Metrics with nonnegative Ricci curvature on convex three-manifolds

Aché, Antonio¹ ; Maximo, Davi² ; Wu, Haotian³

¹ Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, United States
² Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, United States
³ Department of Mathematics, University of Oregon, Fenton Hall, Eugene, OR 97403, United States

Geometry & topology, Tome 20 (2016) no. 5, pp. 2905-2922.

Résumé

We prove that the space of smooth Riemannian metrics on the three-ball with nonnegative Ricci curvature and strictly convex boundary is path-connected, and, moreover, that the associated moduli space (ie modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes and Smith (to appear in J. Differential Geom.), we show the existence of a properly embedded free boundary minimal annulus on any three-ball with nonnegative Ricci curvature and strictly convex boundary.

Détail

DOI : 10.2140/gt.2016.20.2905

Classification : 53C21
Keywords: moduli space of metrics, gluing positive Ricci curvature, Ricci flow, manifolds with convex boundary

Bibliographie

[1] T Aubin, Métriques riemanniennes et courbure, J. Differential Geom. 4 (1970) 383

[2] R L Bishop, Infinitesimal convexity implies local convexity, Indiana Univ. Math. J. 24 (1975) 169 | DOI

[3] B Botvinnik, P B Gilkey, Metrics of positive scalar curvature on spherical space forms, Canad. J. Math. 48 (1996) 64 | DOI

[4] B Botvinnik, B Hanke, T Schick, M Walsh, Homotopy groups of the moduli space of metrics of positive scalar curvature, Geom. Topol. 14 (2010) 2047 | DOI

[5] S Brendle, A monotonicity formula for mean curvature flow with surgery, preprint (2013)

[6] R Carr, Construction of manifolds of positive scalar curvature, Trans. Amer. Math. Soc. 307 (1988) 63 | DOI

[7] J Cerf, La nullité de π0(DiffS3), from: "Séminaire Henri Cartan", Secrétariat mathématique (1964) 25

[8] P Ehrlich, Metric deformations of curvature, I : Local convex deformations, Geometriae Dedicata 5 (1976) 1 | DOI

[9] A E Fischer, The theory of superspace, from: "Proceedings of the Relativity Conference in the Midwest" (editors M Carmeli, S I Fickler, L Witten), Plenum (1970) 303

[10] A Fraser, M M C Li, Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary, J. Differential Geom. 96 (2014) 183

[11] P Gianniotis, The Ricci flow on manifolds with boundary, preprint (2012)

[12] M Gromov, H B Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. (1983) 83

[13] R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255

[14] B Hanke, T Schick, W Steimle, The space of metrics of positive scalar curvature, Publ. Math. Inst. Hautes Études Sci. 120 (2014) 335 | DOI

[15] N Hitchin, Harmonic spinors, Advances in Math. 14 (1974) 1 | DOI

[16] G Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985) 47

[17] J Jost, The geometric calculus of variations: a short survey and a list of open problems, Exposition. Math. 6 (1988) 111

[18] M Kreck, S Stolz, Nonconnected moduli spaces of positive sectional curvature metrics, J. Amer. Math. Soc. 6 (1993) 825 | DOI

[19] J Lott, Mean curvature flow in a Ricci flow background, Comm. Math. Phys. 313 (2012) 517 | DOI

[20] F C Marques, Deforming three-manifolds with positive scalar curvature, Ann. of Math. 176 (2012) 815 | DOI

[21] D Maximo, I Nunes, G Smith, Free boundary minimal annuli in convex three-manifolds, To appear in J. Differential Geom.

[22] W Meeks Iii, L Simon, S T Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. 116 (1982) 621 | DOI

[23] J Milnor, On spaces having the homotopy type of a CW–complex, Trans. Amer. Math. Soc. 90 (1959) 272 | DOI

[24] G Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers, from: "Comparison geometry" (editors K Grove, P Petersen), Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press (1997) 157

[25] J Rosenberg, S Stolz, Metrics of positive scalar curvature and connections with surgery, from: "Surveys on surgery theory, 2" (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 353

[26] M Walsh, The space of positive scalar curvature metrics on a manifold with boundary, preprint (2014)

[27] H H Wang, Boundary convexity of manifolds with nonnegative Ricci curvature, PhD thesis, Indiana University (1997)

[28] H Weyl, Über die Gleichverteilung von Zahlen mod, I, Math. Ann. 77 (1916) 313 | DOI

[29] D J Wraith, On the moduli space of positive Ricci curvature metrics on homotopy spheres, Geom. Topol. 15 (2011) 1983 | DOI