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Register a new userA New Proof of the Gromovs Theorem on Almost Flat Manifolds
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Authors
Xiaochun Rong
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We will present a new proof for the Gromovs theorem on almost flat manifolds ([Gr], [Ru]).
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We give a new proof of Gromovs theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $ge n-1$. In cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)to 0$; 2) the volume of $M$ ${text{Vol}}(M)to{text{Vol}}(S^n)$; 3) the radius of $M$ ${text{rad}}(M)topi$ are equivalent. In cite{peter}, Peter Petersen, by developing a different technique, gave the 4-th equivalent condition, namely he proved that the $n+1$-th eigenvalue of $M$ $lambda_{n+1}(M)to n$ is also equivalent to the radius of $M$ ${text{rad}}(M)topi$, and hence the other two. In this note, we give a new proof of Petersens theorem by utilizing Coldings techniques.
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We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence Kahler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the conjecture of Draghici-Li-Zhang in the almost-Kahler case
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