A GAP THEOREM FOR THE COMPLEX GEOMETRY OF CONVEX DOMAINS

作者：Zimmer Andrew^*

来源：Transactions of the American Mathematical Society, 2018, 370(10): 7489-7509.

DOI：10.1090/tran/7284

• Strongly pseudoconvex domains
• squeezing function
• Bergman metric
• holomorphic sectional curvature

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex.
One consequence of our general result is the following: for any dimension there exists some epsilon > 0 so that if the squeezing function on a smoothly bounded convex domain is greater than 1 - epsilon outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension d there exists some epsilon > 0 so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within epsilon of -4/(d+1) outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to -4/(d + 1) on the boundary).

• 出版日期2018-10