Remarks on the Yang–Mills flow on a compact Kahler manifolds
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RIS BIB ENDNOTEPublication date: 09.2014
Universitatis Iagellonicae Acta Mathematica, 2013, Volume 51, pp. 14-43
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Remarks on the Yang–Mills flow on a compact Kahler manifolds
We study the Yang–Mills ow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the ow, and introduce some techniques to study the blow- up of the curvature along the ow. Making some technical assumptions, we show how our techniques can be used to prove that the curvature of the evolved connection is uniformly bounded away from an analytic subvariety determined by the Harder-Narasimhan-Seshadri ltration of E. We also discuss how our assumptions are related to stability in some simple cases.
1. Bando S., Siu Y.-T., Stable sheaves and Einstein–Hermitian metrics, Geometry and Analysis on Complex Manifolds, World Sci. Publ., River Edge, NJ, (1994), 39–50.
2. Daskalopoulos G., Wentworth R., Convergence properties of the Yang–Mills flow on K¨ahler surfaces, J. Reine Angew. Math., 575 (2004), 69–99.
3. Daskalopoulos G., Wentworth R., On the blow-up set of the Yang–Mills flow on K¨ahler surfaces, Math. Z., 256, no. 2 (2007), 301–310.
4. Donaldson S. K., Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3), 50 (1985), 1–26.
5. Donaldson S. K., Infinite determinants, stable bundles, and curvature, Duke Math. J., 54 (1987), 231–247.
6. G´omez T. L., Sols I., Zamora A., A GIT interpretation of the Harder–Narasimhan filtra- tion, preprint, arXiv:1112.1886v2, [math.AG].
7. Han Q., Lin F., Elliptic Partial Differential Equations, Courant Lecture Notes in Mathe- matics, 1. Courant Institute of Mathematical Sciences, New York; American Mathemat- ical Society, Providence, RI, 2000.
8. Hong M.-C., Tian G., Asymptotical behaviour of the Yang–Mills flow and singular Yang– Mills connections, Math. Ann., 330 (2004), 441–472.
9. Jacob A., Existence of approximate Hermitian–Einstein structures on semi-stable bun- dles, Asian J. Math., (to appear).
10. Jacob A., The limit of the Yang–Mills flow on semi-stable bundles, J. Reine Angew. Math., (to appear).
11. Jacob A., The Yang–Mills flow and the Atiyah–Bott formula on compact K¨ahler mani- folds, arXiv:1109.1550v2, [math.DG].
12. Kobayashi S., Differential geometry of complex vector bundles, Publications of the Math- ematical Society of Japan, 15. Kanˆo Memorial Lectures, 5. Princeton University Press, Princeton, NJ, 1987.
13. McFeron D., Remarks on some non-linear heat flows in K¨ahler geometry, Ph.D. thesis, Columbia University, 2009.
14. Narasimhan M. S., Seshadri C. S., Stable and unitary bundles on a compact Riemann surface, Math. Ann., 82 (1965), 540–564.
15. Simpson C., Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc., 1 (1988), no. 4, 867–918.
16. Siu Y.-T., Lectures on Hermitian–Einstein metrics for stable bundles and K¨ahler– Einstein metrics, Birk¨auser Verlag, Basel, 1987.
17. Uhlenbeck K., Yau S.-T., On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Comm. Pure and Appl. Math., 39-S (1986), 257–293.
18. Ye R., Curvature estimates for the Ricci flow I, arXiv:0509142.
Information: Universitatis Iagellonicae Acta Mathematica, 2013, Volume 51, pp. 14-43
Article type: Original scientific article
Columbia University
United States of America
Harvard University
United States of America
Published at: 09.2014
Article status: Open
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