[Paper Review] Estimates for Minimal Volume and Minimal Curvature on 4-dimensional compact manifolds

Ezio Costa, Rafael Diógenes+1 more|arXiv (Cornell University)|Jul 30, 2014

Geometric Analysis and Curvature Flows35 references3 citations

TL;DR

This paper establishes sharp estimates for minimal curvature and minimal volume on 4-dimensional compact Riemannian manifolds using curvature decomposition and topological invariants. It proves that if the minimal curvature equals $ 2 au au au \sqrt{2|\chi(M)|} $, then the manifold is Einstein with constant Weyl curvature, and under nonnegative sectional curvature, it must be isometric to $ \mathbb{S}^4 $, $ \mathbb{CP}^2 $, or $ \mathbb{S}^2 \times \mathbb{S}^2 $. The key contribution is a sharp lower bound for minimal curvature in terms of the self-dual part of the curvature functional.

ABSTRACT

In a remarkable article published in 1982, M. Gromov introduced the concept of minimal volume, namely, the minimal volume of a manifold $M^n$ is defined to be the greatest lower bound of the total volumes of $M^n$ with respect to complete Riemannian metrics whose sectional curvature is bounded above in absolute value by 1. While the minimal curvature, introduced by G. Yun in 1996, is the smallest pinching of the sectional curvature among metrics of volume 1. The goal of this article is to provide estimates to minimal volume and minimal curvature on 4-dimensional compact manifolds involving some differential and topological invariants. Among these ones, we get some sharp estimates for minimal curvature.

Motivation & Objective

• To establish sharp lower bounds for minimal curvature on 4-dimensional compact Riemannian manifolds using curvature decomposition and topological invariants.
• To relate minimal curvature to minimal volume and the Euler characteristic via curvature functionals.
• To determine when minimal curvature achieves equality, identifying rigidity conditions for the manifold's geometry.
• To analyze the role of the Weyl tensor and self-dual curvature components in constraining the minimal curvature value.

Proposed method

• Decomposes the Riemann curvature tensor into self-dual and anti-self-dual parts using the Hodge star operator on 4-manifolds.
• Applies the curvature decomposition formula involving scalar curvature, Ricci curvature, and Weyl tensor to bound the $ L^\infty $-norm of the Riemann tensor.
• Uses the identity $ |Rm|^2 = \frac{s^2}{24} + |W|^2 + \frac{1}{2}|\mathring{Ric}|^2 $ to relate the total curvature to intrinsic geometric invariants.
• Introduces a functional $ \mathcal{R}^\infty(g) = |Rm(g)|_\infty $ and defines minimal curvature as the infimum over volume-one metrics.
• Employs variational techniques and integral estimates over $ M^4 $, leveraging the volume normalization $ \text{Vol}(M,g) = 1 $ to derive global inequalities.
• Applies the Gauss-Bonnet formula and topological constraints (e.g., $ \chi(M) = 2 + b_2 $) to link curvature bounds to Euler characteristic.

Experimental results

Research questions

• RQ1 Under what conditions does the minimal curvature achieve equality with $ 2\pi\sqrt{2|\chi(M)|} $?
• RQ2 How are minimal curvature and minimal volume related on 4-manifolds, and what topological invariants control this relationship?
• RQ3 What geometric structure arises when the minimal curvature is realized and the Weyl tensor is constant?
• RQ4 Can the minimal curvature be bounded below in terms of the self-dual curvature component $ K_1^\perp $?
• RQ5 What rigidity results follow when minimal curvature is minimized and sectional curvature is nonnegative?

Key findings

• The minimal curvature satisfies $ [\text{Mincur}(M)]^2 \geq \frac{|\mathcal{Y}_1^\perp(M)|^2}{72} $, providing a sharp lower bound in terms of the self-dual curvature functional.
• If $ \text{Mincur}(M) = 2\pi\sqrt{2|\chi(M)|} $, then the manifold is Einstein with constant Weyl curvature and scalar curvature.
• The equality case implies $ \int_M |\mathring{\text{Ric}}|^2 dV_g = 0 $, forcing the Ricci tensor to be proportional to the metric.
• Under nonnegative sectional curvature and equality in the minimal curvature bound, the manifold is isometric to $ \mathbb{S}^4 $, $ \mathbb{CP}^2 $, or $ \mathbb{S}^2 \times \mathbb{S}^2 $.
• The proof establishes that $ |Rm|^2 \geq 2|K_1^\perp|^2 $ pointwise, leading to the global $ L^\infty $-bound on minimal curvature.
• The equality case in the curvature decomposition implies $ w_3^\pm = w_2^\pm $, which forces the Weyl tensor components to be balanced.