Almost non-negatively curved 4-manifolds with torus symmetry

Harvey, John; Searle, Catherine

doi:10.1090/proc/15093

Almost non-negatively curved $4$-manifolds with torus symmetry
HTML articles powered by AMS MathViewer

by John Harvey and Catherine Searle PDF

Proc. Amer. Math. Soc. 148 (2020), 4933-4950 Request permission

Abstract:

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.

References

K. Bezdek, Ein elementarer Beweis für die isoperimetrische Ungleichung in der Euklidischen und hyperbolischen Ebene, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 27 (1984), 107–112 (1985) (German). MR 823098
Raoul Bott, Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231–244. MR 211416
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
Philip T. Church and Klaus Lamotke, Almost free actions on manifolds, Bull. Austral. Math. Soc. 10 (1974), 177–196. MR 370632, DOI 10.1017/S000497270004082X
Ronald Fintushel, Circle actions on simply connected $4$-manifolds, Trans. Amer. Math. Soc. 230 (1977), 147–171. MR 458456, DOI 10.1090/S0002-9947-1977-0458456-6
Fernando Galaz-Garcia, Nonnegatively curved fixed point homogeneous manifolds in low dimensions, Geom. Dedicata 157 (2012), 367–396. MR 2893494, DOI 10.1007/s10711-011-9615-y
Fernando Galaz-Garcia and Martin Kerin, Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension, Math. Z. 276 (2014), no. 1-2, 133–152. MR 3150196, DOI 10.1007/s00209-013-1190-5
Karsten Grove and Catherine Searle, Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra 91 (1994), no. 1-3, 137–142. MR 1255926, DOI 10.1016/0022-4049(94)90138-4
Karsten Grove and Burkhard Wilking, A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry, Geom. Topol. 18 (2014), no. 5, 3091–3110. MR 3285230, DOI 10.2140/gt.2014.18.3091
John Harvey and Catherine Searle, Orientation and symmetries of Alexandrov spaces with applications in positive curvature, J. Geom. Anal. 27 (2017), no. 2, 1636–1666. MR 3625167, DOI 10.1007/s12220-016-9734-7
Wu-Yi Hsiang and Bruce Kleiner, On the topology of positively curved $4$-manifolds with symmetry, J. Differential Geom. 29 (1989), no. 3, 615–621. MR 992332
Bruce Alan Kleiner, Riemannian four-manifolds with nonnegative curvature and continuous symmetry, ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Ph.D.)–University of California, Berkeley. MR 2685331
Shoshichi Kobayashi, Transformation groups in differential geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. MR 1336823
Zhong-dong Liu and Zhong Min Shen, On the Betti numbers of Alexandrov spaces, Ann. Global Anal. Geom. 12 (1994), no. 2, 123–133. MR 1277664, DOI 10.1007/BF02108293
Peter Orlik and Frank Raymond, Actions of the torus on $4$-manifolds. I, Trans. Amer. Math. Soc. 152 (1970), 531–559. MR 268911, DOI 10.1090/S0002-9947-1970-0268911-3
Jeff Parker, $4$-dimensional $G$-manifolds with $3$-dimensional orbits, Pacific J. Math. 125 (1986), no. 1, 187–204. MR 860758
G. Perelman, Alexandrov spaces with curvature bounded from below II, preprint (1991), available at https://anton-petrunin.github.io/papers/alexandrov/perelmanASWCBFB2+.pdf.
G. Perelman and A. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint (1995), available at https://anton-petrunin.github.io/papers/qg_ams.pdf.
Anton Petrunin, Applications of quasigeodesics and gradient curves, Comparison geometry (Berkeley, CA, 1993–94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203–219. MR 1452875, DOI 10.2977/prims/1195166129
Anton Petrunin, Semiconcave functions in Alexandrov’s geometry, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137–201. MR 2408266, DOI 10.4310/SDG.2006.v11.n1.a6
Conrad Plaut, Metric spaces of curvature $\geq k$, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 819–898. MR 1886682
Catherine Searle and DaGang Yang, On the topology of non-negatively curved simply connected $4$-manifolds with continuous symmetry, Duke Math. J. 74 (1994), no. 2, 547–556. MR 1272983, DOI 10.1215/S0012-7094-94-07419-X
Katsuhiro Shiohama, An introduction to the geometry of Alexandrov spaces, Lecture Notes Series, vol. 8, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. MR 1320267
Takashi Shioya and Takao Yamaguchi, Collapsing three-manifolds under a lower curvature bound, J. Differential Geom. 56 (2000), no. 1, 1–66. MR 1863020
Jeremy Wong, An extension procedure for manifolds with boundary, Pacific J. Math. 235 (2008), no. 1, 173–199. MR 2379775, DOI 10.2140/pjm.2008.235.173
Andreas Wörner, A splitting theorem for nonnegatively curved Alexandrov spaces, Geom. Topol. 16 (2012), no. 4, 2391–2426. MR 3033520, DOI 10.2140/gt.2012.16.2391
Takao Yamaguchi, Manifolds of almost nonnegative Ricci curvature, J. Differential Geom. 28 (1988), no. 1, 157–167. MR 950560