D(S) > 0.
Conjecture 8.7.2is known only when n = 1 as mentioned above.
8.7.3. Theorem8.4provides a connection between singular Liouville equa-tions and the branch points of the associated hyperelliptic curve. We expect the phenomenon to hold true for other related equations. For example we might ask the following question on Chern–Simons–Higgs equation:
Suppose that u is a sequence of bubbling solutions of the Chern–Simons–Higgs equation
u + 1
2eu(1− eu) = 8πnδ0 in E.
Is the bubbling set{p1, . . . , pn}, as → 0, a branch point of the hyper elliptic curve C2= n(B)?
This has recently been answered affirmatively for n = 1 and for E a rectangular torus [45,46].
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Ching-Li Chai
Department of Mathematics University of Pennsylvania Philadelphia
USA
E-mail address: [email protected]
Chang-Shou Lin
Department of Mathematics and
Center for Advanced Studies in Theoretic Sciences (CASTS) National Taiwan University
Taipei Taiwan
E-mail address: [email protected]
Chin-Lung Wang
Department of Mathematics and
Taida Institute of Mathematical Sciences (TIMS) National Taiwan University
Taipei Taiwan
E-mail address: [email protected] Received April 27, 2014