# The Yang-Mills flow and the Atiyah-Bott formula on compact Kähler manifolds

@article{Jacob2011TheYF, title={The Yang-Mills flow and the Atiyah-Bott formula on compact K{\"a}hler manifolds}, author={Adam Jacob}, journal={American Journal of Mathematics}, year={2011}, volume={138}, pages={329 - 365} }

We study the Yang-Mills flow on a holomorphic vector bundle $E$ over a compact K\"ahler manifold $X$. Along a solution of the flow, we show that the curvature endomorphism $i\Lambda F(A_t)$ approaches in $L^2$ 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… Expand

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#### References

SHOWING 1-10 OF 35 REFERENCES

Convergence properties of the Yang-Mills flow on Kaehler surfaces

- Mathematics
- 2004

Let $E$ be a hermitian complex vector bundle over a compact K\"ahler surface $X$ with K\"ahler form $\omega$, and let $D$ be an integrable unitary connection on $E$ defining a holomorphic structure… Expand

The limit of the Yang-Mills flow on semi-stable bundles

- Mathematics
- 2011

By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kahler manifold X, the Yang-Mills flow converges away from an analytic singular set. If E is… Expand

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

- Mathematics
- 1988

The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex… Expand

Stable and unitary vector bundles on a compact Riemann surface

- Mathematics
- 1965

Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation of the fundamental group of X. We prove in this… Expand

Lower bounds on the Calabi functional

- Mathematics
- 2005

The main result of this paper shows that "test configurations" give new lower bounds on the $L^{2}$ norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the… Expand

A simple proof of a theorem by Uhlenbeck and Yau

- Mathematics
- 2003

Abstract.A subbundle of a Hermitian holomorphic vector bundle (E, h) can be metrically and differentially defined by the orthogonal projection onto itself. A weakly holomorphic subbundle of (E, h)… Expand

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

- Mathematics
- 1978

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the… Expand

The Yang-Mills equations over Riemann surfaces

- Mathematics
- Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1983

The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge… Expand

Existence of approximate Hermitian-Einstein structures on semi-stable bundles

- Mathematics
- 2010

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donaldson's functional is… Expand

Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics: Delivered At The German Mathematical Society Seminar In Düsseldorf In June, 1986

- Mathematics
- 1987

1. The heat equation approach to Hermitian-Einstein metrics on stable bundles.- 1. Definition of Hermitian-Einstein metrics.- 2. Gradient flow and the evolution equation.- 3. Existence of solution of… Expand