×

Critical metrics of the volume functional on compact three-manifolds with smooth boundary. (English) Zbl 1372.58010

A Miao-Tam critical metric is a 3-tuple \((M^n,\,g,\,f)\) where \((M^{n},\,g)\) is a compact Riemannian manifold of dimension \(n\geq 3\) with a smooth boundary \(\partial M\) and \(f: M^{n}\to \mathbb{R}\) is a smooth function such that \(f^{-1}(0)=\partial M\) satisfying the overdetermined-elliptic system \(\mathfrak{L}_{g}^{*}(f)=g\) where \(\mathfrak{L}_{g}^{*}(f)=-(\Delta f)g+\mathrm{Hess } f-\mathrm{Ric}\) is the formal \(L^{2}\)-adjoint of the linearization of the scalar curvature operator \(\mathfrak{L}_{g}\). Such a function \(f\) is called a potential function. In [Calc. Var. Partial Differ. Equ. 36, No. 2, 141–171 (2009; Zbl 1175.49043)] P. Miao and L.-F. Tam showed that these critical metrics arise as critical points of the volume functional on \(M^n\) when restricted to the class of metrics \(g\) with prescribed constant scalar curvature such that \(g_{|_{T \partial M}}=h\) for a prescribed Riemannian metric \(h\) on the boundary. In this paper, the authors obtain an estimate to the area of the boundary \(\partial M\) of Miao-Tam critical metrics on compact three-manifolds. In addition, the authors obtain a Böchner type formula which enables to show that a Miao-Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in \(\mathbb{S}^3\).

MSC:

58E11 Critical metrics
58D17 Manifolds of metrics (especially Riemannian)
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C65 Integral geometry

Citations:

Zbl 1175.49043
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Ambrozio, L.: On static three-manifolds with positive scalar curvature. arXiv:1503.03803v1 [math.DG] (2015)
[2] Baltazar, H., Ribeiro Jr., E.: Critical metrics of the volume functional on manifolds with boundary. arXiv:1603.02932 [math.DG] (2016) · Zbl 1368.53035
[3] Barros, A; Diógenes, R; Ribeiro, E, Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary, J. Geom. Anal., 25, 2698-2715, (2015) · Zbl 1335.53042
[4] Besse, A.: Einstein Manifolds. Springer, Berlin (1987) · Zbl 0613.53001
[5] Boucher, W; Gibbons, G; Horowitz, G, Uniqueness theorem for anti-de Sitter spacetime, Phys. Rev. D (3), 30, 2447-2451, (1984)
[6] Cao, H-D; Chen, Q, On bach-flat gradient shrinking Ricci solitons, Duke Math. J., 162, 1149-1169, (2013) · Zbl 1277.53036
[7] Corvino, J, Scalar curvature deformations and a gluing construction for the Einstein constraint equations, Commun. Math. Phys., 214, 137-189, (2000) · Zbl 1031.53064
[8] Corvino, J; Eichmair, M; Miao, P, Deformation of scalar curvature and volume, Math. Annalen., 357, 551-584, (2013) · Zbl 1278.53041
[9] Chow, B., et al.: The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects. Mathematical Surveys and Monographs, vol. 135. American Mathematical Society, Providence (2007)
[10] Fan, X-Q; Shi, Y-G; Tam, L-F, Large-sphere and small-sphere limits of the Brown-York mass, Commun. Anal. Geom., 17, 37-72, (2009) · Zbl 1175.53083
[11] Fischer, A; Marsden, J, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Am. Math. Soc., 80, 479-484, (1974) · Zbl 0288.53040
[12] Gilbard, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Heidelberg (1983) · Zbl 0361.35003
[13] Hijazi, O; Montiel, S; Raulot, S, Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant, Ann. Glob. Anal. Geom., 47, 167-178, (2014) · Zbl 1318.53054
[14] Kobayashi, O, A differential equation arising from scalar curvature function, J. Math. Soc. Jpn., 34, 665-675, (1982) · Zbl 0486.53034
[15] Kobayashi, O., Obata, M.: Conformally-Flatness and Static Space-Time, Manifolds and Lie Groups. (Notre Dame, IN, 1980). Progress in Mathematics, vol. 14, pp. 197-206. Birkhäuser, Boston (1981) · Zbl 0485.58020
[16] Lafontaine, J, Sur la géométrie d’une généralisation de l’équation différentielle d’obata, J. Math. Pures Appl., 62, 63-72, (1983) · Zbl 0513.53046
[17] Miao, P; Tam, L-F, On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. PDE, 36, 141-171, (2009) · Zbl 1175.49043
[18] Miao, P; Tam, L-F, Einstein and conformally flat critical metrics of the volume functional, Trans. Am. Math. Soc., 363, 2907-2937, (2011) · Zbl 1222.53041
[19] Okumura, M, Hypersurfaces and a pinching problem on the second fundamental tensor, Am. J. Math., 96, 207-213, (1974) · Zbl 0302.53028
[20] Shen, Y, A note on fischer-marsden’s conjecture, Proc. Am. Math. Soc., 125, 901-905, (1997) · Zbl 0867.53035
[21] Viaclovsky, J.: Topics in Riemannian Geometry. Notes of Course Math 865, Fall 2007. http://www.math.wisc.edu/ jeffv/courses/865_Fall_2007
[22] Wald, R.: General Relativity. University of Chicago Press, Chicago (1984) · Zbl 0549.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.