THE FIRST EIGENVALUE OF AN IRREDUCIBLE HOMOGENEOUS MANIFOLD

Let M be an n-dimensional compact minimal submanifold in the unit sphere. It is shown that the diameter and volume of M satisfyd≥π/2+C(n)d~n/(d~n+V).An application is that if M is an n-dimensional compact irreducible homogeneous manifold, the first eigenvalue λ_1 of M satisfiesλ_1≥n/d~2(π/2+C(n)d~n/(d~n+V))~2.In the above two cases, C(n)'s are the same constants depending only on n.

THE FIRST EIGENVALUE OF AN IRREDUCIBLE HOMOGENEOUS MANIFOLD

Let $M$ be an n-dimensional compact minimal submanifold in the unit sphere. It is shown that the dismeter and volnme of $M$ satisfy $$\d \ge \frac{\pi }{2} + C(n)\frac{{{d^n}}}{{{d^n} + V}}\]$$ An application is that if $M$ is an n-dimensional compact irreducible homogeneous manifold, the first eigenvalue $\{\lambda _1}\]$ of $M$ satisfies In the above two eases, $\C{(n)^'}\]$ are the same constants depending only on n.