MARC 主機 00000nam  2200313   4500 
001    AAI9127090 
005    20051208095304.5 
008    051208s1991                        eng d 
035    (UnM)AAI9127090 
040    UnM|cUnM 
100 1  Steinke, John Martin 
245 14 The second variation of normal currents 
300    115 p 
500    Source: Dissertation Abstracts International, Volume: 52-
       04, Section: B, page: 2084 
500    Adviser: Fred Almgren, Jr 
502    Thesis (Ph.D.)--Princeton University, 1991 
520    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 one-parameter 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 non-compactness 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 
520    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 well-defined and arise 
520    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 $\delta >$ 
       ${2(m-1)\over 8m-5}$ (${1\over4})$ 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 non-
       existence theorem for simple Euclidean norm minimizers in 
       real homology classes of any dimension 0 $$ 0 
590    School code: 0181 
590    DDC 
650  4 Mathematics 
690    0405 
710 20 Princeton University 
773 0  |tDissertation Abstracts International|g52-04B 
856 40 |u