scispace - formally typeset
Search or ask a question

Showing papers on "Scalar curvature published in 1986"



Journal ArticleDOI
TL;DR: In this article, it was shown that the mass of an asymptotically flat n-manifold is a geometric invariant and the proof was based on harmonic coordinates.
Abstract: We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic operators with rough coefficients on weighted Sobolev spaces are summarised. Some relations between the mass, scalar curvature and harmonic maps are described and the positive mass theorem for n-dimensional spin manifolds is proved.

826 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that if the ambient space N is a general Riemannian manifold, the curvature of N will interfere with the mot ion of the surfaces M o o R, + 1.
Abstract: which shrink towards the center of the initial sphere in finite time. It was shown in [3], that this behaviour is very typical: If the initial hypersurface M o o R , + 1 is uniformly convex, then the surfaces M t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction. If the ambient space N is a general Riemannian manifold, the curvature of N will interfere with the mot ion of the surfaces M r We want to show here that the contract ion first to a small sphere and then to a single point is still

276 citations


Journal ArticleDOI
TL;DR: In this paper, a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator) was given.
Abstract: On a Riemannian spin manifold, we give a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator). We prove, in the limiting case, that the eigenspinor field is a killing spinor, i.e., parallel with respect to a natural connection. In particular, if the scalar curvature is positive, the eigenspinor field is annihilated by harmonic forms and the metric is Einstein.

272 citations



Journal ArticleDOI
TL;DR: In this article, the authors discuss the problem of trouver a fonction on a variete de Riemann compacte and the difficulty of enforcing it in a Courbure scalaire.
Abstract: Soit (M n ,g) une variete de Riemann compacte et R une fonction reguliere sur M. On considere le probleme de trouver une fonction v sur M telle que la metrique exp(2v)•g a une courbure scalaire K. On considere seulement le cas ou la courbure scalaire de g est non negative

187 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduce the concept of space of curvature in the Riemannian structure and smoothness of the intrinsic metric in spaces of bounded curvature both and in parallel translation.
Abstract: CONTENTS Introduction Chapter I. Basic concepts connected with the intrinsic metric § 1. Basic definitions § 2. General propositions on upper angles Chapter II. Spaces of curvature () § 3. Basic properties of an -domain § 4. Constructions § 5. Equivalent definitions of an -domain § 6. Area and the isoperimetric inequality § 7. Plateau's problem § 8. Spaces of curvature Chapter III. The space of directions § 9. The space of directions at a point in § 10. The tangent space Chapter IV. Spaces of bounded curvature § 11. Spaces of curvature both and § 12. Introduction of the Riemannian structure Chapter V. Smoothness of the metric in spaces of bounded curvature § 13. Parallel translation § 14. Smoothness of the metric References

182 citations





Journal ArticleDOI
TL;DR: In this paper, a metrique auto-duale a courbure scalaire positive sur une variete compacte simplement connexe qui n'est pas conformement equivalente to ces metriques standard sur S 4 ou CP 2.
Abstract: On etudie l'existence d'une metrique auto-duale a courbure scalaire positive sur une variete compacte simplement connexe qui n'est pas conformement equivalente a ces metriques standard sur S 4 ou CP 2

Journal ArticleDOI
A. J. Tromba1
TL;DR: In this paper, the sectional curvature of Teichmuller space with respect to the Weil-Petersson metric is derived in terms of the Laplace-Beltrami operator on functions.
Abstract: A formula for the sectional curvature of Teichmuller space with respect to the Weil-Petersson metric is derived in terms of the Laplace-Beltrami operator on functions. It will be shown that the sectional curvature as well as the holomorphic sectional curvature and Ricci curvature are negative. Bounds on the holomorphic and the Ricci curvature are given.

Journal ArticleDOI
TL;DR: In this article, the covariant derivative of the fundamental 4-form of Riemannian manifolds with structure group Spin (7) is studied, and it is shown that there are precisely four classes of such manifolds.
Abstract: Riemannian manifolds with structure group Spin (7)are 8-dimensional and have a distinguished 4 -form. In this paper, the covariant derivative of the fundamental 4 -form is studied, and it is shown that there are precisely four classes of such manifolds.

