Positive mass theorem for the Yamabe problem on spin manifolds
TL;DR: In this paper, a modification of Witten's argument is presented, but no analysis on asymptotically flat spaces is needed, and the proof is simplified by Witten with the help of spinors.
Abstract: Let (M, g) be a compact connected spin manifold of dimension n ≥ 3 whose Yamabe invariant is positive. We assume that (M, g) is locally conformally flat or that n ∈ {3, 4, 5}. According to a positive mass theorem by Schoen and Yau the constant term in the asymptotic development of the Green’s function of the conformal Laplacian is positive if (M, g) is not conformally equivalent to the sphere. The proof was simplified by Witten with the help of spinors. In our article we will give a proof which is even considerably shorter. Our proof is a modification of Witten’s argument, but no analysis on asymptotically flat spaces is needed.
Citations
More filters
Book•
[...]
06 Jun 2009
TL;DR: In this article, the Dirac spectrum on non-compact manifolds has been studied in the context of spin geometry and lower and upper eigenvalue bounds on closed manifolds.
Abstract: Basics of spin geometry.- Explicit computations of spectra.- Lower eigenvalue estimates on closed manifolds.- Lower eigenvalue estimates on compact manifolds with boundary.- Upper eigenvalue bounds on closed manifolds.- Prescription of eigenvalues on closed manifolds.- The Dirac spectrum on non-compact manifolds.- Other topics related with the Dirac spectrum.
151 citations
TL;DR: In this paper, the first positive eigenvalue of the Dirac operator on a compact Riemannian spin manifold of dimension ≥ 2 was shown for the case n ≥ 3 and for the remaining case n = 2, ker D = {0}.
Abstract: Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric \(\tilde g\) conformal to g, we denote by \(\tilde\lambda\) the first positive eigenvalue of the Dirac operator on \((M, \tilde g, \sigma)\) . We show that
$${\rm inf}_{\tilde{g} \in [g]} \tilde\lambda {\rm Vol}(M,\tilde g)^{1/n} \leq (n/2) {\rm Vol}(S^n)^{1/n}.$$
This inequality is a spinorial analogue of Aubin’s inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D ≠ {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with \(2\tilde\lambda^2 \leq \tilde\mu\) , where \(\tilde\mu\) denotes the first positive eigenvalue of the Laplace operator.
54 citations
TL;DR: In this article, it was shown that the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold is positive unless the manifold is conformally diffeomorphic to the standard sphere.
Abstract: We prove that under suitable assumptions, the constant term in the Green function of the Paneitz–Branson operator on a compact Riemannian manifold (M, g) is positive unless (M, g) is conformally diffeomorphic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann and Humbert (Geom Func Anal 15(3):567–576, 2005 [1]).
37 citations
Posted Content•
TL;DR: The first positive eigenvalue of the Dirac operator on a Riemannian spin manifold was shown in this paper, where the authors showed that for any metric (m,g,si) conformal to a spin manifold, there exists a metric with dimension 2.
Abstract: Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tilde\lambda$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tilde\lambda \Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ This inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case $n \geq 3$ and in the case $n = 2$, $\ker D=\{0\}$. Our proof also works in the remaining case $n=2$, $\ker D
eq \{0\}$. With the same method we also prove that any conformal class on a Riemann surface contains a metric with $2\tilde\lambda^2\leq \tilde\mu$, where $\tilde\mu$ denotes the first positive eigenvalue of the Laplace operator.
36 citations
Journal Article•
TL;DR: In this paper, the authors studied the dependence of the zero sets of eigenspinors of the Dirac operator on the Riemannian metric and proved that on closed spin manifolds of dimension 2 or 3 for a generic RiemANNian metric the nonharmonic eigenpinors have no zeros.
Abstract: We consider a Riemannian spin manifold (M, g) with a fixed spin structure. The zero sets of solutions of generalized Dirac equations on M play an important role in some questions arising in conformal spin geometry and in mathematical physics. In this setting the mass endomorphism has been defined as the constant term in an expansion of Green's function for the Dirac operator. One is interested in obtaining metrics, for which it is not zero.
In this thesis we study the dependence of the zero sets of eigenspinors of the Dirac operator on the Riemannian metric. We prove that on closed spin manifolds of dimension 2 or 3 for a generic Riemannian metric the nonharmonic eigenspinors have no zeros. Furthermore we prove that on closed spin manifolds of dimension 3 the mass endomorphism is not zero for a generic Riemannian metric.
21 citations
References
More filters
TL;DR: In this article, a new proof of the positive energy theorem of classical general relativity was given and it was shown that there are no asymptotically Euclidean gravitational instantons.
Abstract: A new proof is given of the positive energy theorem of classical general relativity. Also, a new proof is given that there are no asymptotically Euclidean gravitational instantons. (These theorems have been proved previously, by a different method, by Schoen and Yau.) The relevance of these results to the stability of Minkowski space is discussed.
1,714 citations
TL;DR: In this paper, it was shown that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat, which is the assumption of the existence of a maximal spacelike hypersurface.
Abstract: LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.
1,473 citations
[...]
1,317 citations
TL;DR: In this paper, a new global idea was introduced to solve the Yamabe problem in dimensions 3, 4, and 5, and the existence of a positive solution u on M of the equation was proved in all remaining cases.
Abstract: A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 that Yamabe's paper was seriously incorrect. Trudinger was able to correct Yamabe's proof in case the scalar curvature is nonpositive. Progress was made on the case of positive scalar curvature by T. Aubin [1] in 1976. Aubin showed that if dim M > 6 and M is not conformally flat, then M can be conformally changed to constant scalar curvature. Up until this time, Aubin's method has given no information on the Yamabe problem in dimensions 3, 4, and 5. Moreover, his method exploits only the local geometry of M in a small neighborhood of a point, and hence could not be used on a conformally flat manifold where the Yamabe problem is clearly a global problem. Recently, a number of geometers have been interested in the conformally flat manifolds of positive scalar curvature where a solution of Yamabe's problem gives a conformally flat metric of constant scalar curvature, a metric of some geometric interest. Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of S X S~. In this paper we introduce a new global idea into the problem and we solve it in the affirmative in all remaining cases; that is, we assert the existence of a positive solution u on M of the equation
1,303 citations
Journal Article•
944 citations