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Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2013

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.
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Dates et versions

hal-01270168 , version 1 (08-02-2016)

Identifiants

Citer

Oussama Hijazi, Sebastián Montiel. A spinorial characterization of hyperspheres. Calculus of Variations and Partial Differential Equations, 2013, 48, pp.527-544. ⟨10.1007/s00526-012-0560-x⟩. ⟨hal-01270168⟩
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