1997/11/08 by Alexander Nabutovsky, Shmuel Weinberger, Nabutovsky, Alexander +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #advanced mathematical theories #math.DG
paper · pdf · doi:10.48550/arxiv.math/9711225
arxiv created 1997/11/08 · openalex publication_date 1997/11/08 · arxiv updated 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce a new approach to variational problems on the space Riem(Mn) of Riemannian structures (i.e. isometry classes of Riemannan metrics) on any fixed compact manifold Mn of dimension n >= 5. This approach often enables one to replace the considered variational problem on Riem(Mn) (or on some subset of Riem(Mn)) by the same problem but on spaces Riem(Nn) for every manifold Nn from a class of compact manifolds of the same dimension and with the same homology as Mn but with the following two useful properties: (1) If νis any Riemannian structure on any manifold Nn from this class such that Ric_(Nn,ν) >= -(n-1), then the volume of (Nn,ν) is greater than one; and (2) Manifolds from this class do not admit Riemannian metrics of non-negative scalar curvature. As a first application we prove a theorem which can be informally explained as follows: Let M be any compact connected smooth manifold of dimension greater than four, M et(M) be the space of isometry classes of compact metric spaces homeomorphic to M endowed with the Gromov-Hausdorff topology, Riem1(M) in M et(M ) be the space of Riemannian structures on M such that the absolute values of sectional curvature do not exceed one, and R1(M) denote the closure of Riem1(M) in M et(M ). Then diameter regarded as a functional on R1(M) has infinitely many "very deep" local minima.