Adam Jacob

THE ISOPERIMETRIC PROBLEM ON PLANES WITH DENSITY

We discuss the isoperimetric problem in planes with density. In particular, we examine planes with generalized curvature zero. We solve the isoperimetric problem on the plane with density e x , as well as on the plane with density r p for p

Limiting properties of certain geometric flows in complex geometry

In this thesis, we study convergence results of certain non-linear geometric flows on vector bundles over complex manifolds. First we consider the case of a semi-stable vector bundle E over a compact Kahler manifold X of arbitrary dimension. We show that in this case Donaldsons functional is bounded from below. This allows us to construct an approximate Hermitian-Einstein 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 Yang-Mills flow, the trace of the curvature approaches in L2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of E. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills 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 Harder-Narasimhan-Seshadri filtration, verifying a conjecture of Bando and Siu. Our work on semi-stable 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 Hermitian-Einstein connection. This allows us to extend to the non-Kahler case the correspondence between stable Higgs bundles and (possibly) non-unitary Hermitian-Einstein connections first proven by Simpson on Kahler manifolds.