Geometry Seminar
Almost geodesic compact holomorphic curves on quotients of bounded symmetric domains
Professor Ngaiming MOK
The University of Hong
Kong
Abstract
Let D be a bounded symmetric domain, X be a complex manifold uniformized by D, and S be a compact complex submanifold of X. For e > 0 we will say that S is e-geodesic if the second fundamental form of S in X is bounded by e at every point.
I will show
that, when D is the 2-disk, for every e
> 0 there exists an e-geodesic compact holomorphic curve on
some X uniformized by D. This implies the same statement for
a bounded symmetric domain of rank ³ 2. This
results from a recent construction by Eyssidieux and myself of
examples of holomorphic maps f between compact Riemann
surfaces of higher genus such that the sup norm 
with respect to Poincaré
metrics is smaller than any pre-assigned constant.
Given D, there are only a finite number of complex totally geodesic submanifolds D' up to holomorphic isometries of D, and one can raise a more refined question for each such pair (D, D'), asking whether e-geodesic compact holomorphic submanifolds S modelled on D' are necessarily totally geodesic. The examples in the above show that for certain D' of dimension 1 this fails to be the case. I will explain why for certain pairs (D, D') with dim(D') = 1 the answer is in fact positive. We say in this case that (D, D') exhibits gap rigidity.
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Date: |
October 7, 2004 (Thursday) |
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Time: |
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Place: |
Room 517, Meng Wah Complex |
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All are welcome |
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