Limiting Properties of Certain Geometric Flows in Complex Geometry

Title:

Limiting Properties of Certain Geometric Flows in Complex Geometry

Author(s):

Jacob, Adam Joshua

Date:

2012

Type:

Dissertations

Department:

Mathematics

Permanent URL:

http://hdl.handle.net/10022/AC:P:13047

Notes:

Ph.D., Columbia University.

Abstract:

In this thesis, we study convergence results of certain nonlinear geometric flows on vector bundles over complex manifolds. First we consider the case of a semistable vector bundle E over a compact Kahler manifold X of arbitrary dimension. We show that in this case Donaldson's functional is bounded from below. This allows us to construct an approximate HermitianEinstein structure on E along the Donaldson heat flow, generalizing a classic result of Kobayashi for projective manifolds to the Kahler case. Next we turn to general unstable bundles. We show that along a solution of the YangMills flow, the trace of the curvature approaches in L2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the HarderNarasimhan filtration of E. This proves a sharp lower bound for the HermitianYangMills functional and thus the YangMills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. Furthermore, we show any reflexive extension to all of X of the limiting bundle is isomorphic to the double dual of the graded quotients from the HarderNarasimhanSeshadri filtration, verifying a conjecture of Bando and Siu. Our work on semistable bundles plays an important part of this result. For the final section of this thesis, we show that, in the case where X is an arbitrary Hermitian manifold equipped with a Gauduchon metric, given a stable Higgs bundle the Donaldson heat flow converges along a subsequence of times to a HermitianEinstein connection. This allows us to extend to the nonKahler case the correspondence between stable Higgs bundles and (possibly) nonunitary HermitianEinstein connections first proven by Simpson on Kahler manifolds.

Subject(s):

Mathematics
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