[Paper Review] Ricci flow and nonnegativity of curvature

Lei Ni|ArXiv.org|May 16, 2003

Geometric Analysis and Curvature Flows24 references3 citations

TL;DR

This paper establishes a topological splitting theorem for simply-connected, complete Riemannian manifolds with bounded nonnegative sectional curvature under the assumption that Ricci flow preserves the nonnegativity of sectional curvature. By developing a time-dependent tensor maximum principle and analyzing the heat equation deformation of Busemann functions, the authors prove such manifolds split isometrically as a product of a compact nonnegatively curved manifold and Euclidean space, and construct the first known examples in dimension ≥4 where Ricci flow fails to preserve sectional curvature nonnegativity.

ABSTRACT

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. The example is the first of this type. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow.

Motivation & Objective

• To investigate the topological consequences of Ricci flow preserving nonnegative sectional curvature on complete noncompact manifolds.
• To resolve the technical challenge of applying maximum principles to non-differentiable Busemann functions under time-dependent Ricci flow.
• To establish a time-dependent tensor maximum principle for the Hessian of solutions to the heat equation under evolving metrics.
• To construct explicit examples of complete Riemannian manifolds in dimension ≥4 with bounded nonnegative sectional curvature where Ricci flow does not preserve nonnegativity of sectional curvature.
• To provide a classification of complete manifolds with bounded nonnegative curvature operator using heat flow methods.

Proposed method

• Develops a time-dependent version of the optimal tensor maximum principle from [NT3], adapted to the Ricci flow evolution of metrics.
• Analyzes the heat equation for the Hessian of Busemann functions under the Ricci flow, deriving a parabolic PDE for the Hessian components.
• Establishes a Li-Yau type heat kernel estimate and Harnack inequality for the time-dependent heat equation, generalizing methods from Grigoryan [Gr1] and Saloff-Coste [Sa].
• Uses the heat equation deformation of Busemann functions to study the asymptotic geometry of level sets and infer splitting structure.
• Applies the maximum principle to show that the Hessian of the Busemann function remains nonnegative definite under the flow.
• Constructs a complete Riemannian metric on a noncompact manifold (via Riemannian submersion) with bounded nonnegative sectional curvature, proving it fails to preserve nonnegativity under Ricci flow.

Experimental results

Research questions

• RQ1 Does the Ricci flow preserve the nonnegativity of sectional curvature on complete noncompact manifolds with bounded nonnegative sectional curvature?
• RQ2 Can the heat equation deformation of Busemann functions be used to derive topological splitting results under Ricci flow?
• RQ3 What are the structural implications when Ricci flow preserves nonnegative sectional curvature on a simply-connected complete manifold?
• RQ4 Are there examples of complete manifolds with bounded nonnegative sectional curvature where Ricci flow fails to preserve this curvature condition in dimension ≥4?
• RQ5 Can the curvature operator nonnegativity condition be classified using heat flow techniques on complete manifolds?

Key findings

• On a simply-connected, complete Riemannian manifold with bounded nonnegative sectional curvature, if Ricci flow preserves the nonnegativity of sectional curvature, then the manifold splits isometrically as a product of a compact manifold with nonnegative sectional curvature and a manifold diffeomorphic to Euclidean space.
• The paper constructs the first known example of a complete Riemannian manifold of dimension ≥4 with bounded nonnegative sectional curvature such that the Ricci flow does not preserve the nonnegativity of sectional curvature.
• The heat kernel estimate and Harnack inequality for the time-dependent heat equation are established under Ricci flow, extending prior results to a broader class of equations.
• The curvature of the constructed example is uniformly bounded, and the metric is explicitly described using Riemannian submersions from SO(3) and T(S²).
• The fiber of the Riemannian submersion π′ is shown to be totally geodesic, supporting the curvature bounds and geometric structure.
• The result provides an alternative proof of the classification of complete manifolds with bounded nonnegative curvature operator, previously established via different methods.