Mean curvature flow and Riemannian submersions
Tóm tắt
We prove that a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion is that the submersion has minimal fibers. We then lift some results taken from the literature (i.e., Andrews and Baker in J Differ Geom 85:357–395, 2010; Baker in The mean curvature flow of submanifolds of high codimension, 2011; Huisken in J Differ Geom 20:237–266, 1984; Math Z 195:205–219, 1987; Pipoli and Sinestrari in Mean curvature flow of pinched submanifolds of
$${\mathbb {CP}}^{n}$$
) to create new examples of evolution by mean curvature flow. In particular we consider the evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg group and of the tangent sphere bundle equipped with the Sasaki metric.
Tài liệu tham khảo
Andrews, B., Baker, C.: Mean curvature flow of pinched submanifolds to spheres. J. Differ. Geom. 85, 357–395 (2010)
Baker, C.: The mean curvature flow of submanifolds of high codimension. Ph.D. thesis. Australian National University. arXiv:1104.4409v1 [math.DG] (2011)
Besse, A.L.: Einstein Manifolds. Springer, Hidelberg (1987)
Escobales Jr., R.H.: Riemannian submersions with totally geodesic fibers. J. Differ. Geom. 10, 253–276 (1975)
Escobales Jr., R.H.: Riemannian submersions from complex projective space. J. Differ. Geom. 13, 93–107 (1978)
Falcitelli, M., Ianus, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific, River Edge, NJ (2004)
Gudmundsson, S., Kappos, E.: On the geometry of tangent bundles. Expos. Math. 20, 1–41 (2002)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)
Huisken, G.: Deforming hypersurfaces of the sphere by their mean curvature. Math. Z. 195, 205–219 (1987)
Kowalski, O., Sekizawa, M.: On tangent sphere bundles with small or large constant radius. Ann. Glob. Anal. Geom. 18, 207–219 (2000)
Marenich, V.: Geodesics in Heisenberg groups. Geom. Dedicata 66(2), 175–185 (1997)
Nguyen, H.T.: Convexity and cylindrical estimates for mean curvature flow in the sphere. Trans. Am. Math. Soc. 367, 4517–4536 (2015)
O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)
Pacini, T.: Mean curvature flow, orbits, moment maps Trans. Am. Math. Soc. 355(8), 3343–3357 (2003)
Pipoli, G., Sinestrari, C.: Mean curvature flow of pinched submanifolds of \({\mathbb{CP}}^{n}\), arXiv:1502.00519 [math.DG]
Smoczyk, K.: Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature. Calc Var Partial Differ Equ 4(2), 155–170 (1996)