Linear stability of Einstein metrics and Perelman's lambda-functional for manifolds with conical singularities
- Degree Grantor:
- University of California, Santa Barbara. Mathematics
- Degree Supervisor:
- Xianzhe Dai
- Place of Publication:
- [Santa Barbara, Calif.]
- Publisher:
- University of California, Santa Barbara
- Creation Date:
- 2016
- Issued Date:
- 2016
- Topics:
- Mathematics
- Genres:
- Online resources and Dissertations, Academic
- Dissertation:
- Ph.D.--University of California, Santa Barbara, 2016
- Description:
In this thesis, we study linear stability of Einstein metrics and develop the theory of Perelman's lambda-functional on compact manifolds with isolated conical singularities. The thesis consists of two parts. In the first part, inspired by works in [DWW05], [GHP03], and [Wan91], by using a Bochner type argument, we prove that complete Riemannian manifolds with non-zero imaginary Killing spinors are stable, and provide a stability condition for Riemannian manifolds with non-zero real Killing spinors in terms of a twisted Dirac operator. Regular Sasaki-Einstein manifolds are essentially principal circle bundles over Kahler-Einstein manifolds. We prove that if the base space of a regular Sasaki-Einstein manifold is a product of at least two Kahler-Einstein manifolds, then the regular Sasaki-Einstein manifold is unstable. More generally, we show that Einstein metrics on principal torus bundles constructed in [WZ90] are unstable, if the base spaces are products of at least two Kahler-Einstein manifolds.
In the second part, we prove that the spectrum of --4Delta + R consists of discrete eigenvalues with finite multiplicities on a compact Riemannian manifold of dimension n with a single conical singularity, if the scalar curvature of cross section of conical neighborhood is greater than n -- 2. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectrum properties, we extend the theory of Perelman's lambda-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.
- Physical Description:
- 1 online resource (119 pages)
- Format:
- Text
- Collection(s):
- UCSB electronic theses and dissertations
- Other Versions:
- http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:10159748
- ARK:
- ark:/48907/f3028rmp
- ISBN:
- 9781369146769
- Catalog System Number:
- 990046969240203776
- Copyright:
- Changliang Wang, 2016
- Rights:
In Copyright
- Copyright Holder:
- Changliang Wang
File | Description |
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Access: Public access | |
Wang_ucsb_0035D_13023.pdf | pdf (Portable Document Format) |