Space of Kähler potentials on singular and non-compact manifolds
Space of Kähler potentials on singular and non-compact manifolds
| dc.identifier.uri | http://hdl.handle.net/11401/76387 | |
| dc.description.sponsorship | This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree. | en_US |
| dc.format | Monograph | |
| dc.format.medium | Electronic Resource | en_US |
| dc.language.iso | en_US | |
| dc.publisher | The Graduate School, Stony Brook University: Stony Brook, NY. | |
| dc.type | Dissertation | |
| dcterms.abstract | Let H be the space of Kähler metrics in a fixed cohomology class. This space may be endowed with a Weil-Petersson-type metric, referred to as the Mabuchi metric, which allows one to study the geometry of H. It is now well-known that the geometry of the space of Kähler potentials, in particular, the geodesics in H, may be used for studying `canonical metrics' on the base manifold. In order to be interpreted as the potential of a Kähler metric, however, one needs to prove certain regularity for such solutions. In the first part, I shall discuss deriving of weighted estimates for the space and time derivatives of solutions in the case of ALE Kähler potentials, and further, prove results regarding the Mabuchi energy and the uniqueness of metrics of constant scalar curvature. In the latter part of the talk I will discuss certain weighted estimates for the solutions to the geodesic equation when the end points have conical singularities. The results may also be seen as X.-X. Chen's fundamental work on the geodesic convexity of H in the case of smooth compact manifolds. | |
| dcterms.abstract | Let H be the space of Kähler metrics in a fixed cohomology class. This space may be endowed with a Weil-Petersson-type metric, referred to as the Mabuchi metric, which allows one to study the geometry of H. It is now well-known that the geometry of the space of Kähler potentials, in particular, the geodesics in H, may be used for studying `canonical metrics' on the base manifold. In order to be interpreted as the potential of a Kähler metric, however, one needs to prove certain regularity for such solutions. In the first part, I shall discuss deriving of weighted estimates for the space and time derivatives of solutions in the case of ALE Kähler potentials, and further, prove results regarding the Mabuchi energy and the uniqueness of metrics of constant scalar curvature. In the latter part of the talk I will discuss certain weighted estimates for the solutions to the geodesic equation when the end points have conical singularities. The results may also be seen as X.-X. Chen's fundamental work on the geodesic convexity of H in the case of smooth compact manifolds. | |
| dcterms.available | 2017-09-20T16:50:09Z | |
| dcterms.contributor | Varolin, Dror | en_US |
| dcterms.contributor | Chen, Xiu-Xiong | en_US |
| dcterms.contributor | Khuri, Marcus | en_US |
| dcterms.contributor | Bedford, Eric. | en_US |
| dcterms.creator | Aleyasin, Seyed Ali | |
| dcterms.dateAccepted | 2017-09-20T16:50:09Z | |
| dcterms.dateSubmitted | 2017-09-20T16:50:09Z | |
| dcterms.description | Department of Mathematics. | en_US |
| dcterms.extent | 58 pg. | en_US |
| dcterms.format | Monograph | |
| dcterms.format | Application/PDF | en_US |
| dcterms.identifier | http://hdl.handle.net/11401/76387 | |
| dcterms.issued | 2014-12-01 | |
| dcterms.language | en_US | |
| dcterms.provenance | Made available in DSpace on 2017-09-20T16:50:09Z (GMT). No. of bitstreams: 1 Aleyasin_grad.sunysb_0771E_11808.pdf: 436709 bytes, checksum: 22adc26d51c1a4f4a1d14ab10690cebf (MD5) Previous issue date: 1 | en |
| dcterms.publisher | The Graduate School, Stony Brook University: Stony Brook, NY. | |
| dcterms.subject | Degenerate Comple Monge-Ampère Equation, Kähler metrics, Metrics of Constant Scalar Curvature, Several Complex Variables | |
| dcterms.subject | Theoretical mathematics | |
| dcterms.subject | Degenerate Comple Monge-Ampère Equation, Kähler metrics, Metrics of Constant Scalar Curvature, Several Complex Variables | |
| dcterms.title | Space of Kähler potentials on singular and non-compact manifolds | |
| dcterms.title | Space of Kähler potentials on singular and non-compact manifolds | |
| dcterms.type | Dissertation |
