin{array}{lll} (-Δ_g)^s_p u(x)+ \left| u \right|^{p-2} u= f(x,u) & \text { in }& Ω, \hspace{3,4cm} u=0 & \text{in }& M\setminusΩ, \end{array}\right. \end{gathered} \end{eqnarray} and $Ω$ is an open bounded subset of M with a smooth boundary.
We obtain necessary and sufficient existence conditions for solutions of the boundary value problem $$ Δ_p u = f \quad \mbox{on } M, \quad \left. \left| \nabla u \right|^{p - 2} \frac{\partial u}{\partial ν} \right|_{ \partial M } = h, $$ where $p > 1$ is a real number, $M$ is a connected oriented complete Riemannian manifold with boundary, and $ν$ is the external normal vector to $\partial M$.
Sectoriality of the Laplacian on Asymptotically Hyperbolic Spaces
本文证明了渐近双曲度量下函数拉普拉斯算子与对称2-张量Lichnerowicz拉普拉斯算子在加权Hölder空间中的扇形性，并由此利用解析半群理论得到抛物型发展方程的适定性。
On any closed Riemannian manifold of dimension greater than $7$, we construct examples of background physical coefficients for which the Einstein-Lichnerowicz equation possesses a non-compact set of positive solutions. This yields in particular the existence of an infinite number of positive solutions in such cases.
Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds
本文研究了完备黎曼流形上爱因斯坦-标量场利钦诺维奇方程的有限能量解的存在性，通过正则化问题、山峰点论证和Harnack不等式，证明了在特定条件下非负解和正解的存在性。
We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Riemannian manifolds with lower Ricci curvature bounds, and isoperimetric inequalities on Gromov-Hausdorff limit spaces. We also establish a new Lichnerowicz-Obata type theorem.
Large-time behavior of two families of operators related to the fractional Laplacian on certain Riemannian manifolds
本文研究两类含Poisson半群的分数阶拉普拉斯相关算子的大时间行为，证明非负Ricci曲率流形上L¹初值解渐近为质量乘基本解，双曲空间等秩一对称空间中Poisson半群收敛性失效但径向初值下仍成立。
The Logarithmic Laplacian on Complete Riemannian Manifolds
本文首次在一般完备黎曼流形上引入对数拉普拉斯算子的Bochner积分公式，统一了紧致与非紧致情形，并在里奇曲率下界下推导显式公式，比较谱与热核定义，最终在双曲空间上给出精确估计并建立Lp连续性。
On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric tensors, a similar result is proven for (convex cocompact) quotients of hyperbolic space.
We consider the first eigenvalue $λ_1(Ω,σ)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $Ω$ with smooth boundary, $σ\in\bf R$ being the Robin boundary parameter. When $σ>0$ we give a positive, sharp lower bound of $λ_1(Ω,σ)$ in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of $Ω$, a lower bound of the mean curvature of $\partialΩ$ and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean do…
In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and L1 and L2 energy for drifting heat equation on closed manifolds with weighted measure. In some sense, this study is about the eigenvalue estimate on Ricci solitons.
We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.
An overdetermined problem related to the p-Laplacian on Riemannian manifolds
本文研究了在具有正里奇曲率的紧致黎曼流形上p-拉普拉斯方程的超定问题，通过引入新的P函数获得积分恒等式，并应用该结果得到Heintze-Karcher不等式和肥皂泡定理。
Ejiri gave a negative answer to a conjecture of Lichnerowicz concerning Riemannian manifolds with constant scalar curvature admitting an infinitesimal non isometric conformal transformation. With this aim he constructed a warped product of a circle of lenght $T$ and a compact manifold. But he omitted in his analysis the condition that $T$ must to be big enough. Here we give an explicit sharp bound $T_0 < T$ that will make the proof complete. Our presentation is self-contained and mainly uses bifurcation techniques. Moreover, we show that there are other such examples and contribute some result…
These notes are the basis of a course given at the Institut Henri Poincare in September 2014. We survey some recent results related to the geometric analysis of hypoelliptic diffusion operators on totally geodesic Riemannian foliations. We also give new applications to the study of hypocoercive estimates for Kolmogorov type operators.
Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3
本文研究三维紧致黎曼曲面单位切丛上亚黎曼拉普拉斯算子的亚椭圆自伴扰动，证明在大特征值极限下特征函数集中于量子化水平集并满足守恒性质。
The main technical result of the paper is a Bochner type formula for the sub-laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz' theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the $Sp(n)Sp(1)$ components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a-priori estimates for square integrals of horizontal derivatives of …
Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold
本文为接触次黎曼流形中的光滑超曲面构造了特征点集外的内蕴次拉普拉斯算子，证明其是黎曼近似下Laplace-Beltrami算子的极限，并在三族模型情形中证明其随机完备性及诱导的随机过程几乎必然不命中特征点。
Sub-Laplacian generalized curvature dimension inequalities on Riemannian foliations
本文在不假定度量束状结构和叶的总测地或最小条件的情况下，为一般黎曼流形的水平拉普拉斯算子发展了广义曲率维数不等式，并应用该框架得到多种几何和分析结果。
The Ray-Singer analytic torsion is the zeta-function trace of a certain sum of logarithm operators on the de Rham complex. In this note we examine the residue analytic torsion, defined using the residue-trace instead of the spectral zeta function quasi-trace.
