### Stability of submanifolds with parallel mean curvature in calibrated manifoldsStability of submanifolds with parallel mean curvature in calibrated manifolds

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 Author Salavessa, Isabel M. C. Source arXiv.org Content type Text File Format PDF Date of Submission 2009-11-24 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Natural sciences & mathematics ♦ Mathematics ♦ Physics Subject Keyword Mathematics - Differential Geometry ♦ Mathematical Physics ♦ 35J19 ♦ Primary: 53C42 ♦ 53C38. Secondary: 58E12 ♦ 47A75 ♦ math ♦ physics:math-ph Abstract On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $\Omega$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $\Omega$-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when $n=1$ and $\Omega$ is the volume element of $\bar{M}$. To the second variation we associate an $\Omega$-Jacobi operator and define $\Omega$-stablility. Under natural conditions, we prove that the Euclidean $m$-spheres are the unique $\Omega$-stable submanifolds of $\mathbb{R}^{m+n}$. We study the $\Omega$-stability of geodesic $m$-spheres of a fibred space form $M^{m+n}$ with totally geodesic $(m+1)$-dimensional fibres. Description Reference: Bull Braz Math Soc (NS) 41(4)(2010), 495-530 Educational Use Research Learning Resource Type Article