Thumbnail
Access Restriction
Open

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


Open content in new tab

   Open content in new tab