MASS ENDOMORPHISM AND SPINORIAL YAMABE TYPE PROBLEMS ON CONFORMALLY FLAT MANIFOLDS
MASS ENDOMORPHISM AND SPINORIAL YAMABE TYPE PROBLEMS ON CONFORMALLY FLAT MANIFOLDS
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DOI：
10.4310/cag.2006.v14.n1.a7
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发表时间：
2005-03
影响因子：
0.7
通讯作者：
B. Ammann;E. Humbert;B. Morel
中科院分区：
文献类型：
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作者：
B. Ammann;E. Humbert;B. Morel
设M是具有黎曼度量g和自旋结构的紧致流形。设[g]\lambda(M，[g]，\si)=\inf_(\tilde{g})Vol(M，\tilde{g})^{1/n}$其中$\lambda_1^+(\tilde{g})$是度规$\tilde{g}$中Dirac算子D的最小正本征值。以前的结果表明，$\lambda(M，[g]，\si)\leq\lambda(\ms^n)=\frac{n}{2}\om_n^{{1/n}}$其中\om_n表示标准n球体的体积。本文研究了D可逆的n维共形平坦流形的这一问题。例如，我们证明了如果某一自同态不消失，则严格的不等式在维$n\等价于0，1，2\mod 4$成立。由于它与广义相对论中的ADM质量密切相关，所以这种自同态称为质量自同态。我们将严格的不等式应用到自旋共形谱理论中，证明了最小正狄拉克本征值在g的扩展体积-1-共形类内达到其下确界。
Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $\lambda (M,[g],\si)= \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $\lambda_1^+(\tilde{g})$ is the smallest positive eigenvalue of the Dirac operator D in the metric $\tilde{g}$. A previous result stated that $\lambda(M,[g],\si) \leq \lambda(\mS^n) =\frac{n}{2} \om_n^{{1/n}}$ where \om_n stands for the volume of the standard n-sphere. In this paper, we study this problem for conformally flat manifolds of dimension n \geq 2 such that D is invertible. E.g. we show that strict inequality holds in dimension $n\equiv 0,1,2\mod 4$ if a certain endomorphism does not vanish. Because of its tight relations to the ADM mass in General Relativity, the endomorphism will be called mass endomorphism. We apply the strict inequality to spin-conformal spectral theory and show that the smallest positive Dirac eigenvalue attains its infimum inside the enlarged volume-1-conformal class of g.