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Degree Theory of Immersed Hypersurfaces
About this Title
Harold Rosenberg and Graham Smith
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 265, Number 1290
ISBNs: 978-1-4704-4185-2 (print); 978-1-4704-6148-5 (online)
DOI: https://doi.org/10.1090/memo/1290
Published electronically: April 8, 2020
Keywords: Degree theory,
immersions,
convexity,
prescribed curvature,
non-linear elliptic PDEs
MSC: Primary 58D10; Secondary 58B05, 58C40, 58J05
Table of Contents
Chapters
- 1. Introduction
- 2. Degree theory
- 3. Applications
- A. Weakly smooth maps
- B. Prime immersions
Abstract
We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where $K$ is mean curvature, extrinsic curvature and special Lagrangian curvature, and we show that in all these cases, this number is equal to $-\chi (M)$, where $\chi (M)$ is the Euler characteristic of the ambient manifold $M$.- N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249. MR 92067
- Marcel Berger, A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003. MR 2002701
- Luis Caffarelli, Louis Nirenberg, and Joel Spruck, Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Appl. Math. 41 (1988), no. 1, 47–70. MR 917124, DOI 10.1002/cpa.3160410105
- Tobias H. Colding and Camillo De Lellis, The min-max construction of minimal surfaces, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 75–107. MR 2039986, DOI 10.4310/SDG.2003.v8.n1.a3
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- K. D. Elworthy and A. J. Tromba, Degree theory on Banach manifolds, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 86–94. MR 0277009
- José M. Espinar and Harold Rosenberg, When strictly locally convex hypersurfaces are embedded, Math. Z. 271 (2012), no. 3-4, 1075–1090. MR 2945598, DOI 10.1007/s00209-011-0904-9
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR 0473443
- Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0348781
- Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. MR 656198, DOI 10.1090/S0273-0979-1982-15004-2
- Gerhard Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463–480. MR 837523, DOI 10.1007/BF01388742
- Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
- F. Labourie, Problèmes de Monge-Ampère, courbes holomorphes et laminations, Geom. Funct. Anal. 7 (1997), no. 3, 496–534 (French, with English summary). MR 1466336, DOI 10.1007/s000390050017
- François Labourie, Immersions isométriques elliptiques et courbes pseudo-holomorphes, Geometry and topology of submanifolds (Marseille, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 131–140 (French). MR 1116136
- Davi Maximo, Ivaldo Nunes, and Graham Smith, Free boundary minimal annuli in convex three-manifolds, J. Differential Geom. 106 (2017), no. 1, 139–186. MR 3640009
- John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640
- F. Pacard and X. Xu, Constant mean curvature spheres in Riemannian manifolds, Manuscripta Math. 128 (2009), no. 3, 275–295. MR 2481045, DOI 10.1007/s00229-008-0230-7
- A. Robeday, Masters Thesis, Univ. Paris VII
- Harold Rosenberg and Matthias Schneider, Embedded constant-curvature curves on convex surfaces, Pacific J. Math. 253 (2011), no. 1, 213–218. MR 2869442, DOI 10.2140/pjm.2011.253.213
- Matthias Schneider, Closed magnetic geodesics on $S^2$, J. Differential Geom. 87 (2011), no. 2, 343–388. MR 2788659
- Matthias Schneider, Closed magnetic geodesics on closed hyperbolic Riemann surfaces, Proc. Lond. Math. Soc. (3) 105 (2012), no. 2, 424–446. MR 2959932, DOI 10.1112/plms/pds013
- F. R. Smith, On the existence of embedded minimal $2$-spheres in the $3$-sphere, endowed with an arbitrary metric, Bull. Austral. Math. Soc., 28, (1983), 159–160
- Graham Smith, An Arzela-Ascoli theorem for immersed submanifolds, Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), no. 4, 817–866 (English, with English and French summaries). MR 2789720
- G. Smith, Constant curvature hyperspheres and the Euler Characteristic, arXiv:1103.3235
- G. Smith, The Plateau Problem for General Curvature Functions, arXiv:1008.3545
- Graham Smith, Special Lagrangian curvature, Math. Ann. 355 (2013), no. 1, 57–95. MR 3004576, DOI 10.1007/s00208-011-0773-x
- Graham Smith, Bifurcation of solutions to the Allen-Cahn equation, J. Lond. Math. Soc. (2) 94 (2016), no. 3, 667–687. MR 3614923, DOI 10.1112/jlms/jdw053
- Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532832
- A. J. Tromba, The Euler characteristic of vector fields on Banach manifolds and a globalization of Leray-Schauder degree, Adv. in Math. 28 (1978), no. 2, 148–173. MR 493919, DOI 10.1016/0001-8708(78)90061-0
- Brian White, The space of $m$-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J. 36 (1987), no. 3, 567–602. MR 905611, DOI 10.1512/iumj.1987.36.36031
- Brian White, Every three-sphere of positive Ricci curvature contains a minimal embedded torus, Bull. Amer. Math. Soc. (N.S.) 21 (1989), no. 1, 71–75. MR 994891, DOI 10.1090/S0273-0979-1989-15765-0
- Brian White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom. 33 (1991), no. 2, 413–443. MR 1094464
- Brian White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200. MR 1101226, DOI 10.1512/iumj.1991.40.40008
- Rugang Ye, Foliation by constant mean curvature spheres, Pacific J. Math. 147 (1991), no. 2, 381–396. MR 1084717