We prove a CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Panietz operator which satisfies a Lichnerowicz type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three dimensional unit sphere.
Equivalence of sub-Laplacian on Polarized groups
本研究刻画了次黎曼李群间与次拉普拉斯算子可交换的光滑映射，证明其为次黎曼淹没映射，还明确了Carnot群中次拉普拉斯算子可决定次黎曼结构，澄清了相关前期分析。
We compute the Green's function for the Hodge Laplacian on the symmetric spaces M\timesΣ, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and Σis a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green's function is governed by a fourth order analogue of the Heun equation.
The aim of this paper is give a simple proof of some results in \cite{Jun Ling-2006-IJM} and \cite{JunLing-2007-AGAG}, which are very deep studies in the sharp lower bound of the first eigenvalue in the Laplacian operator on compact Riemannian manifolds with nonnegative Ricci curvature. We also get a result about lower bound of the first Neumann eigenvalue in a special case. Indeed, our estimate of lower bound in the this case is optimal. Although the methods used in here due to \cite{Jun Ling-2006-IJM} (or \cite{JunLing-2007-AGAG}) on the whole, to some extent we can tackle the singularity of…
We consider the first Robin eigenvalue $ł_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $ł_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $\a<0$. Our results cover corresponding comparison …
On Courant and Pleijel theorems for sub-Riemannian Laplacians
本文研究次黎曼流形次拉普拉斯算子特征函数结点域数量，将Pleijel定理的有效性从一般流形归结为幂零群情形，并详细分析了Heisenberg群情形，改进了相关不等式的最优常数界限。
Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifolds
本文通过非线性 Φ-Bochner 公式和 Nash-Moser 迭代，在 Ricci 曲率下界和 Φ、G 的适当条件下，建立了 Φ-Laplacian 方程的局部梯度界，进而得到全局解的 Liouville 定理；对于抛物情形，利用最大值原理在紧流形上导出梯度估计并获得 Liouville 型结果。
What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved. MSC 2010: 35Q30, 58J65
Riemannian starshape and capacitary problems
本文证明了经典欧几里得星形环容量势水平集星形结果的Riemannian版本，将其限制在开区间与单位球面的warped product中，并用 $q$-Laplacian 替换 Laplacian 扩展了该结论。
We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.
We show that a compact quaternionic contact manifold of dimension seven that satisfies a Lichnerowicz-type lower Ricci-type bound and has the $P$-function of any eigenfunction of the sub-Laplacian non-negative achieves its smallest possible eigenvalue only if the structure is qc-Einstein. In particular, under the stated conditions, the lowest eigenvalue is achieved if and only if the manifold is qc-equivalent to the standard $3$-Sasakian sphere.
Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions
该研究构造了具有预设节点集的黎曼度量，使得拉普拉斯特征函数的零集与给定配置一致，并证明了特征值与体积的关系。
We study a new class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two. These manifolds admit a canonical horizontal connection and a canonical sub-Laplacian. We construct on these manifolds an analogue of the Riemannian Ricci tensor and prove Bochner type formulas for the sub-Laplacian. As a consequence, it is possible to formulate on these spaces a sub-Riemannian analogue of the so-called curvature dimension inequality. Sub-Riema…
I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension.
In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which affirmatively answers to a conjecture proposed by Chen-Zheng-Yang. In addition, we also exploit the closed eigenvalue problem of Laplacian and obtain a similar optimal upper bound. As some important examples, we investigate the eigenvalues of the eigenvalue problem of the Laplacian on the unit sphere and cylinder, compact homogeneous Riemannian manifolds wi…
In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp-Lieb's inequality and its generalization to compact Riemannian manifolds without boundary.
On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $Δ_ω$, as the divergence of the horizontal gradient, once a volume $ω$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of random walks, where \emph{all} sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement $\mathbf{c}$ to the sub-Riemannian distribution, and is denoted $L^c$. We address the problem of equivalence of the two operators. This problem is int…
We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncom…
This paper is a proceedings version of \cite{CHT-I}, in which we state a Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we establish some properties of the Quantum Limits (QL). We consider a sub-Riemannian (sR) metric on a compact 3D manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We state a QE theorem for the eigenfunctions of any associated sR Laplacian, under the assumption that the Reeb flow is ergodic. The limi…
We study an optimal control problem associated to the conformal Laplacian obstacle problem on closed n-dimensional Riemannian manifolds with n >2. When the Yamabe invariant of the Riemannian manifold is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal metric with constant scalar curvature. For the standard sphere, we prove that the standard bubbles -- namely conformal factor of metrics conformal to the standard one with constant positive scalar curvature -- are the only optimal …
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
Resolving the viscosity operator ambiguity on Riemannian manifolds via a kinematic selection principle
本文研究了在黎曼流形上Navier-Stokes方程的不同粘性算子选择问题，通过运动构造原理唯一选择变形拉普拉斯算子，并证明了在特定边界条件下，变形拉普拉斯算子在超曲面上具有良好的 coercivity 性质，从而解决了分析上的障碍。
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