International Congress of Mathematicians

Recent Progress in Kahler Geometry 2810.8. Some new result with G. TianIn 2001, Donaldson proved the followingTheorem 0.17. [17] (Openness) For any smooth solution to the geodesic equationwith a disc domain, there are always exists a smooth solution to the geodesic equationif we perturb the boundary data in a small open set (of the given boundary data).This is somewhat surprising result since it is very hard to deform any solutionof a homogenous Monge-Ampere equation even locally. However, Donaldson wasable to make clever use of the Fredholm theory of holomorphic discs with totally realboundary in his proof. Then the problem of closed-ness becomes very important inlight of this theorem. Tian and I are able to establish the closed-ness in this case.Theorem 0.18. [13] (Closure property) The defamation of geodesic solution in thepreceding theorem is indeed closed, provided we allow solution to be smooth almosteverywhere.This is a deep theorem and we will not go into detail here due to the expositorynatureof this talk. However, this theorem, along with the ideas of proof, shall haveimplication in both geometry and other Monge-Ampere type equation in the future.References[i[2[3;[4;[5;[6;[7;[8"[9[10;[HS. Bando. On the three dimensional compact Kahler manifolds of nonnegativebisectional curvature. J. D. G., 19:283^297, 1984.S. Bando and T. Mabuchi. Uniquness of Einstein Kahler metrics modulo connectedgroup actions. In Algebraic Geometry, Advanced Studies in Pure Math.,1987.E. Calabi. Extremal Kahler metrics. In Seminar on Differential Geometry,volume 16 of 102, 259^290. Ann. of Math. Studies, University Press, 1982.E. Calabi. Extremal Kahler metrics, II. In Differential geometry and Complexanalysis, 96^114. Springer, 1985.E. Calabi and X. X. Chen. Space of Kahler metrics (II), 1999. to appear inJ.D.G.H. D. Cao. Deformation of Kahler metrics to Kähler-Einstein metrics on compactKahler manifolds. Invent. Math., 81:359^372, 1985.J.Y. Chen and J.Y. Li. mean curvature flow of surfaces in 4-manifolds, 2000.Adv. in Math, (to appear).J.Y. Chen and J.Y. Li. quaternionic maps between hyperkähler manifolds, 2000.J. D. G.X. X. Chen. On lower bound of the Mabuchi energy and its application. InternationalMathematics Research Notices, 12, 2000.X. X. Chen. Space of Kahler metrics. Journal of Differential Geometry,,56:189^234, 2000.X. X. Chen. Calabi flow in Riemann surface revisited: a new point of views.(6):276^297, 2001. "International Mathematics Research Notices".