Minimal Volume Of Manifold

The main work of the thesis is to give a concrete proof of Min Vol(R~n) = 0 for n≥3.We only prove the two cascs:R~3 and R~4 which play an important role in our method.In general case,R~n can be written as a produet manifold of several R~3's and R~4's and R~2.We first show that there is no any rotational symmetrie: complete metrics with bounded curvatures on R~3 making the volumes tending to zero(R~3 seems not to admit,a complete finite-volume metrie of bounded curvature and 'cusp-like' end.See[Bowditch1993]).By using the topological decomposition constructed in[Cheeger-Gromov1985],we construet the required metrics on every small piece and gluing them together in certain order.It is worth to point out that such metrics are complete.For R~4,we view it as R~3×R and consider the wrapped product metrics on it.Here the metrics on R~3 are based on the metrics constructed as before.We also prove that Min Vol(R~n#R~n) = 0 for n≥2.Of course,we need to use the method of smooth gluing of metrics discussed as before.In chapter 0,we also introduce the other two invariants:Gromov volume (also called simplicial volume) and minimal entropy,which are closely related to minimal volume.We also introduce the definitions of(?)-strueture,F-structure and T-structure.[Gromov1982,Appendix I]is explained in detail in Appendix A.Appendix B gives an introduetion to holonomy action on principal bundle and the structure equation on principal bundle which appeared in section 0.3 and 3.3 respectively.