Scalar Curvature Gap Study The Eigenvalue Problem

In this paper, we mainly study the gap of scalar curvature of a minimal hypersurface in a unit sphere and the eigenvalue problem of Riemannian manifolds.Let Mⁿ be an n-dimensional compact minimal hypersurface in the unit sphere Sⁿ⁺¹. Denote by 5 the square of the length of the second fundamental form of M and it can be recognized as a function on M. In 1983, Peng, C.K. and Terng, C.L found the second gap of S in the case that n ≤ 5. The proof of theorem 1 in the third section shows the gap when n ≤ 7:Main Theorem A: Let Mⁿ be an n-dimensional compact minimal hypersurface in the unit sphere Sⁿ⁺¹. Then there exists a positive constant 5(n) depending only on n such that when n ≤ S(x) ≤ n + δ(n), S(x) ≡ n.For the above gap problem, Ogiue and Sun claimed to have proved the theorem for arbitrary n case. But there is a fatal mistake in their proof. In the fourth section of this thesis, we cite their sketch of the proof, pointing out its mistake and give a corresponding counter example.In the fifth section of this thesis, we add some special conditions for the mean curvature under which a second gap can be found.We have already had many wonderful estimates of higher eigenvalues on minimal submanifolds as stated in [29]. In the final section, we estimate the higher eigenvalues on compact Riemannian manifolds. Theorem 3 mainly studies the eigenvalue problems on a class of manifolds with boundary. Theorem 4 estimates the lower bounds of higher eigenvalues of compact submanifolds without boundary in a complete, simply connected Riemannian manifold with nonpositive scalar curvature. Theorem 5 gives a Lieb-Li-Yau's type estimate of the Dirichlet boundary value problems of a small domain with boundary in Riemannian manifolds:Main Theorem B: Let Mⁿ(n ≥ 3) be an-dimensional Riemannian manifold with K_m ≤ b, Ric_m ≥ — (n —1) a where a,b > 0. D (?) M is a compact connected domain with boundary. If D is included in a geodesic...