[Paper Review] Gap theorem on Kähler manifold with nonnegative orthogonal bisectional curvature

Lei Ni, Yanyan Niu|arXiv (Cornell University)|Aug 11, 2017

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper establishes a gap theorem for complete noncompact Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, proving that such manifolds are flat if the scalar curvature decays sufficiently fast in average. It also proves a Liouville theorem for plurisubharmonic functions under these curvature conditions, generalizing prior results by relaxing curvature assumptions from nonnegative bisectional to nonnegative orthogonal bisectional curvature.

ABSTRACT

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author. We also prove a Liouville theorem for plurisubharmonic functions on such a manifolds, which generalizes a previous result of L.-F. Tam and the first author.

Motivation & Objective

• To generalize the gap theorem for Kähler manifolds from nonnegative bisectional curvature to the weaker condition of nonnegative orthogonal bisectional curvature.
• To extend the Liouville theorem for plurisubharmonic functions from nonnegative holomorphic bisectional curvature to nonnegative orthogonal bisectional curvature.
• To establish a connection between the decay of scalar curvature and the vanishing of curvature via the solution of the Poincaré-Lelong equation.
• To demonstrate that the curvature condition (NOB) is strictly weaker than nonnegative bisectional curvature but stronger than nonnegative quadratic orthogonal bisectional curvature.
• To construct explicit examples of Kähler metrics with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature that do not have nonnegative bisectional curvature.

Proposed method

• Utilizes the heat flow of the Hodge-Laplacian on (1,1)-forms to evolve the Ricci form and study the asymptotic behavior of the solution.
• Applies the monotonicity formula from [11] and [13] to exploit the nonnegativity of orthogonal bisectional curvature in the proof.
• Employs the Poincaré-Lelong equation with optimal solution constructed via heat flow, relying on integral decay conditions on the norm of the form.
• Implements a parabolic method and partial maximum principle to control the growth of plurisubharmonic functions under curvature constraints.
• Uses a radial model with U(n)-invariant metrics to derive necessary and sufficient conditions for nonnegative orthogonal bisectional and Ricci curvatures.
• Analyzes the asymptotic behavior of curvature components via integral inequalities involving the volume growth and curvature decay.

Experimental results

Research questions

• RQ1 Does the gap theorem for flatness hold under the weaker assumption of nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature?
• RQ2 Can the Liouville theorem for plurisubharmonic functions be extended from nonnegative holomorphic bisectional curvature to nonnegative orthogonal bisectional curvature?
• RQ3 What is the precise relationship between nonnegative orthogonal bisectional curvature and nonnegative bisectional curvature in terms of curvature decay and geometric rigidity?
• RQ4 Can one construct explicit examples of Kähler metrics with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature that fail to have nonnegative bisectional curvature?
• RQ5 How do the integral decay conditions on the scalar curvature relate to the vanishing of curvature in the gap theorem?

Key findings

• The gap theorem holds for complete noncompact Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, provided the average scalar curvature decays as o(r⁻²).
• A Liouville theorem for plurisubharmonic functions is established: if u is continuous and satisfies lim_{x→∞} u(x)/log r(x) = 0, then u is constant under nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature.
• The solution to the Poincaré-Lelong equation for a (1,1)-form ρ with norm f satisfies pointwise estimates involving integrals of k_f(r), with explicit constants depending on dimension and ε.
• For U(n)-invariant Kähler metrics on ℂⁿ, nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature are equivalent to the conditions A+C≥0, A+(n−1)B≥0, B≥0, and C≥0.
• The paper constructs examples showing that nonnegative orthogonal bisectional curvature is strictly weaker than nonnegative bisectional curvature, but stronger than nonnegative quadratic orthogonal bisectional curvature.
• The proof simplifies Lemma 4.3(ii) of [7] by refining the analysis of the function B(r) and showing it remains positive for large r under curvature conditions.