Exploring new extrinsic upper bounds on the first eigenvalue of the Laplace operator for compact submanifolds in Euclidean spaces

Abstract

Upper bounds of the first non-trivial eigenvalue ${\lambda }_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space ${\mathbb{R}}^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of ${\lambda }_1$ depends on the length of mean curvature vector field, the dimension $n$, the volume of $M^n$, and of a vector of ${\mathbb{R}}^{m+1}$. When $M^n$ does not lie minimally in a hypersphere of ${\mathbb{R}}^{m+1}$, classical Reilly's inequality \cite{Re} is improved and new upper bounds are explicitly computed. For instance, considering a torus of revolution whose generating circle has a radius of $1$ and is centered at distance $\sqrt{2}$ from the axis of revolution, we find ${\lambda }_1<\frac{4}{3}(\sqrt{2}-1)\approx 0.552284$, whereas Reilly's upper bound gives ${\lambda }_1<1/\sqrt{2}\approx 0.707106$.