Mean curvature flow of compact spacelike submanifolds in higher codimension

Guilfoyle, Brendan; Klingenberg, Wilhelm

doi:10.1090/tran/7766

Mean curvature flow of compact spacelike submanifolds in higher codimension
HTML articles powered by AMS MathViewer

by Brendan Guilfoyle and Wilhelm Klingenberg PDF

Trans. Amer. Math. Soc. 372 (2019), 6263-6281 Request permission

Abstract:

We prove the longtime existence for mean curvature flow of a smooth $n$-dimensional spacelike submanifold of an ($n+m$)-dimensional manifold whose metric satisfies the timelike curvature condition.

References

Dmitri V. Alekseevsky, Brendan Guilfoyle, and Wilhelm Klingenberg, On the geometry of spaces of oriented geodesics, Ann. Global Anal. Geom. 40 (2011), no. 4, 389–409. MR 2836829, DOI 10.1007/s10455-011-9261-5
Luigi Ambrosio and Halil Mete Soner, A measure-theoretic approach to higher codimension mean curvature flows, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 27–49 (1998). Dedicated to Ennio De Giorgi. MR 1655508
Henri Anciaux, Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2699–2718. MR 3165652, DOI 10.1090/S0002-9947-2013-05972-7
Robert Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Comm. Math. Phys. 94 (1984), no. 2, 155–175. MR 761792
Jingyi Chen and Jiayu Li, Mean curvature flow of surface in $4$-manifolds, Adv. Math. 163 (2001), no. 2, 287–309. MR 1864836, DOI 10.1006/aima.2001.2008
Klaus Ecker, Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space, J. Differential Geom. 46 (1997), no. 3, 481–498. MR 1484889
Klaus Ecker and Gerhard Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys. 135 (1991), no. 3, 595–613. MR 1091580
Nikos Georgiou and Brendan Guilfoyle, On the space of oriented geodesics of hyperbolic 3-space, Rocky Mountain J. Math. 40 (2010), no. 4, 1183–1219. MR 2718810, DOI 10.1216/RMJ-2010-40-4-1183
N. Georgiou and B. Guilfoyle, A new geometric structure on tangent bundles, Preprint, https://arxiv.org/abs/1806.05440, 2018.
Claus Gerhardt, Hypersurfaces of prescribed mean curvature in Lorentzian manifolds, Math. Z. 235 (2000), no. 1, 83–97. MR 1785072, DOI 10.1007/s002090000124
Yamile Godoy and Marcos Salvai, Global smooth geodesic foliations of the hyperbolic space, Math. Z. 281 (2015), no. 1-2, 43–54. MR 3384862, DOI 10.1007/s00209-015-1474-z
Brendan Guilfoyle and Wilhelm Klingenberg, An indefinite Kähler metric on the space of oriented lines, J. London Math. Soc. (2) 72 (2005), no. 2, 497–509. MR 2156666, DOI 10.1112/S0024610705006605
B. Guilfoyle and W. Klingenberg, Proof of the Carathéodory conjecture, https://arxiv.org/abs/0808.0851, 2013.
Kuo-Wei Lee and Yng-Ing Lee, Mean curvature flow of the graphs of maps between compact manifolds, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5745–5759. MR 2817407, DOI 10.1090/S0002-9947-2011-05204-9
Guanghan Li and Isabel M. C. Salavessa, Mean curvature flow of spacelike graphs, Math. Z. 269 (2011), no. 3-4, 697–719. MR 2860260, DOI 10.1007/s00209-010-0768-4
Yasuo Matsushita, Fields of $2$-planes and two kinds of almost complex structures on compact $4$-dimensional manifolds, Math. Z. 207 (1991), no. 2, 281–291. MR 1109666, DOI 10.1007/BF02571388
Marcos Salvai, Global smooth fibrations of $\Bbb R^3$ by oriented lines, Bull. Lond. Math. Soc. 41 (2009), no. 1, 155–163. MR 2482001, DOI 10.1112/blms/bdn115
R. Schoen and J. Wolfson, Minimizing area among Lagrangian surfaces: the mapping problem, J. Differential Geom. 58 (2001), no. 1, 1–86. MR 1895348
Knut Smoczyk, Self-shrinkers of the mean curvature flow in arbitrary codimension, Int. Math. Res. Not. 48 (2005), 2983–3004. MR 2189784, DOI 10.1155/IMRN.2005.2983
R. P. Thomas and S.-T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113. MR 1957663, DOI 10.4310/CAG.2002.v10.n5.a8
Benjamin Stuart Thorpe, A regularity theorem for graphic spacelike mean curvature flows, Pacific J. Math. 255 (2012), no. 2, 463–487. MR 2928561, DOI 10.2140/pjm.2012.255.463
Mu-Tao Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension, Invent. Math. 148 (2002), no. 3, 525–543. MR 1908059, DOI 10.1007/s002220100201