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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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[1] Besse, A.: Einstein Manifolds. Springer, Berlin (1987) · Zbl 0613.53001
[2] Bray, H; Lee, DA, On the Riemannian Penrose inequality in dimensions less than eight, Duke Math. J., 148, 81-106, (2009) · Zbl 1168.53016
[3] Brendle, S; Marques, F; Neves, A, Deformations for the hemisphere that increase the scalar curvature, Invent. Math., 185, 175-197, (2010) · Zbl 1227.53048
[4] Chruściel, PT; Delay, E, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr. (N.S.), 94, 1-103, (2003) · Zbl 1058.83007
[5] Chruściel, PT; Isenberg, J; Pollack, D, Initial data engineering, Comm. Math. Phys., 257, 29-42, (2005) · Zbl 1080.83002
[6] Chruściel, PT; Pacard, F; Pollack, D, Singular Yamabe metrics and initial data with exactly kotter-Schwarzschild-de Sitter ends. II. generic metrics, Math. Res. Lett., 16, 157-164, (2009) · Zbl 1170.83004
[7] Corvino, J, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys., 214, 137-189, (2000) · Zbl 1031.53064
[8] Corvino, J., Pollack, D.: Scalar curvature and the Einstein constraint equations. In: Bray, H.L., Minicozzi, W.P. II (eds.) Surveys in Geometric Analysis and Relativity. Adv. Lect. Math. 20, 145-188 (2011) · Zbl 1268.53048
[9] Corvino, J; Schoen, RM, On the asymptotics of the vacuum Einstein constraint equations, J. Differ. Geom., 73, 185-217, (2006) · Zbl 1122.58016
[10] Delay, E, Localized gluing of Riemannian metrics in interpolating their scalar curvature, Differ. Geom. Appl., 29, 433-439, (2011) · Zbl 1219.53041
[11] Foote, R, Regularity of the distance function, Proc. Amer. Math. Soc., 92, 153-155, (1984) · Zbl 0528.53005
[12] Gromov, M., Lawson, H.B. Jr.: The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. (2) 111(3), 423-434 (1980). · Zbl 0463.53025
[13] Isenberg, J; Mazzeo, R; Pollack, D, Gluing and wormholes for the Einstein constraint equations, Comm. Math. Phys., 231, 529-568, (2002) · Zbl 1013.83008
[14] Isenberg, J; Maxwell, D; Pollack, D, A gluing construction for non-vacuum solutions of the Einstein constraint equations, Adv. Theor. Math. Phys., 9, 129-172, (2005) · Zbl 1101.83005
[15] Joyce, D, Constant scalar curvature metrics on connected sums, Int. J. Math. Math. Sci., 7, 405-450, (2003) · Zbl 1026.53019
[16] Kazdan, J.L., Warner, F.W.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math. 101(2), 317-331 (1975) · Zbl 0297.53020
[17] Kazdan, JL; Warner, FW, A direct approach to the determination of Gaussian and scalar curvature functions, Invent. Math., 28, 227-230, (1975) · Zbl 0297.53021
[18] Mazzeo, R; Pollack, D; Uhlenbeck, K, Connected sum constructions for constant scalar curvature metrics, Topol. Methods Nonlinear Anal., 6, 207-233, (1995) · Zbl 0866.58069
[19] Miao, P; Shi, Y-G; Tam, L-F, On geometric problems related to Brown-York and Liu-Yau quasilocal mass, Comm. Math. Phys., 298, 437-459, (2010) · Zbl 1200.58018
[20] Miao, P; Tam, L-F, On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. Partial Differ. Equ., 36, 141-171, (2009) · Zbl 1175.49043
[21] Miao, P; Tam, L-F, Einstein and conformally flat critical metrics of the volume functional, Trans. Amer. Math. Soc., 363, 2907-2937, (2011) · Zbl 1222.53041
[22] Miao, P; Tam, L-F, Scalar curvature rigidity with a volume constraint, Comm. Anal. Geom., 20, 1-30, (2012) · Zbl 1250.53041
[23] Min-Oo, M.: Scalar curvature rigidity of certain symmetric spaces. In: Lalonde, F. (ed.) Geometry, topology, and dynamics (Montreal, 1995). CRM Proc. Lecture Notes (Amer. Math. Soc., Providence RI), vol. 15, pp. 127-136 (1998) · Zbl 0911.53032
[24] Obata, M, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Jpn., 14, 333-340, (1962) · Zbl 0115.39302
[25] Schoen, RM, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math., 41, 317-392, (1988) · Zbl 0674.35027
[26] Schoen, R; Yau, S-T, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65, 45-76, (1979) · Zbl 0405.53045
[27] Schoen, RM; Yau, S-T, On the structure of manifolds with positive scalar curvature, Manuscripta Math., 28, 159-183, (1979) · Zbl 0423.53032
[28] Schneider, R., Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge (1993) · Zbl 1080.83002
[29] Shen, Y, A note on fischer-marsden’s conjecture, Proc. Amer. Math. Soc., 125, 901-905, (1997) · Zbl 0867.53035
[30] Shi, Y-G; Tam, L-F, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differ. Geom., 62, 79-125, (2002) · Zbl 1071.53018
[31] Tam, L.-F.: Private communication (2010) · Zbl 0528.53005
[32] Witten, E, A new proof of the positive energy theorem, Comm. Math. Phys., 80, 381-402, (1981) · Zbl 1051.83532
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