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Geometry of spherical minimal submanifolds
Abstract
There are three parts in this dissertation: "First eigenvalue and volume estimate for a compact minimal submanifold", "Isometric immersion of minimal spherical submanifolds via the second standard immersion of a sphere" and "Some pinching results about compact submanifolds of a sphere". In the first part, let $M\sp{n}$ be a n-dimensional submanifold of a Riemannian manifold $\bar M\sp{m}$ of dimension m, we introduce two isoperimetric constants $c\sb{k}$ and $d\sb{k}$, and estimate the upper bound of the first eigenvalue and the lower bound of the volume for a compact connected k-dimensional minimal submanifold $K\sp{k}$ of $M\sp{n}$ in terms of these two constants. For many important special manifolds such as a minimal submanifold of a sphere, a Kaehler Einstein submanifold of the complex projective space $CP\sp{m}$, a submanifold of a compact symmetric space of rank one and a submanifold of finite type, the isoperimetric constants $c\sb{k}$ and $d\sb{k}$ can be estimated explicitly. As a byproduct, we also generalize some well-known results about the first eigenvalue and volume estimate, and generalize some inequalities for total mean curvature. In the second part, let $M\sp{n}$ be a n-dimensional compact connected minimal submanifold of the sphere $S\sp{n + p}$(1), we study the isometric immersion of $M\sp{n}$ into SM(n + p + 1) via the second standard immersion of $S\sp{n + p}$(1). We obtain some integral inequalities in terms of the spectrum of the Laplace operator of $M\sp{n}$, and find some obstructions for the existence of such immersions. In the last part, we generalize some rigidity theorems of minimal submanifolds of a sphere to compact submanifolds(not necessary minimal) of a sphere. The main results include an improved Simons' inequality and a global pinching theorem for a compact hypersurface of a sphere.
Subject Area
Mathematics
Recommended Citation
Zhang, Xin-Min, "Geometry of spherical minimal submanifolds" (1990). Doctoral Dissertations Available from Proquest. AAI9110235.
https://scholarworks.umass.edu/dissertations/AAI9110235