## 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.

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 58E11 Critical metrics
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### References:

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