In this paper, we prove that there exists a universal constant C, depending only on positive integers n ≥ 3 and p ≤ n − 1, such that if M n is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball B n+k whose size of the traceless second fundamental form is less than C, then the pth cohomology group of M n vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball B 2+k .

In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem [9] to self-shrinkers, that is:Let P be a hyperplane passing through the origin. The only properly immersed self-shrinker Σ contained in one of the closed half-space determined by P is Σ = P .Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller [11]. Also, using a similar geometric idea, we obtain that the only complete self-shrinker properly immersed in an closed cylinder B k+1 (R) × R n−k ⊂ R n+1 , for some k ∈ {1, . . . , n} and radius R, R ≤ √ 2k, is the cylinder S k ( √ 2k) × R n−k . We also extend the above results for λ−hypersurfaces.

Let x : M m →M , m ≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply-connected manifoldM with sectional curvature satisfying −k 2 ≤ KM ≤ 0, for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L m -norm. If KM ≡ 0, assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L 2 harmonic 1-forms on M has finite dimension. Moreover there exists a constant Λ > 0, explicitly computed, such that if the total curvature is bounded from above by Λ then there is no nontrivial L 2 -harmonic 1-forms on M .

Abstract. In this paper we obtain lower bound estimates of the spectrum of Laplace-Beltrami operator on complete submanifolds with bounded mean curvature, whose ambient space admits a Riemannian submersion over a Riemannian manifold with negative sectional curvature. Our main theorem generalizes many previous known estimates and applies for both immersions and submersions.

No abstract