[Paper Review] On the three-circle theorem and its applications in Sasakian manifolds

Shu-Cheng Chang, Yingbo Han+1 more|arXiv (Cornell University)|Jan 25, 2018

Geometric Analysis and Curvature Flows39 references3 citations

TL;DR

This paper establishes the CR three-circle theorem for complete noncompact pseudohermitian manifolds of vanishing torsion with nonnegative pseudohermitian sectional curvature, proving a sharp dimension estimate for CR holomorphic functions of polynomial growth and confirming the first CR Yau’s uniformization conjecture. The result relies on a CR sub-Laplacian comparison theorem and extends key Kähler-theoretic results to the CR setting.

ABSTRACT

This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau's uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yau's uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.

Motivation & Objective

• To establish a CR analogue of the three-circle theorem in pseudohermitian manifolds with vanishing torsion.
• To confirm the first CR Yau’s uniformization conjecture under nonnegative pseudohermitian sectional curvature.
• To derive a sharp dimension estimate for CR holomorphic functions of polynomial growth in the CR setting.
• To lay the foundation for the second and third CR Yau’s uniformization conjectures by proving foundational comparison and convexity results.
• To extend Liouville-type theorems to positive pseudoharmonic functions under nonnegative pseudohermitian Ricci curvature.

Proposed method

• Derives a CR sub-Laplacian comparison theorem using curvature assumptions and auxiliary functions of the form $ u(r) = \frac{1}{2r} + \frac{A}{(1+r)^{1+\epsilon}} $.
• Applies the Hessian comparison principle and maximum principle adapted to the CR setting to establish convexity of $ \log M_f(r) $ in $ \log r $.
• Uses the Poincaré-Siegel map to embed the space of CR holomorphic functions of degree $ d $ into a finite-dimensional complex space.
• Constructs a comparison function $ h(r) $ satisfying $ h'(r) = \frac{\exp\left(\frac{2A}{\epsilon(1+r)^\epsilon}\right)}{r} \exp\left(-\frac{2A}{\epsilon}\right) $ to control growth rates.
• Applies the CR three-circle theorem to show that $ \frac{M_f(r)}{\exp(dh(r))} $ is increasing, leading to lower bounds on $ M_f(r) $.
• Employs the asymptotic behavior of $ h(r) $ to derive dimension estimates via injectivity of the jet map $ \Phi $.

Experimental results

Research questions

• RQ1 Does the three-circle theorem hold in the CR setting for pseudohermitian manifolds with nonnegative pseudohermitian sectional curvature?
• RQ2 Can the first CR Yau’s uniformization conjecture be confirmed under nonnegative curvature and vanishing torsion?
• RQ3 What is the sharp dimension estimate for the space of CR holomorphic functions of polynomial growth in such manifolds?
• RQ4 Does a Liouville theorem hold for positive pseudoharmonic functions on complete noncompact CR manifolds with nonnegative pseudohermitian Ricci curvature?
• RQ5 Can the CR sub-Laplacian comparison theorem be used to derive sharp growth estimates for CR holomorphic functions?

Key findings

• The CR three-circle theorem holds for complete noncompact pseudohermitian $ (2n+1) $-manifolds with vanishing torsion and nonnegative pseudohermitian sectional curvature.
• The first CR Yau’s uniformization conjecture is confirmed: $ \dim_{\mathbb{C}}(\mathcal{O}_d^{CR}(M)) \leq \dim_{\mathbb{C}}(\mathcal{O}_d^{CR}(\mathbb{H}^n)) $, with equality implying isometric biholomorphism to the Heisenberg group.
• A sharp dimension estimate $ \dim_{\mathbb{C}}(\mathcal{O}_d^{CR}(M)) \leq C(\lambda,n) d^n $ holds when curvature is nonnegative outside a compact set and bounded below.
• For sufficiently small $ \lambda = a(d_c(K))^2 $, the dimension of $ \mathcal{O}_d^{CR}(M) $ is bounded by that of $ \mathcal{O}_d^{CR}(\mathbb{H}^n) $, implying rigidity.
• A Liouville theorem holds: any positive pseudoharmonic function on a complete noncompact pseudohermitian manifold with vanishing torsion and nonnegative pseudohermitian Ricci curvature is constant.
• The dimension estimate is sharp: there exists $ \delta(\lambda) > 0 $ such that $ \dim_{\mathbb{C}}(\mathcal{O}_{\delta(\lambda)}^{CR}(M)) = 1 $, indicating minimal growth for nontrivial functions.