Convex hypersurfaces with pinched second fundamental form

Convex Hypersurfaces with Pinched Principal Curvatures and Flow of Convex Hypersurfaces by High Powers of Curvature

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous ...

On Submanifolds with Tamed Second Fundamental Form

Based on the ideas of Bessa-Jorge-Montenegro [4] we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as subma...

Hypersurfaces of a Sasakian space form with recurrent shape operator

Let $(M^{2n},g)$ be a real hypersurface with recurrent shapeoperator and tangent to the structure vector field $xi$ of the Sasakian space form$widetilde{M}(c)$. We show that if the shape operator $A$ of $M$ isrecurrent then it is parallel. Moreover, we show that $M$is locally a product of two constant $phi-$sectional curvaturespaces.

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

On Locally Convex Hypersurfaces with Boundary

In this paper we give some geometric characterizations of locally convex hypersurfaces. In particular, we prove that for a given locally convex hypersurface M with boundary, there exists r > 0 depending only on the diameter of M and the principal curvatures of M on its boundary, such that the r-neighbourhood of any given point on M is convex. As an application we prove an existence theorem for ...