Geometric Applications of Dirac Operator and Atiyah-Singer Index Theory

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Abstract Award: DMS-1007041 Principal Investigator: Xianzhe Dai This proposal concerns several problems in geometry that are related to Dirac operators and Atiyah-Singer index theory. It includes the use of Dirac operator and spinors in the study of stability problem for Einstein metrics. The PI will seek better understanding of the stability of Einstein metrics with positive scalar curvature by exploring the connection with Killing spinors and Sasakian-Einstein metrics. Another problem involves the study of heat kernel and Bergman kernel using local index theory technique and the study of their relation with canonical metrics. The PI would like to study the spectral gap of Dirac operators and the asymptotic expansion of Bergman kernel in the semipostive case. The PI will also study Ricci flows on a class of noncompact manifolds, the ALE spaces. The question of long time convergence will be the main focus. Finally, the PI will investigate the behavior of geometric invariants under metric degeneration, including adiabatic limit and conical degeneration. In particular, one of the applications will be the Ray-Singer conjecture for manifolds with conical singularities. Einstein's General relativity geometrizes gravity, one of the four fundamental forces in nature and the dominating one in shaping our universe. Einstein manifolds play essential role in mathematics and physics. It is important to understand the stability of Einstein manifolds. Stability issue is also important in the study of geometric evolution equations such as the Ricci flow. Recent development shows the extraordinary power of the Ricci flow. Dirac operators and related geometric invariants are playing significant and important role in diverse fields of mathematics and physics. This proposal aims for better understanding of the stability of Einstein manifolds, of the Ricci flow on noncompact manifolds, and of geometric invariants.