1995 Fiscal Year Final Research Report Summary
Convergence Theory for Alexandrov Spaces
| Project/Area Number |
06640155
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
Geometry
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
YAMAGUCHI Takao Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究科, 教授 (00182444)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOYA Takashi Kyushu Univ., Graduate School of Math., Associ.Prof., 大学院・数理学研究科, 助教授 (90235507)
TANAKA Shunichi Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究科, 教授 (00028127)
SHIOHAMA Katsuhiro Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究科, 教授 (20016059)
CHOU Kanchi Kyushu Univ., Graduate School of Math., Associ.Prof., 大学院・数理学研究科, 助教授 (10197634)
SATOU Hiroshi Kyushu Univ., Graduate School of Math., Professor, 大学院・数理学研究科, 教授 (30037254)
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| Project Period (FY) |
1994 – 1995
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| Keywords | Alexandrov space / Gromov imariant / Singular space / Hausdorff convergence / singular set / isometry group |
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| Research Abstract |
In the case where the singularities of Alexandrov spaces with curvature bounded below are not so big, under convergence of spaces. we were able to construct Lipschitz homeomorphisms between spaces. In particalar, the continuity of volumes of Alexandrov spaces follows from this result. Moreover, we proved that the Hausdorff measure of the singular set of an Alexandrov space is zero, and that one can define a natural Riemannian structure on the regular set. We also proved that the isometry group of an Alexandrov space with curvature bounded below is a lie group, which has some applications to Riemannian geometry. On the other hand, we extended the notion of the Gromov invariant to Alexandrov spaces, and clarified the relation between the curvature, volume and the Gromov invariant. First, making use of the Alexander-Spanier cohomology theory, we proved the existence of the fundamental class [X] of X, and defined the Gromov invariant of X.Next, we proved that the mass of the fundamental class [X] coincides with the volume of X.In the proof of this face, we used geometric measure theory to approximate a chain representing [X] in the mass topology by a Lipschitz chain with nice properties, and developed a cancellation technique which might be considered as a replacement of Stokes' theorem. And we proved that the Gromov invariant of a negatively curved Alexandrov space can be estimated below interms of the upper bound of curvature and the volume. In the case of Alexandrov surfaces, we obtained a sharp estimate for the Gromov invariant with the type of singularities. For Alexandrov spaces with curvature bounded below, we bave an estimate for the Gromov invariant from above in terms of the volume and the lower bound of curvature. Thus it turned out that the appearanceo of singularities of such a space does not affect the Gromov invariant so much. This shows the big difference between the two cases, spaces curved above and spaces curved below.
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