International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 1, Pages 43-48
doi:10.1155/S0161171200003306
First eigenvalue of submanifolds in Euclidean space
Department of Mathematics, Hangzhou Teachers' College, 96 Wen Yi Road, Hangzhou 310012, China
Received 23 April 1999
Copyright © 2000 Kairen Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.