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
naturally
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 |uhttp://pqdd.sinica.edu.tw/twdaoapp/servlet/
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