A spinorial characterization of hyperspheres
Résumé
Let $M$ be a compact orientable $n$-dimensional hypersurface, with nowhere vanishing mean
curvature $H$, immersed in
a Riemannian spin manifold $\overline{M}$ admitting a non trivial parallel spinor field. Then the first eigenvalue $\lambda_1(D_M^H)$ (with the lowest absolute value) of the Dirac operator $D_M^H$ corresponding to the conformal metric $\langle\;,\;\rangle^H=H^2\,\langle\;,\;\rangle$, where $\langle\;,\;\rangle$ is the induced metric on $M$, satisfies $\left|\lambda_1(D_M^H)\right|\le \frac{n}{2}$. By applying
the Bourguignon-Gauduchon first variational formula, we obtain a necessary condition for $\left|\lambda_1(D_M^H)
\right|=\frac{n}{2}$. As a consequence, we prove that round hyperspheres are the only hypersurfaces of the Euclidean space satisfying the equality in the B\"ar inequality$$
\lambda_1(D_M)^2\le \frac{n^2}{4\hbox{vol}(M)}\int_M H^2\, dV,$$
where $D_M$ stands now for the Dirac operator of the induced metric.
Domaines
Géométrie différentielle [math.DG]
Origine : Fichiers produits par l'(les) auteur(s)
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