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Stability of submanifolds with parallel mean curvature in calibrated manifolds

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Abstract

On a Riemannian manifold \( \bar M^{m + n} \) with an (m + 1)-calibration Ω, we prove that an m-submanifold M with constant mean curvature H and calibrated extended tangent space ℝH ⋇ TM is a critical point of the area functional for variations that preserve the enclosed Ω-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when n = 1 and Ω is the volume element of \( \bar M \). To the second variation we associate an Ω-Jacobi operator and define Ω-stability. Under natural conditions, we show that the Euclidean m-spheres are the unique Ω-stable submanifolds of ℝm+n. We study the Ω-stability of geodesic m-spheres of a fibred space form M m+n with totally geodesic (m + 1)-dimensional fibres.

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Authors and Affiliations

  1. Centro de Física das Interações Fundamentais Instituto Superior Técnico, Technical University of Lisbon, Edifício Ciência, Piso 3 Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Isabel M. C. Salavessa

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  1. Isabel M. C. Salavessa
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Correspondence to Isabel M. C. Salavessa.

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Partially supported by FCT through program PTDC/MAT/101007/2008.

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Salavessa, I.M.C. Stability of submanifolds with parallel mean curvature in calibrated manifolds. Bull Braz Math Soc, New Series 41, 495–530 (2010). https://doi.org/10.1007/s00574-010-0023-y

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  • DOI: https://doi.org/10.1007/s00574-010-0023-y

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