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On manifolds with Ricci curvature lo...

Liu, Gang.

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On manifolds with Ricci curvature lower bound and Kahler manifolds with nonpositive bisectional curvature.

Record Type:	Language materials, printed : Monograph/item
Title/Author:	On manifolds with Ricci curvature lower bound and Kahler manifolds with nonpositive bisectional curvature./
Author:	Liu, Gang.
Description:	112 p.
Notes:	Source: Dissertation Abstracts International, Volume: 74-10(E), Section: B.
Contained By:	Dissertation Abstracts International74-10B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3567448
ISBN:	9781303192562

On manifolds with Ricci curvature lower bound and Kahler manifolds with nonpositive bisectional curvature.
Liu, Gang.

On manifolds with Ricci curvature lower bound and Kahler manifolds with nonpositive bisectional curvature. - 112 p.

Source: Dissertation Abstracts International, Volume: 74-10(E), Section: B.

Thesis (Ph.D.)--University of Minnesota, 2013.

The relation between curvature and topology is a fundamental problem in differential geometry. For example, the Gauss-Bonnet theorem says the sign of curvature could determine the genus of the surface. Brendle and Schoen [8] proved that if a compact manifold has sectional curvature between ¼ and 1, then it is a space form.

ISBN: 9781303192562Subjects--Topical Terms:

515831
Mathematics.

On manifolds with Ricci curvature lower bound and Kahler manifolds with nonpositive bisectional curvature.
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020 $a 9781303192562
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100 1 $a Liu, Gang. $3 1272216
245 1 0 $a On manifolds with Ricci curvature lower bound and Kahler manifolds with nonpositive bisectional curvature.
300 $a 112 p.
500 $a Source: Dissertation Abstracts International, Volume: 74-10(E), Section: B.
500 $a Adviser: Jiaping Wang.
502 $a Thesis (Ph.D.)--University of Minnesota, 2013.
520 $a The relation between curvature and topology is a fundamental problem in differential geometry. For example, the Gauss-Bonnet theorem says the sign of curvature could determine the genus of the surface. Brendle and Schoen [8] proved that if a compact manifold has sectional curvature between ¼ and 1, then it is a space form.
520 $a In the thesis, first, we classify complete noncompact three dimensional manifold with nonnegative Ricci curvature. As a corollary, we confirms a conjecture of Milnor in dimension three. Note that in the compact case, the classification was done by R. Hamilton by using the Ricci flow. Also, previously, there are some partial classifications assuming additional conditions. Our proof will be based on the minimal surface theory developed by Schoen and Yau [74], Schoen and Fischer Colbrie [24].
520 $a Next we study compact Kahler manifolds with nonpositive bisectional curvature. In particular, we confirm a conjecture of Yau which states that for there is a canonical fibration structure for these manifolds. More relating results will be proved.
520 $a In the third part, we generalize the classical volume comparison theorem to the Kahler setting. We prove a few gap theorems which tells us some differences between Kahler geometry and Riemannian geometry. We also show that locally, the volume of a Kahler-Einstein manifold is no greater than that of the complex space forms. Note that when the bisectional curvature is bounded from below, the sharp volume comparison was obtained by Li and Wang.
520 $a Then we prove a rigidity result for volume entropy. This was first proved by Leddrapier and Wang. Our proof is much shorter and simpler.
520 $a Finally, we study complete manifolds with nonnegative Bakry-Emery Ricci curvature. It turns out that when the potential f is bounded, geometrically these manifolds will be very similar with manifolds of nonnegative Ricci curvature. In particular, we partially classify complete three dimensional manifold with nonnegative Bakry-Emery Ricci curvature.
590 $a School code: 0130.
650 4 $a Mathematics. $3 515831
650 4 $a Applied Mathematics. $3 1669109
690 $a 0405
690 $a 0364
710 2 $a University of Minnesota. $b Mathematics. $3 1675975
773 0 $t Dissertation Abstracts International $g 74-10B(E).
790 $a 0130
791 $a Ph.D.
792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3567448