@MISC{Berard94simons'equation, author = {Pierre Berard}, title = {Simons' Equation Revisited}, year = {1994} }

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Abstract

In this paper we establish the generalized Simons' equations satisfied by the second fundamental form of a general immersion M p ,! M n . We then specialize these equations to the case of an immersion with constant mean curvature (or parallel mean curvature vector) in a space form. 1 Anais Acad. Brasil. Ciencias 66:4 (1994), 397--403 SIMONS' EQUATION REVISITED Pierre B'erard 1. Introduction In 1968, J. Simons (Simons, 1968) established partial differential equations satisfied by the second fundamental form of a minimal immersion. The second order equation enabled him to prove a gap phenomenon for minimal submanifolds of the sphere. Simons' equation, as it is now called, has been used by various authors in order to study minimal immersions and, more recently, immersions with constant mean curvature. In (B'erard et al. 1994), we needed a similar equation for the second fundamental form of an immersion with constant mean curvature and we thought that it might be useful to make the ...