## Duke Mathematical Journal

### Topology of Riemannian submanifolds with prescribed boundary

#### Abstract

We prove that a smooth compact submanifold of codimension $2$ immersed in $\mathbf{R}^{n},n\geq 3,$ bounds at most finitely many topologically distinct, compact, nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman

#### Article information

Source
Duke Math. J. Volume 152, Number 3 (2010), 533-565.

Dates
First available in Project Euclid: 20 April 2010

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1271783598

Digital Object Identifier
doi:10.1215/00127094-2010-018

Mathematical Reviews number (MathSciNet)
MR2654222

Zentralblatt MATH identifier
05708812

#### Citation

Alexander, Stephanie; Ghomi, Mohammad; Wong, Jeremy. Topology of Riemannian submanifolds with prescribed boundary. Duke Math. J. 152 (2010), no. 3, 533--565. doi:10.1215/00127094-2010-018. http://projecteuclid.org/euclid.dmj/1271783598.

#### References

• S. B. Alexander and R. J. Currier, Non-negatively curved ends of Euclidean hypersurfaces, Geom. Dedicata 40 (1991), 29--43.
• S. B. Alexander and M. Ghomi, The convex hull property and topology of hypersurfaces with nonnegative curvature, Adv. Math. 180 (2003), 324--354.
• —, The convex hull property of noncompact hypersurfaces with positive curvature, Amer. J. Math. 126, 2004, 891--897.
• S. B. Alexander, V. Kapovitch, and A. Petrunin, An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds, Illinois J. Math. 52 (2008), 1031--1033.
• A. D. Alexandrov and Y. G. Reshetnyak, General Theory of Irregular Curves, trans. L. Y. Yuzina, Math. Appl. (Soviet Series) 29, Kluwer, Dordrecht, 1989.
• A. D. Alexandrov and V. A. Zalgaller, Intrinsic geometry of surfaces, trans. J. M. Danskin, Trans. Math. Monogr. 15, Amer. Math. Soc., Providence, 1967.
• I. Belegradek and V. Kapovitch, Finiteness theorems for nonnegatively curved vector bundles, Duke Math. J. 108 (2001), 109--134.
• D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, Grad. Studies in Math. 33, Amer. Math. Soc., Providence, 2001.
• Y. D. Burago and S. Z. Shefel, The geometry of surfaces in Euclidean spaces'' in Geometry, III, Encyclopaedia Math. Sci. 48, Springer, Berlin, 1992, 1--85. 251--256. ;$\!$
• H. Busemann and W. Feller, Krümmungseigenschaften Konvexer Flächen, Acta Math. 66 (1936), 1--47.
• M. Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 371--377.
• J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413--443.
• M. P. Do Carmo, and F. W. Warner, Rigidity and convexity of hypersurfaces in spheres, J. Differential Geometry 4 (1970), 133--144.
• M. Ghomi, Strictly convex submanifolds and hypersurfaces of positive curvature, J. Differential Geom. 57 (2001), 239--271.
• —, Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature, Topology 41 (2002), 107--117.
• M. Ghomi and R. Greene, Relative isometric embeddings of Riemannian manifolds, to appear in Trans. Amer. Math. Soc.
• M. Ghomi and M. Kossowski, $h$-principles for hypersurfaces with prescribed principal curvatures and directions, Trans. Amer. Math. Soc. 358 (2006), 4379--4393.
• H. Gluck and L.-H. Pan, Embedding and knotting of positive curvature surfaces in $3$-space, Topology 37 (1998), 851--873.
• D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75--90.
• B. Guan and J. Spruck, Locally convex hypersurfaces of constant curvature with boundary, Comm. Pure Appl. Math. 57 (2004), 1311--1331.
• P. Hartman and L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901--920.
• L. Hauswirth, Bridge principle for constant and positive Gauss curvature surfaces, Comm. Anal. Geom. 7 (1999), 497--550.
• M. W. Hirsch, The imbedding of bounding manifolds in euclidean space, Ann. of Math. (2) 74 (1961), 494--497.
• V. Kapovitch, Perelman's stability theorem'' in Surveys in Differential Geometry, Vol. XI, Surv. Differ. Geom. 11, Int. Press, Somerville, Mass., 2007.
• N. N. Kosovskiĭ, Gluing of Riemannian manifolds of curvature $\geq \kappa$ (in Russian), Algebra i Analiz 14 (2002), 140--157.; English translation in St. Petersburg J. Math. 14 (2005), 467--478.
• J. Mccuan, Positively curved surfaces with no tangent support plane, Proc. Amer. Math. Soc. 133 (2005), 263--273.
• B. O'Neill, Semi-Riemannian Geometry: With Applications to Relativity, Pure and Appl. Math. 103, Academic Press, New York, 1983.
• G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. 40 (1994), 209--212.
• —, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers'' in Comparison Geometry (Berkeley, Calif., 1993--94.), Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press, Cambridge, 1997, 157--163.
• —, Alexandrov's spaces with curvature bounded from below, II, preprint, 1991.
• H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), 211--239.
• R. Sacksteder, On hypersurfaces with no negative sectional curvatures, Amer. J. Math. 82 (1960), 609--630.
• K. Shiohama, T. Shioya, and M. Tanaka. The Geometry of Total Curvature on Complete Open Surfaces, Cambridge Tracts in Math. 159, Cambridge Univ. Press, Cambridge, 2003.
• M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. IV, 2nd ed., Publish or Perish, Wilmington, Del., 1979.
• N. S. Trudinger and X.-J. Wang, On locally convex hypersurfaces with boundary, J. Reine Angew. Math. 551 (2002), 11--32.
• V. V. Usov, The three-dimensional swerve of curves on convex surfaces, (in Russian), Sibirsk. Mat. Ž. 17 (1976), 1427--1430.; English translation in Siberian Math. J. 17 (1976), 1427--1430.
• J. Van Heijenoort, On locally convex manifolds, Comm. Pure Appl. Math. 5 (1952), 223--242.
• H.-H. Wang, Boundary convexity on manifolds with nonnegative Ricci curvature, Ph.D. dissertation, Indiana University, Bloomington, Ind., 1997.
• J. Wong, An extension procedure for manifolds with boundary, Pacific J. Math. 235 (2008), 173--199.
• S.-T. Yau, Open problems in geometry'' in Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, Calif. 1990), Proc. Sympos. Pure Math. 54, Part 1, Amer. Math. Soc., Providence, 1993, 1--28.
• E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471--495.