Andrew Zimmer

Associate Professor
Department of Mathematics
University of Wisconsin-Madison
Office: 611 Van Vleck
e-mail: amzimmer2[at]wisc.edu

Research:

I enjoy thinking about geometric problems in several complex variables and discrete subgroups of Lie groups.

Publications / Preprints:

Papers by area: all, several complex variables, discrete subgroups of Lie groups.

The papers below are listed by the year they were made publicly available.

2025:

1. Weakly holomorphic homogeneous regular manifolds
   Submitted, 26 pages.
2. Geometry and Dynamics of Transverse Groups (with R. Canary, T. Zhang).
   Submitted, 24 pages.
   Expository.

2024:

3. Gromov hyperbolicity of intrinsic metrics from isoperimetric inequalities (with T. Wang).
   Submitted, 44 pages.
4. Bi-Lipschitz rigidity of discrete subgroups (with R. Canary, H. Oh).
   Submitted, 18 pages.
5. Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groups (with P.-L. Blayac, R. Canary, F. Zhu).
   Submitted, 53 pages.
6. Patterson-Sullivan theory for coarse cocycles (with P.-L. Blayac, R. Canary, F. Zhu).
   Submitted, 72 pages.

2023:

7. Smooth equivalence of families of strongly pseudoconvex domains (with H. Gaussier, X. Gong).
   Submitted, 31 pages.
8. Patterson-Sullivan measures for relatively Anosov groups (with R. Canary, T. Zhang).
   Mathematische Annalen, to appear, 41 pages.
9. A rigidity result for proper holomorphic maps between balls (with E. Gevorgyan, H. Wang).
   Proceedings of the AMS, 152: 1573-1585, 2024.
   REU Project!
10. Patterson-Sullivan measures for transverse subgroups (with R. Canary, T. Zhang).
    Journal of Modern Dynamics, 20: 319-377, 2024.

2022:

11. Relatively Anosov representations via flows II: examples (with F. Zhu).
    Journal of the London Math. Society, 109: e12949, 2024 (61 pages).
12. Relatively Anosov representations via flows I: theory (with F. Zhu).
    Groups, Geometry, and Dynamics, to appear, 73 pages.
13. Unbounded visibility domains, the end compactification, and applications (with G. Bharali).
    Transactions of the AMS, 376: 5949-5988, 2023.
14. The structure of relatively hyperbolic groups in convex real projective geometry (with M. Islam).
    Algebraic and Geometric Topology, to appear, 31 pages.
15. Entropy rigidity for cusped Hitchin representations (with R. Canary, T. Zhang).
    Submitted, 57 pages.

2021:

16. A metric analogue of Hartogs' theorem (with H. Gaussier).
    Geometric and Functional Analysis, 32: 1041-1062, 2022.
    An earlier version was titled: When is the Kobayashi metric a K\"ahler metric?
17. Hankel operators on domains with bounded intrinsic geometry.
    Journal of Geometric Analysis, 33: Article number: 176, 2023 (29 pages).
18. Convex co-compact groups with one dimensional boundary faces (with M. Islam).
    Groups, Geometry, and Dynamics, 18: 1145-1183, 2024.
19. Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups (with R. Canary, T. Zhang).
    Advances in Mathematics, 404, Part B: 108439, 2022 (67 pages).

2020:

20. Convex co-compact representations of 3-manifold groups (with M. Islam).
    Journal of Topology, 17: e12332, 2024 (56 pages).
21. Compactness of the dbar-Neumann problem on domains with bounded intrinsic geometry.
    Journal of Functional Analysis, 281: 108992, 2021 (47 pages).
22. Kobayashi hyperbolic convex domains not biholomorphic to bounded convex domains.
    Mathematische Zeitschrift, 300: 1905-1916, 2022.
23. A lower bound for the K\"ahler-Einstein distance from the Diederich-Fornaess index.
    Proceedings of the AMS, 149: 1641-1646, 2021.
24. The space of convex domains in complex Euclidean space (with H. Gaussier).
    Journal of Geometric Analysis (special issue), 30: 1312-1358, 2020.
    Expository.
25. A higher rank rigidity theorem for convex real projective manifolds.
    Geometry and Topology, 27: 2899-2936, 2023.

2019:

26. Convex co-compact actions of relatively hyperbolic groups (with M. Islam).
    Geometry and Topology, 27: 417-511, 2023.
27. Smoothly bounded domains covering compact manifolds.
    Indiana University Mathematics Journal, 70: 2653-2676, 2021.
28. A flat torus theorem for convex co-compact actions of projective linear groups (with M. Islam).
    Journal of the London Math. Society, 103: 470-489, 2021.
29. Subelliptic estimates from Gromov hyperbolicity.
    Advances in Mathematics, 402: 108334, 2022 (94 pages).
30. Regularity of limit sets of Anosov representations (with T. Zhang).
    Journal of Topology, 17: e12355, 2024 (72 pages).

