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Abstract. Let M be a complete immersed minimal hypersurface in a hyperbolic space. In this paper we establish conditions on the first eigenvalue of the stability and super stability operators and the L d norm of the length of the second fundamental form of M to imply that M is totally geodesic. Similar results for minimal submanifolds in a hyperbolic space are also proven.

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In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold M n in the hyperbolic space H n+m in order to show that M n is totally geodesic. We also obtain sharp upper bounds estimates for the first eigenvalue of the super stability operator in the case of M is a surface in H 4 .

The purpose of this article is to study the geometry of gradient almost Yamabe solitons immersed into warped product manifolds I × f M n whose potential is given by the height function from the immersion. First, we present some geometric rigidity on compact solitons due to a curvature condition on the warped product manifold. In the sequel, we investigate conditions for the existence of totally geodesic, totally umbilical and minimal solitons. Furthermore, in the scope of constant angle immersions, a classification of rotational gradient almost Yamabe soliton immersed into R × f R n is also made.