Deformation of scalar curvature and volume. (English) Zbl 1278.53041

Let \(M\) be a closed manifold which has dimension at least 3. Let \(M_c\) be the space of Riemannian metrics on \(M\) with constant scalar curvature. Let \(V:M_c\rightarrow(0,\infty)\) be the volume function. For \(c\neq0\), critical points of the restricted volume map are the stationary points of the total scalar curvature \(R(g)=\int_MR(g)\) restricted to \(M_c\). Let \(L_g\) be the linearization of the scalar curvature operator and let \(L_g^*\) be the formal adjoint. Critical metrics for \(V_c\) admit non-trivial solutions \((f,\kappa)\) to the overdetermined elliptic system \(L_g^*f=\kappa g\) for \(\kappa\) constant. The authors localize this analysis to the case where the metric deformation is supported on the closure of a bounded domain \(\Omega\subset M\). The obstruction to finding such a deformation is the existence of a non-trivial solution \((f,\kappa)\) of the system \(L_g^*f=\kappa g\) on \(\Omega\); if such perturbation exists then the metric is called \(V\)-static with \(V\)-static potential \(f\).
The authors give a variational characterization of such metric – the boundary values play a central role. Section 1 contains a statement of results. Section 2 gives a variational characterization of \(V\)-static metrics. Section 3 contains the proof of one of the theorems in the introduction. Section 4 deals with constant scalar curvature gluing with a volume constraint. In Section 5, the gluing is localized. Section 6 gives counterexamples to a conjecture of Min-Oo with non-trivial topology and arbitrary large volume. Appendix A treats Schauder theory.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E11 Critical metrics
Full Text: DOI arXiv


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