Journal ArticleDOI
TL;DR: In this article, it was shown that if M is a complete Riemannian manifold with Ricci curvatures all nonnegative, then it has the Brownian coupling property.
Abstract: This paper shows that if M is a complete Riemannian manifold with Ricci curvatures all nonnegative than M has the Brownian coupling property. From this one may immediately draw deductions concerning the nonexistence of certain harmonic maps


Journal ArticleDOI
01 Apr 1986
TL;DR: For axisymmetric f E C? (S2) with | = 0, the best constant C = Vol(S 2) as mentioned in this paper, where C is the largest constant in the inequality.
Abstract: For axisymmetric f E C? (S2) we find conditions to make f the scalar curvature of a metric pointwise conformal to the standard metric of S2. Closely related to these results, we prove that in the inequality (Moser [8]) 2e" Ce1 Vt&2/167F Vu E H,(S2) with | = 0, the best constant C = Vol(S2).

Book ChapterDOI
Yum-Tong Siu1
01 Jan 1986
TL;DR: For compact Riemann surfaces of genus at least two, using Petersson's Hermitian pairing for automorphic forms, Royden as mentioned in this paper showed that the holomorphic sectional curvature of the Weil-Petersson metric is bounded away from zero.
Abstract: For compact Riemann surfaces of genus at least two, using Petersson’s Hermitian pairing for automorphic forms, Weil introduced a Hermitian metric for the Teiclmuller space, now known as the Weil-Petersson metric. Ahlfors [1,2] showed that the Weil-Petersson metric is Kahler and that its Ricci and holomorphic section curvatures are negative. By using a different method of curvature computation, Royden [8] later showed that the holomorphic sectional curvature of the Weil-Petersson metric is bounded away from zero and conjectured the best bound to be \(- \frac{1}{{2\pi \left( {g - 1} \right)}}\) , where g is the genus. Recently Wolpert [12] and also Royden proved Royden’s conjecture on the bound of the holomorphic sectional curvature and obtained in addition the negativity of the Riemannian sectional curvature. Wolpert’s method used some SL(2,R) invariant first-order differential operators obtained by Maass [7]. Royden’s computation is based on the fact that the Pbincare metric on a compact Riemann surface of genus at least two is Einstein.




Journal ArticleDOI
TL;DR: In this paper, the authors investigated the Einstein equation for a class of (4n+4)-dimensional SU(2)-invariant metrics on S4 and R4 fiber bundles over quaternionic Kahler base manifolds.
Abstract: The authors investigate the Einstein equation for a class of (4n+4)-dimensional SU(2)-invariant metrics on S4 and R4 fibre bundles over quaternionic Kahler base manifolds. Using numerical techniques, they establish the existence of complete compact inhomogeneous Einstein spaces of this form with positive Ricci curvature, and complete non-compact inhomogeneous Einstein spaces with zero or negative Ricci curvature.

Journal ArticleDOI
TL;DR: There is a unique Lagrangian quadratic in the curvature tensor which yields second-order field equations in dimensions greater than four as discussed by the authors, which is applied to a Kaluza-Klein theory.
Abstract: There is a unique Lagrangian quadratic in the curvature tensor which yields second-order field equations in dimensions greater than four. This Lagrangian is applied to a Kaluza-Klein theory and its cosmological implications are investigated.







Journal ArticleDOI
TL;DR: In this article, the curvature of the Weil-Petersson metric of the moduli space of general compact polarized Kahler-Einstein manifolds of zero first Chern class was studied.
Abstract: We study the problem of computing the curvature of the Weil-Petersson metric of the moduli space of general compact polarized Kahler-Einstein manifolds of zero first Chern class. We use canonical lifting of vector fields from the moduli space to the total deformation space to obtain a formula for the curvature of the Weil-Petersson metric. From this formula we obtain negative bisectional curvature for certain directions. This formula also reprove and explain the recent result of Schumacher that the holomorphic sectional curvature of the Weil-Petersson metric for K3-surfaces and symplectic manifolds are negative.