σk-Scalar curvature and eigenvalues of the Dirac operator

Wang, Guofang

doi:10.1007/s10455-006-9022-z

Cited by 5 publications

(5 citation statements)

References 19 publications

(36 reference statements)

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“…By the Chern-Gauss-Bonnet formula, it follows that the total Branson's Q-curvature is a multiple of the total σ 2 -curvature (the classical second elementary function associated with the Schouten tensor). Hence Corollary 1 is exactly the result obtained by G. Wang [22] but it seems that there is a serious gap in his proof regarding the prescription, in a conformal class, of the σ 2 -curvature function.…”

Section: The Case N ≥supporting

confidence: 65%

Branson's Q-curvature in Riemannian and Spin Geometry

Hijazi

Raulot²

2007

SIGMA

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Abstract. On a closed n-dimensional manifold, n ≥ 5, we compare the three basic conformally covariant operators: the Paneitz-Branson, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. Equality cases are also characterized.

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Section: The Case N ≥supporting

confidence: 65%

Branson's Q-curvature in Riemannian and Spin Geometry

Hijazi

Raulot²

2007

SIGMA

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“…Chang-Gursky-Yang [90,91]) and eigenvalue estimates for Dirac operators (see Guofang Wang [408]). Since χ(M ) is a topological and |W | 2 dS is a pointwise conformal invariant, this shows that M QdS is a conformal invariant, which governs e.g.…”

Section: Paneitz-branson Type Equationsmentioning

confidence: 99%

Polyharmonic Boundary Value Problems

Gazzola

Grunau

Sweers

2010

Lecture Notes in Mathematics

457

309

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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher ServicesPrinted on acid-free paper springer.com Dedicated to our wives Chiara, Brigitte and Barbara.The cover figure displays the solution of Δ 2 u = f in a rectangle with homogeneous Dirichlet boundary condition for a nonnegative function f with its support concentrated near a point on the left hand side. The dark part shows the region where u < 0.x Preface Preface xi in bounded domains is to understand whether the results available in the simplest case m = 1 can also be proved for any m, or whether the results for m = 1 are special, in particular as far as positivity and the use of maximum principles are concerned. The differential equation ( * ) is complemented with suitable boundary conditions. As already mentioned above, if m = n = 2, equation ( * ) may be considered as a nonlinear plate equation for plates subject to nonlinear feedback forces, one may think e.g. of suspension bridges. In this case, ( * ) may also be interpreted as a reactiondiffusion equation, where the diffusion operator Δ 2 refers to (linearised) surface diffusion.The first part of Chapter 7 is devoted to the proof of symmetry results for positive solutions to ( * ) in the ball under Dirichlet boundary conditions. As already mentioned, truncation and reflection methods do not apply to higher order problems so that a suitable generalisation of the moving planes technique is needed here. Equation ( * ) deserves a particular attention when f (u) has a power-type behaviour. In this case, a crucial role is played by the critical power s = (n + 2m)/(n − 2m) which corresponds to the critical (Sobolev) exponent which appears whenever n > 2m. Indeed, subcritical problems in bounded domains enjoy compactness properties as a consequence of the Rellich-Kondrachov embedding theorem. But compactness is lacking when the critical growth is attained and by means of Pohožaev-type identities, this gives rise to many interesting phenomena. The existence theory can be developed similarly to the second order case m = 1 while it becomes immediately quite difficult to prove positivity or nonexistence of certain solutions. Nonexistence phenomena are related to so-called critical dimensions introduced by Pucci-Serrin [348,349]. They formulated an interesting conjecture concerning these critical dimensions. We give a proof of a relaxed form of it in Chapter 7. We also give a functional analytic interpretation of these nonexistence results, which is reflected in the possibility of adding L 2 -remainder terms in Sobolev inequalities with critical exponent and optimal constants. Moreover, the influence of topological and geometrical properties of Ω on the solvability of the equation is investigated. Also applications to conformal geometry, such as the Paneitz-Branson equation, i...

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“…In the last inequality we have used (25). Now, we want to estimate i,j,l F ij w ijl u l and i,j F ij w ij11 respectively.…”

Section: Estimatesmentioning

confidence: 99%

“…Another proof was given by Catino-Djadli in [5]. See also [2], [8], [12], [19], [20], [25], and especially a survey paper [17] for other applications. This paper can be seen as a new application of this analysis.…”

Section: Introductionmentioning

confidence: 99%

A new conformal invariant on 3-dimensional manifolds

Ge¹,

Wang²

2013

Advances in Mathematics

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By improving the analysis developed in the study of σ k -Yamabe problem, we prove in this paper that the De Lellis-Topping inequality is true on 3-dimensional Riemannian manifolds of nonnegative scalar curvature. More precisely, if (M 3 , g) is a 3-dimensional closed Riemannian manifold with non-negative scalar curvature, thenis the average of the scalar curvature R of g. Equality holds if and only if (M 3 , g) is a space form. We in fact study the following new conformal invariantwhere C1([g0]) := {g = e −2u g0 | R > 0} and prove that Y ([g0]) ≤ 1/3, which implies the above inequality.

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σk-Scalar curvature and eigenvalues of the Dirac operator

Abstract: Abstract. On a 4-dimensional closed spin manifold (M 4 , g), the eigenvalues of the Dirac operator can be estimated from below by the total σ2-scalar curvature of M 4 as followsEquality implies that (M 4 , g) is a round sphere and the corresponding eigenspinors are Killing spinors.

Cited by 5 publications

References 19 publications

Branson's Q-curvature in Riemannian and Spin Geometry

Branson's Q-curvature in Riemannian and Spin Geometry

Polyharmonic Boundary Value Problems

A new conformal invariant on 3-dimensional manifolds

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