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On the topology of conformally compact Einstein 4-manifolds
In this paper we study the topology of conformally compact Einstein
4-manifolds. When the conformal infinity has positive Yamabe invariant and the
renormalized volume is also positive we show that the conformally compact
Einstein 4-manifold will have at most finite fundamental group. Under the
further assumption that the renormalized volume is relatively large, we
conclude that the conformally compact Einstein 4-manifold is diffeomorphic to
and its conformal infinity is diffeomorphic to .Comment: 16 page
Applications of degree estimate for subalgebras
Let be a field of positive characteristic and be the free
algebra of rank two over . Based on the degree estimate done by Y.-C. Li and
J.-T. Yu, we extend the results of S.J. Gong and J.T. Yu's results: (1) An
element is a test element if and only if does not
belong to any proper retract of ; (2) Every endomorphism preserving the
automorphic orbit of a nonconstant element of is an automorphism; (3)
If there exists some injective endomorphism of such that
where , then is a coordinate. And
we reprove that all the automorphisms of are tame. Moreover, we also
give counterexamples for two conjectures established by Leonid Makar-Limanov,
V. Drensky and J.-T. Yu in the positive characteristic case.Comment: 12 page
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