[Paper Review] Variational problems for Riemannian functionals and arithmetic groups

Alexander Nabutovsky, Shmuel Weinberger|arXiv (Cornell University)|Nov 8, 1997

advanced mathematical theories3 citations

TL;DR

This paper introduces a novel variational approach for studying Riemannian functionals on the space of isometry classes of metrics on compact manifolds of dimension ≥5. By reducing problems on a given manifold M^n to equivalent problems on a class of manifolds with controlled Ricci curvature and scalar curvature properties, the authors prove that the diameter functional on the closure of Riemannian structures with bounded sectional curvature admits infinitely many 'very deep' local minima.

ABSTRACT

In this paper we introduce a new approach to variational problems on the space Riem(M^n) of Riemannian structures (i.e. isometry classes of Riemannan metrics) on any fixed compact manifold M^n of dimension n >= 5. This approach often enables one to replace the considered variational problem on Riem(M^n) (or on some subset of Riem(M^n)) by the same problem but on spaces Riem(N^n) for every manifold N^n from a class of compact manifolds of the same dimension and with the same homology as M^n but with the following two useful properties: (1) If νis any Riemannian structure on any manifold N^n from this class such that Ric_(N^n,ν) >= -(n-1), then the volume of (N^n,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, Riem_1(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 R_1(M) denote the closure of Riem_1(M) in M et(M ). Then diameter regarded as a functional on R_1(M) has infinitely many "very deep" local minima.

Motivation & Objective

• To develop a new method for analyzing variational problems on the space of Riemannian structures on compact manifolds of dimension n ≥ 5.
• To replace variational problems on Riem(M^n) with equivalent problems on a class of model manifolds N^n sharing M^n's homology but with favorable geometric constraints.
• To establish geometric conditions—specifically Ricci curvature bounded below and absence of non-negative scalar curvature metrics—on the model manifolds to enable stronger analytical control.
• To apply the method to prove the existence of infinitely many 'very deep' local minima for the diameter functional on the closure of Riemannian structures with bounded sectional curvature.
• To demonstrate that such minima are structurally robust and not artifacts of symmetry or low dimensionality.

Proposed method

• Introduce a class of compact n-dimensional manifolds N^n with the same homology as M^n but with Ric(N^n, ν) ≥ -(n−1) implying vol(N^n, ν) > 1 for any such metric ν.
• Utilize the absence of Riemannian metrics with non-negative scalar curvature on these model manifolds to rule out certain geometric degenerations.
• Reduce variational problems on Riem(M^n) to equivalent problems on Riem(N^n) for all N^n in the constructed class, leveraging topological and geometric invariance.
• Apply Gromov-Hausdorff topology to analyze the closure R_1(M) of Riem_1(M) in the space of compact metric spaces homeomorphic to M.
• Use the interplay between curvature bounds, volume control, and scalar curvature obstructions to analyze critical points of the diameter functional.
• Establish that the diameter functional on R_1(M) has infinitely many local minima that are 'very deep'—i.e., not only local but with large energy gaps to nearby critical values.

Experimental results

Research questions

• RQ1 Can variational problems on the space of Riemannian structures on a fixed manifold M^n be reduced to equivalent problems on a class of model manifolds with controlled curvature and volume?
• RQ2 What geometric constraints on model manifolds ensure that volume is bounded below and non-negative scalar curvature is impossible?
• RQ3 Does the diameter functional on the closure of Riemannian structures with bounded sectional curvature admit infinitely many local minima?
• RQ4 Are these local minima 'very deep' in the sense of having significant energy gaps to neighboring critical values?
• RQ5 Can the existence of such minima be established via a reduction to manifolds with specific topological and curvature properties?

Key findings

• For any compact connected smooth manifold M^n of dimension n ≥ 5, the diameter functional on the closure R_1(M) of Riemannian structures with |sectional curvature| ≤ 1 has infinitely many local minima.
• These local minima are 'very deep', meaning they are separated by significant energy gaps from nearby critical values, indicating robustness.
• The reduction method relies on a class of model manifolds N^n with the same homology as M^n, satisfying Ric(N^n, ν) ≥ -(n−1) ⇒ vol(N^n, ν) > 1.
• The model manifolds do not admit any Riemannian metric of non-negative scalar curvature, which is essential for preventing degenerate behavior.
• The method allows transferring variational problems from Riem(M^n) to Riem(N^n) for all N^n in the class, preserving the structure of critical points.
• The result establishes a new mechanism for the proliferation of critical points in geometric variational problems via arithmetic group actions and curvature control.