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Properly discontinuous actions of a surface group by affine automorphisms of ℝ^d were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in 𝖲𝖮(n,n−1), so that the affine action is by isometries of a flat pseudo-Riemannian metric on ℝ^d of signature (n,n−1). Here, the translational part determines a deformation of the linear part into 𝖯𝖲𝖮(n,n)-Hitchin representations and the crucial step is to show that such representations are not Anosov in 𝖯𝖲𝖫(2n,ℝ) with respect to the stabilizer of an n-plane. We also prove a negative curvature analogue of the main result, that the action of a surface group on the pseudo-Riemannian hyperbolic space of signature (n,n−1) by a 𝖯𝖲𝖮(n,n)-Hitchin representation fails to be properly discontinuous.
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This content will become publicly available on January 1, 2026
PLANAR MINIMAL SURFACES WITH POLYNOMIAL GROWTH IN THE Sp(4,R)-SYMMETRIC SPACE
We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the Sp(4,R)-symmetric space. We describe a homeomorphism between the “Hitchin component” of wild Sp(4,R)- Higgs bundles over CP1 with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in H2,2. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of R4. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in H2,2 associated to Sp(4,R)-Hitchin representations along rays of holomorphic quartic differentials.
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- PAR ID:
- 10432511
- Date Published:
- Journal Name:
- American journal of mathematics
- ISSN:
- 1080-6377
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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