First eigenvalue of submanifolds in Euclidean space

Abstract

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

Keywords

• Laplacian
• eigenvalue
• submanifold.