2018:

31. The geometry of domains with negatively pinched K\"ahler metrics (with F. Bracci, H. Gaussier).
    Journal of Differential Geometry, 126: 909-938, 2024.
32. Asymptotic behavior of orbits of holomorphic semigroups (with F. Bracci, M. Contrears, S. Diaz-Madrigal, H. Gaussier).
    Journal de Mathématiques Pures et Appliquées, 133: 263-286, 2020.
33. Two boundary rigidity results for holomorphic maps.
    American Journal of Mathematics, 144: 119-168, 2022.
34. Homeomorphic extension of quasi-isometries for convex domains in C^d and iteration theory (with F. Bracci, H. Gaussier).
    Mathematische Annalen, 379: 691-718, 2021.
35. Smoothly bounded domains covering finite volume manifolds.
    Journal of Differential Geometry, 119: 161-182, 2021.

2017:

36. The automorphism group and limit set of a bounded domain II: the convex case.
    Journal of the London Math. Society, 104: 453-491, 2021.
37. The automorphism group and limit set of a bounded domain I: the finite type case.
    Advances in Mathematics, 366: 107085, 2020 (42 pages).
38. Projective Anosov representations, convex cocompact actions, and rigidity.
    Journal of Differential Geometry, 119: 513-586, 2021.
39. Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents.
    Mathematische Annalen, 374: 1811-1844, 2019.

2016:

40. A gap theorem for the complex geometry of convex domains.
    Transactions of the AMS, 370: 7489-7509, 2018.
41. Generic analytic polyhedron with non-compact automorphism group.
    Indiana University Mathematics Journal, 67: 1299-1326, 2018.
42. Gromov Hyperbolicity of Bounded Convex Domains.
    In: Blanc-Centi, editor, Metrical and dynamical aspects of complex analysis, pp. 67-114
    Lecture Notes in Mathematics, Vol. 2195, New York: Springer, 2017.
43. Goldilocks domains, a weak notion of visibility, and applications (with G. Bharali).
    Advances in Mathematics, 310: 377-425, 2017.

2015:

44. Rigidity of convex divisible domains in flag manifolds (with W. Van Limbeek).
    Geometry and Topology, 23: 171-240, 2019.
45. Entropy rigidity of Hilbert and Riemannian metrics (with T. Barthelmé, L. Marquis).
    International Mathematics Research Notices, 2017 (22): 6841-6866, 2017.
46. Proper quasi-homogeneous domains in flag manifolds and geometric structures.
    Annales de l'Institut Fourier, 68: 2635-2662, 2018.
47. Characterizing the unit ball by its projective automorphism group.
    Geometry and Topology, 20: 2397-2432, 2016.
48. The structure of projective maps between real projective manifolds.
    Geometriae Dedicata, 190: 81-102, 2017.
49. Characterizing domains by the limit set of their automorphism group.
    Advances in Mathematics, 308: 438 - 482, 2017.
    An earlier version was titled: Characterizing polynomial domains by their automorphism group.

2014 and before:

50. Gromov hyperbolicity, the Kobayashi metric, and C-convex sets.
    Transactions of the AMS, 369: 8437-8456, 2017.
51. Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type.
    Mathematische Annalen, 365: 1425-1498, 2016.
52. Rigidity of complex convex divisible sets.
    Journal of Topology and Analysis, 10: 817-85, 2018.
53. Boundaries of non-compact harmonic manifolds.
    Geometriae Dedicata, 168: 339-357, 2014.
54. Compact asymptotically harmonic manifolds.
    Journal of Modern Dynamics, 6: 377-403, 2012.
55. A symplectic proof of a theorem of Franks (with B. Collier, E. Kerman, B. Reiniger, B. Turmunk).
    Compositio Mathematica, 148: 1969-1984, 2012.
    

Other manuscripts:

• Asymptotically harmonic manifolds without focal points. 2011. Not for publication: The main result in this note was extended in ``Compact asymptotically harmonic manifolds.''
• Rigidity in Complex Projective Space. 2014. My Ph.D. thesis. The mathematical content essentially coincides with the paper "Rigidity of complex convex divisible sets" and some of the results were extended in "Characterizing the unit ball by its projective automorphism group."
• Rigidity and convexity of proper geometric structures. 2015. Not for publication: This is an earlier version of ``Proper quasi-homogeneous domains in flag manifolds and geometric structures'' with a different perspective.

Videos:

• Analytic problems on domains with good intrinsic geometry. Fall 2020, Virtual East-West Several Complex Variables seminar.
• Analytic problems on domains with good intrinsic geometry. Fall 2020. Northwestern, Notre-Dame, UIC Complex Geometry Seminar.
• Analytic problems on domains with good intrinsic geometry. Summer 2020. Virtual Conference on Several Complex Variables.
• Convex cocompact actions of non-hyperbolic groups. Winter 2019. Discrete Subgroups of Lie Groups at Banff.
• Characterizing domains by their automorphism group. Summer 2016. Complex Analysis and Complex Geometry at Banff.
• The asymptotic geometry of the Kobayashi metric. Summer 2015. Spring School in Lille: Metrical and dynamical aspects of complex analysis.
• Higher rank geometric structures. Winter 2015. GEAR Seminar at University of Illinois.

Last updated: March 25, 2025.