Conformal geometry, contact geometry, and the calculus of variations
type: Item
instance of: scholarly article
title: Conformal geometry, contact geometry, and the calculus of variations
zbMATH Open document ID: 0990.53035
DOI: 10.1215/S0012-7094-00-10127-5
author: Jeff Viaclovsky
published in: Duke Mathematical Journal
publication date: 2002-08-22T00:00:00Z
review text: Let \((N,g_0)\) be a compact Riemannian manifold of dimension \(n\geq 3\). In this paper, the author considers the curvature equations \[ \sigma_k\left( \text{Ric}_g-{R_g\over 2(n-1)} \cdot g\right)= \text{constant} \tag{1} \] for metrics \(g\) in the conformal class of \(g_0\), where \(\text{Ric}_g\) and \(R_g\) are the Ricci tensor and the scalar curvature of the metric \(g\), respectively, and where \(\sigma_k\) is the \(k\)-th elementary symmetric function of the eigenvalues \((g\) is used to view \((2,0)\)-tensors as endomorphisms of the tangent bundle). The case \(k=1\), \(R_g=\) constant is known as the Yamabe problem. The first theorem shows that unit volume solutions of (1) are critical points of the functional \[ {\mathcal F}_k:g \mapsto\int_N \sigma_k\left( \text{Ric}_g-{R_g\over 2(n-1)} \cdot g\right)d V_g\tag{2} \] when \((N,[g])\) is locally conformally flat and \(k\neq n/2\). If \(N\) is not locally conformally flat, the statement is true for \(k=1,2\). From this result, it follows that, for all \(k\), a positive scalar curvature Einstein metric is a strict local minimum unless \((N,[g_0])\) is conformally equivalent to the standard unit sphere \(S^n\), in which case there is an \((n+1)\)-parameter family of local minima. If \(N\) is not locally conformally flat, again the statement holds for \(k=1,2\). Further, if \((N,g_0)\) is of unit volume and constant sectional curvature \(K\neq 0\), then it is proven that, for \(1\leq k<n-1\), \(g_0\) is the unique unit volume solution of (1) in its conformal class unless \((N,g_0)\) is isometric to the \(S^n\) with the standard metric. In this case there is an \((n+1)\)-parameter family of solutions obtained as images of the standard metric under the conformal transformations of \(S^n\). The methods used are geometric and employ relations between conformal geometry and contact geometry to show the correspondence between the solutions of (1) and the critical points of (2). The characterization of the solutions …
reviewed by: Lorenzo Nicolodi
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 58E11
zbMATH DE Number: 1442144
zbMATH Keywords: locally conformally flat spaces
zbMATH Keywords: compact Riemannian manifold
zbMATH Keywords: curvature equations
zbMATH Keywords: Yamabe problem
zbMATH Keywords: critical points
MaRDI profile type: Publication
cites work: Q4146134
cites work: Q4720334
cites work: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
cites work: Q3262807
cites work: A differential geometric generalization of characteristics
cites work: Q5332396
cites work: Transformation groups in differential geometry.
cites work: The Yamabe problem
cites work: Moduli Spaces of Singular Yamabe Metrics
cites work: Certain conditions for a Riemannian manifold to be isometric with a sphere
cites work: The conjectures on conformal transformations of Riemannian manifolds
cites work: The principle of symmetric criticality
cites work: On the Hessian of a function and the curvature of its graph
cites work: Q3480816
cites work: Conformally invariant Monge-Ampère equations: Global solutions
cites work: Q4300995
description: scientific article; zbMATH DE number 1442144