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Paper IPM / M / 127  


Abstract:  
In this paper we study hypersurfaces M_{s}^{n}
in \BbbR_{1}^{n+1} (or in \BbbR^{n+1}) verifying the equation ∆x = Ax+B and the condition that the principal
curvatures of the surface is not (−n ∈ ) times the mean curvature at the points where the mean curvature is nonzero. We prove that the mean curvature of the surface is constant and as a result, either M_{s}^{n} has zero mean curvature or when M_{s}^{n} has at most two different principal curvatures, it is isoparametric. Here M_{s}^{n} is a (pseudo) Riemannian manifold with metric of signature s,s=0,1,\BbbR_{1}^{n+1} is the (n+1)dimensional flat Lorentzian space, A is an endomorphism of \BbbR_{1}^{n+1} and B ∈ \BbbR_{1}^{n+1}, ∆ is the Laplacian operator on M_{s}^{n}, x: M_{s}^{n} → \BbbR_{1}^{n+1} is an isometric immersion. Download TeX format 

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