中科院数学与系统科学研究院

数学研究所

学术报告

多复变与复几何研讨班

报告人：蒋云平（The City University of New York-Queens College and Graduate Center）

题  目：Teichmüller Spaces of Circle Endomorphisms

时  间：2024.03.27（星期三）16:00-17:00

地    点：数学院南楼N933

摘  要：Consider two circle endomorphisms f and g of degree d ≥ 2, both C1+α and expanding. There is a circle homeomorphism h such as h ◦ f = g ◦ h. The map h is not only a homeomorphism but also a quasisymmetric homeomorphism. Consequently, we introduce a Teichm¨uller distance between them, utilizing the boundary quasiconformal dilatation of h. It furnishes a pseudo-metric on the space of all C1+α expanding circle endomorphisms for degree d, thereby inducing a Teichm¨uller metric on the Teichm¨uller space of smooth conjugacy classes. Every smooth conjugacy class admits a unique representation (up to rotations), preserving the Lebesgue measure. However, this Teichm¨uller space remains incomplete. Its completion manifests as the Teichm¨uller space of all symmetric conjugacy classes of uniformly symmetric circle endomorphisms. While every symmetric conjugacy class possesses a representation preserving the Lebesgue measure, the question of uniqueness has been open for a long time. Collaborating with my Ph.D. students John Adamski, Yunchun Hu, and Zhe Wang, we recently resolved this issue, establishing a broad symmetric rigidity theorem. Specifically, for two topological circle endomorphisms f and g (both with f(1) = g(1) = 1), exhibiting bounded geometry and the Lebesgue measure preservation, and suppose h (h(1) = 1) is a symmetric homeomorphism satisfying h ◦ f = g ◦ h, we ascertain that h must be the identity. This symmetric rigidity theorem facilitates the completion of our project concerning complex manifold structures on the Teichm¨uller space of symmetric conjugacy classes of uniformly symmetric circle endomorphisms, as well as the Teichm¨uller space of symmetric conjugacy classes of uniformly quasisymmetric circle endomorphisms that preserve the Lebesgue measure. Lastly, I posit a conjecture regarding the equivalence of the Kobayashi and Teichm¨uller metrics on these two complex Banach manifolds.