A Yamabe type problem on compact spin manifolds
Let \((M,g,\sigma)\) be a compact spin manifold of dimension \(n\geq 2\). For a metric \(\tilde{g}\) in the conformal class \([g]\) of \(g\), let \(\lambda_1^+(\tilde{g})\) [resp. \(\lambda_1^-(\tilde{g})\)] be the smallest positive (resp. largest negative) eigenvalue of the Dirac operator \(D\) in the metric \(\tilde{g}\). The conformal invariants \(\lambda_{\min}^+(M,[g],\sigma):=\inf_{\tilde{g}\in [g]}\lambda_1^+(\tilde{g}) \text{Vol}(M,\tilde{g})^{1/n}\) and \(\lambda_{\min}^-(M,[g],\sigma):=\inf_{\tilde{g}\in [g]}| \lambda_1^-(g)|\text{Vol}(M,\tilde{g})^{1/n}\) have been studied by many authors [for example, see \textit{O. Hijazi}, Commun. Math. Phys. 104, 151--162 (1986; Zbl 0593.58040); \textit{J. Lott}, Pac. J. Math. 125, 117--126 (1986; Zbl 0605.58044); \textit{C. Bär}, Math. Ann. 293, 39--46 (1992; Zbl 0741.58046); the first author, ``The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions'' (Preprint, arXiv math. DG/0309061) and ``A variational problem in conformal spin geometry'' (Habilitationsschrift, Univ. Hamburg, May 2003)]. The authors show that the inequalities \[ \lambda_{\min}^+(M,[g],\sigma)\leq \lambda_{\min}^+(S^n)=\frac{n}{2}\omega_n^{1/n}\tag{1} \] and \[ \lambda_{\min}^-(M,[g],\sigma)\leq \lambda_{\min}^-(S^n)=\lambda_{\min}^+(S^n)\tag{2} \] are true, where \(\omega_n\) denotes the volume of the unit standard sphere \(S^n\) (see also the first author, loc. cit.). Moreover, if \((M,g)\) is nonconformally flat and \(n\geq 7\), then both inequalities are strict. If \((M,g)\) is conformally flat, \(D\) is invertible and the mass endomorphism [for this concept, see also the authors, ``Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds'' (Preprint Inst. É. Cartan, Nancy 2003/58)] possesses a negative (resp. positive) eigenvalue, then inequality (1) [resp. (2)] is also strict.
scientific article; zbMATH DE number 2093999
scientific article