The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

DOI10.1016/j.na.2015.06.024zbMath1325.53041arXiv1504.03259OpenAlexW2963229633MaRDI QIDQ494203

Dimiter Vassilev, Stefan Ivanov

Publication date: 31 August 2015

Published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1504.03259

zbMATH Keywords

sub-Riemannian geometrySobolev inequalityYamabe equationCR and quaternionic contact structuresLichnerowicz eigenvalue estimateObata theorem

Mathematics Subject Classification ID

Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Sub-Riemannian geometry (53C17)

Related Items (9)

On conformal qc geometry, spherical qc manifolds and convex cocompact subgroups of \(\mathrm{Sp}{(n+1,1)}\) ⋮ Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds ⋮ The Lichnerowicz-Obata theorem for the Kohn Laplacian in three dimensions ⋮ Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group ⋮ Quaternionic contact hypersurfaces in hyper-Kähler manifolds ⋮ A survey on inverse mean curvature flow in ROSSes ⋮ The qc Yamabe problem on non-spherical quaternionic contact manifolds ⋮ SUB-RIEMANNIAN RICCI CURVATURES AND UNIVERSAL DIAMETER BOUNDS FOR 3-SASAKIAN MANIFOLDS ⋮ The Obata first eigenvalue theorem on a seven-dimensional quaternionic contact manifold

