Intrinsic CR normal coordinates and the CR Yamabe problem

Jerison, David; Lee, John M.

doi:10.4310/jdg/1214442877

Cited by 148 publications

(156 citation statements)

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“…2 we recall the definition of pseudohermitian normal coordinates for an abstract CR manifold given by D. Jerison and J.M. Lee in [2]. This definition is a refined version of the notion of normal coordinates (see [1]).…”

Section: Theorem 1: Let (M θ) Be a Compact Cr 3-dimensional Manifoldmentioning

confidence: 99%

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The CR Yamabe conjecture the case n = 1

Gamara

2001

J. Eur. Math. Soc.

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Section: Theorem 1: Let (M θ) Be a Compact Cr 3-dimensional Manifoldmentioning

confidence: 99%

“…Lee formulated the CR Yamabe problem in [1], [2] and [3] and developed the analogy between it and the Yamabe problem in conformal Riemannian geometry which had already been solved by T. Aubin [10] and R. Schoen [11]. is a Folland and Stein space,…”

Section: Introductionmentioning

confidence: 99%

The CR Yamabe conjecture the case n = 1

Gamara

2001

J. Eur. Math. Soc.

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“…Since the problem is CR invariant they had to choose θ so as to simplify the curvature and the torsion at a base point q as much as possible. Using the results of [58], we proved the following estimates in pseudo-Hermitian normal coordinates near a base point…”

Section: Cr Yamabe Problem: Ideas Of the Proof Of Theorem 417mentioning

confidence: 99%

“…In order to make the calculation as easy as possible Jerison and Lee refined their notion of normal coordinates, defined in [56], by constructing in [58] new intrinsic CR normal coordinates for an abstract CR manifold. These coordinates are called pseudoHermitian normal coordinates.…”

Section: The Cr Yamabe Problemmentioning

confidence: 99%

“…Here, we have to use the Green's function associated with L to derive a good asymptotic expansion of the Yamabe funtional J near the sets of its critical points at infinity. Finally, to compute the numerator and the denominator of J, we used the approach of Jerison and Lee who refined in [58] the notion of normal coordinates by constructing the so-called pseudo-Hermitian normal coordinates. In pseudo-Hermitian normal coordinates, Jerison and Lee gave the Taylor series of θ and {θ α } to high order at a base point q ∈ M, in terms of the pseudo-Hermitian curvature and torsion.…”

Section: Cr Yamabe Problem: Ideas Of the Proof Of Theorem 417mentioning

confidence: 99%

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Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Gamara

2017

Arab. J. Math.

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Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.Mathematics Subject Classification 58E05 · 57R70 · 53C21 · 53C17

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Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂⁿ

Cohn

2004

Comm Pure Appl Math

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The main results of this paper concern sharp constants for the Moser-Trudinger inequalities on spheres in complex space C n . We derive Moser-Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser-Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser-Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser-Onofri-type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4).

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Intrinsic CR normal coordinates and the CR Yamabe problem

Cited by 148 publications

References 14 publications

The CR Yamabe conjecture the case n = 1

The CR Yamabe conjecture the case n = 1

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂⁿ

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Intrinsic CR normal coordinates and the CR Yamabe problem

Cited by 148 publications

References 14 publications

The CR Yamabe conjecture the case n = 1

The CR Yamabe conjecture the case n = 1

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

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Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂⁿ