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作者 Steinke, John Martin
書名 The second variation of normal currents
說明 115 p
附註 Source: Dissertation Abstracts International, Volume: 52-04, 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 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
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
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
School code: 0181
Host Item Dissertation Abstracts International 52-04B
主題 Mathematics
Alt Author Princeton University
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