Cites Work

• The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold
• Normal and integral currents
• Conformal changes of Riemannian metrics
• A symmetry problem in potential theory
• The conjectures on conformal transformations of Riemannian manifolds
• Contact and conformal maps on Iwasawa \(N\) groups
• Compactness of solutions to the Yamabe problem. II
• Certain conditions for a Riemannian manifold to be isometric with a sphere
• The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven
• On conformally-flat spaces in the large
• Einstein spaces with a conformal group
• Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics
• The diameter estimate and its application to CR Obata’s Theorem on closed pseudohermitian $(2n+1)$-manifolds
• A note on the uniqueness of solutions for the Yamabe problem
• Blow-up phenomena for the Yamabe equation
• $L^{p}$ estimates and asymptotic behavior for finite energy solutions of extremals to Hardy-Sobolev inequalities
• Contact and 1-quasiconformal maps on Carnot groups
• Conformal maps of Carnot groups
• Another Approach to Liouville Theorem
• The dirichlet problem for the bezgman laplacian. II
• Isoperimetric inequalities & volume comparison theorems on CR manifolds
• Rigidity of Carnot groups relative to multicontact structures
• Einstein Metrics with Anisotropic Boundary Behavior
• The first eigenvalue of a sublaplacian on a pseudohermitian manifold
• The Fefferman Metric and Pseudohermitian Invariants
• Scalar Curvature Functions in a Conformal Class of Metrics and Conformal Transformations
• The Yamabe problem
• Extremals for the Sobolev Inequality on the Heisenberg Group and the CR Yamabe Problem
• Psuedo-Einstein Structures on CR Manifolds
• The geometry of a semi-direct extension of a Heisenberg type nilpotent group
• The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation
• Rings and Quasiconformal Mappings in Space
• A class of nonsymmetric harmonic Riemannian spaces
• On Three-Dimensional Cauchy-Riemann Manifolds
• Estimates for the \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \partial \limits^ - _b $\end{document} complex and analysis on the heisenberg group
• Harmonic forms and Riesz transforms for rank one symmetric spaces
• On manifolds with Sasakian $3$-structure over quaternion Kaehler manifolds
• Applications of analysis on nilpotent groups to partial differential equations
• Boundary Behavior of Harmonic Forms on a Rank One Symmetric Space
• Fundamental Solutions for a Class of Hypoelliptic PDE Generated by Composition of Quadratic Forms
• Équations différentielles caractéristiques de la sphère
• An embedding theorem and the harnack inequality for nonlinear subelliptic equations
• Regularity of quasi-linear equations in the Heisenberg group
• The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold
• Positive and negative 3-K-contact structures
• Curvature: A Variational Approach
• Recent progress on the Yamabe problem
• A new proof of the $J^2$-condition for real rank one simple Lie algebras and their classification
• Yang-Mills fields on quaternionic spaces
• On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of fefferman's equation
• A Local Slice Theorem for 3-Dimensional CR Structures
• A NOTE ON GENERALIZED LICHNEROWICZ-OBATA THEOREMS FOR RIEMANNIAN FOLIATIONS
• Another proof of the Yamabe conjecture for locally conformally flat manifolds
• AN OBATA-TYPE THEOREM ON A THREE-DIMENSIONAL CR MANIFOLD
• A fourth order curvature flow on a CR 3-manifold
• Embedded three-dimensional CR manifolds and the non-negativity of Paneitz operators
• Complete Riemannian Manifolds and Some Vector Fields
• Strong Rigidity of Locally Symmetric Spaces. (AM-78)
• The CR Yamabe conjecture. The case \(n=1\)
• Rigidity of \(H\)-type groups
• Some properties of boundaries of symmetric spaces of rank one
• Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Isometries of Carnot groups and sub-Finsler homogeneous manifolds
• Sharp Hardy-Littlewood-Sobolev inequalities on the octonionic Heisenberg group
• Non-Kaehler heterotic string solutions with non-zero fluxes and non-constant Dilaton
• An Obata-type theorem in CR geometry
• Volume and distance comparison theorems for sub-Riemannian manifolds
• Sharp constants in several inequalities on the Heisenberg group
• Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds
• The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold
• A positive mass theorem in three dimensional Cauchy-Riemann geometry
• Quaternionic contact hypersurfaces in hyper-Kähler manifolds
• Quaternionic contact Einstein manifolds
• Generalized Obata theorem and its applications on foliations
• The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven
• Compact supersymmetric solutions of the heterotic equations of motion in dimensions 7 and 8
• CR Yamabe conjecture -- the conformally flat case.
• A decomposition of the space \(\mathcal M\) of Riemannian metrics on a manifold
• Einstein spaces admitting a one-parameter group of conformal transformations
• On isometries and on a theorem of Liouville
• A characteristic property of spheres
• On the CR analogue of Obata's theorem in a pseudohermitian 3-manifold
• Nonnegativity of CR Paneitz operator and its application to the CR Obata's theorem
• There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics
• Conformal paracontact curvature and the local flatness theorem
• Conformality and \(Q\)-harmonicity in Carnot groups
• An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem
• Quaternionic contact structures in dimension 7
• Compactness of solutions to the Yamabe problem. III
• Sharp Hardy-Littlewood-Sobolev inequalities on quaternionic Heisenberg groups
• Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations. I
• Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations. II
• Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains
• A sufficient condition for nonrigidity of Carnot groups
• On the estimate of the first eigenvalue of a sub-Laplacian on a pseudo-Hermitian 3-manifold
• Conformal quaternionic contact curvature and the local sphere theorem
• The entropy formulas for the CR heat equation and their applications on pseudohermitian \((2n+1)\)-manifolds
• Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem
• Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups
• The Yamabe problem on quaternionic contact manifolds
• A compactness theorem for the Yamabe problem
• Blow-up phenomena for the Yamabe equation. II
• Quasiconformal mappings on the Heisenberg group
• Conformal deformation of a Riemannian metric to constant scalar curvature
• Geometric properties of Heisenberg-type groups
• The Yamabe problem on CR manifolds
• Intrinsic CR normal coordinates and the CR Yamabe problem
• Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
• On the pseudo-conformal geometry of hypersurface of the space of \(n\) complex variables
• Symmetry and related properties via the maximum principle
• Proof of the positive mass theorem. II
• Two notes on conformal geometry
• Einstein metrics with prescribed conformal infinity on the ball
• Classification of solutions of some nonlinear elliptic equations
• On complete quaternionic-Kähler manifolds
• Harmonic analysis on solvable extensions of H-type groups
• Pluriharmonic boundary values
• Real hypersurfaces in complex manifolds
• Subelliptic estimates and function spaces on nilpotent Lie groups
• Eigenvalue comparison theorems and its geometric applications
• Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
• Hypoelliptic differential operators and nilpotent groups
• Best constant in Sobolev inequality
• On the characterization of complex projective space by differential equations
• Spherical hypersurfaces in complex manifolds
• Problèmes isoperimetriques et espaces de Sobolev
• Pseudo-hermitian structures on a real hypersurface
• Noncompact homogeneous Einstein spaces
• Riemannian foliations and eigenvalue comparison
• The large \(N\) limit of superconformal field theories and supergravity
• An approach to symmetric spaces of rank one via groups of Heisenberg type
• Differential forms on contact manifolds
• Generalized Heisenberg groups and Damek-Ricci harmonic spaces
• Remarks on a triple of \(K\)-contact structures
• Killing spinor equations in dimension 7 and geometry of integrable \(G_{2}\)-manifolds
• On a differential equation characterizing Euclidean spheres.
• A proof of the Liouville theorem of conformal geometry in the analytic case.
• Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups.
• Characterizing specific Riemannian manifolds by differential equations
• Parallel spinors and connections with skew-symmetric torsion in string theory.
• \(H\)-type groups and Iwasawa decompositions
• Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type
• Lichnerowicz and Obata theorems for foliations.
• The Cayley transform and uniformly bounded representations
• The geometric Sobolev embedding for vector fields and the isoperimetric inequality
• Sub-Riemannian geometry. Proceedings of the satellite meeting of the first European congress of mathematics `Journées nonholonomes: géométrie sous-riemannienne, théorie du contrôle, robotique', Paris, France, June 30--July 1, 1992
• Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere
• An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence-free torsion
• Bounds for the first eigenvalue of the horizontal Laplacian in positively curved sub-Riemannian manifolds
• Bishop and Laplacian comparison theorems on three-dimensional contact sub-Riemannian manifolds with symmetry
• Holomorphic mappings from the ball and polydisc
• On a remarkable formula of Jerison and Lee in CR geometry
• Quaternionic Heisenberg group and heterotic string solutions with non-constant dilaton in dimensions 7 and 5
• Differential geometry and analysis on CR manifolds
• Finsler conformal Lichnerowicz-Obata conjecture
• Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves
• Quaternionic Kähler and \(\mathrm{Spin}(7)\) metrics arising from quaternionic contact Einstein structures
• Curvature dimension inequalities and subelliptic heat kernel gradient bounds on contact manifolds
• The Lichnerowicz theorem on CR manifolds
• Contact and conformal maps in parabolic geometry
• Classification of solutions of a critical Hardy--Sobolev operator
• A priori estimates for the Yamabe problem in the non-locally conformally flat case

This page was built for publication: The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds