Reilly-type spinorial inequalities
\textit{C. Bär} [Ann. Glob. Anal. Geom. 16, 573--596 (1998; Zbl 0921.58065)] showed that the first eigenvalue (and in fact a certain number of eigenvalues) of the Dirac operator on an oriented hypersurface \(M\) of \({\mathbb R}^{n+1}\) satisfies the upper bound \[ \lambda_1^2 \leq {n^2 \over 4 \text{vol}(M)} \int_M H^2\, dV \] where \(H\) denotes the mean curvature. Similarly, for an oriented hypersurface \(M \subset S^{n+1}\) of the sphere of constant sectional curvature \(1\), one has \[ \lambda_1^2 \leq {n^2 \over 4 \text{vol}(M)} \int_M (H^2+1)\, dV . \] These estimates are sharp since they are equalities for geodesic spheres. One can now conjecture that for an oriented hypersurface of hyperbolic space one has the estimate \[ \lambda_1^2 \leq {n^2 \over 4 \text{vol}(M)} \int_M (H^2-1)\, dV . \tag{*} \] Bär only showed the weaker estimate \[ | \lambda_1| \leq {n \over 2}(1+ \| H \| _{L^\infty}) \] in this case. For the proof one uses Killing spinors on the ambient manifolds, restricts them to the hypersurface and uses them as test spinors for which the Rayleigh quotient can be computed. In the present paper the author improves the last estimate to \[ \lambda_1^2 \leq {n^2 \over 4 \text{\,vol}(M)} \int_M (H^2-1)\, dV + {1 \over 4 \,\text{vol}(M)} \| du \| _{L^2}^2 \] where \(u\) is the restriction to \(M\) of the function on hyperbolic space for which the metric \({\bar g} = e^{2u} g_h\) has constant curvature \(+1\). Here \(g_h\) denotes the metric of hyperbolic space. The proof uses again Killing spinors as test spinors plus formulas for Dirac operators under conformal change of the metric. The author performs the conformal change of the metric in order to work with Killing spinors on the sphere which have constant length in contrast to those on hyperbolic space. This is a technical advantage. The estimate is sharp for some geodesic spheres. Whether or not \((*)\) holds remains an open question.
scientific article; zbMATH DE number 1922560