The scalar curvature and the biorthogonal curvature: A pinching problem

2012/12/28 by Ezio Costa, Costa, Ezio Araujo
Mathematics · Physics and Astronomy · #53C25 #53c24 #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1212.6516

openalex publication_date 2012/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The famous pinching problem says that on a compact simply connected n-manifold if its sectional curvature satisfies K_min > (1/4)K_max > 0, then the manifold is homeomorphic to the sphere. In [8, problem 12], S. T. Yau proposed the following problem: If we replace K_max by the scalar curvature, can we deduce similar pinching theorems? In our present note we give an answer to this question in dimension n = 4.

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