說明 
115 p 
附註 
Source: Dissertation Abstracts International, Volume: 5204, Section: B, page: 2084 

Adviser: Fred Almgren, Jr 

Thesis (Ph.D.)Princeton University, 1991 

In this thesis we develop a new notion of second variation of area. The classical use of second variation as a tool in geometry has been most successful for either curves or hypersurfaces. This thesis exploits the normal currents of geometric measure theory. The basic construction is a natural oneparameter family of variations of normal currents which preserve real homology class. Intuitively, the current is varied in all directions simultaneously by smearing it out into a t radius neighborhood of itself. The use of normal currents avoids the two main difficulties of the classical second variation in general codimension, which are the noncompactness of the space of ordinary surfaces and the inability to find suitable variational vector fields. Additionally, normal currents enable one to vary surface energies based on either the Euclidean norm or the more subtle mass norm. The usual compactness theorems for currents guarantee the existence of mass and Euclidean norm minimizers 

At the heart of this work is a crucial cancellation estimate. In the second variational formula for normal currents, curvature terms appear with special spatial symmetries. These symmetries are exploited by means of estimates on the local product structure of normal currents without boundary to show that some of the important terms in the formula in fact cancel each other. The terms which cancel are welldefined and arise naturally 

The main cancellation theorem and other techniques lead to the following results for compact Riemannian manifolds: (1) a new proof of Berger's theorem that the 2$\sp{nd}$ and ($n$ $$ 2)$\sp{nd}$ betti numbers vanish in a $\delta$pinched manifold ${\cal N}\sp{n}$ if $\delta >$ ${2(m1)\over 8m5}$ (${1\over4})$ where n = 2m + 1 (2m), (2) a new proof of Meyer's positive definite curvature operator theorem and a new theorem that positive semidefinite curvature operator with positive sectional curvature implies the vanishing of the 2$\sp{nd}$ and ($n$ $$ 2)$\sp{nd}$ betti numbers for odd n, (3) a nonexistence theorem for simple Euclidean norm minimizers in real homology classes of any dimension 0 $$ 0 

School code: 0181 

DDC 
Host Item 
Dissertation Abstracts International 5204B

主題 
Mathematics


0405

Alt Author 
Princeton University

