Some recent developments in differential geometry
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Keywords
Curvature Flow Isoperimetric Inequality Simple Closed Curve Round Sphere Soap Bubble
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References
- 1.A. D. Alexandrov, Uniqueness theorems for surfaces in the large.Vestik Leningrad Univ. 13 (1958), 5–8; English trans.,Amer. Math. Soc. Transl. (2) 21 (1962), 412-16.Google Scholar
- 2.F. J. Almgren, Jr., Optimal isoperimetric inequalities.Indiana University Math. J. 35 (1986), 451–547.CrossRefMATHMathSciNetGoogle Scholar
- 3.K. Brakke, The motion of a surface by its mean curvature,Math. Notes No. 20, (1978) Princeton Univ. Press.Google Scholar
- 4.J. Eells, The surfaces of Delaunay,Math. Intelligencer 9 (1987), 53–57.CrossRefMATHMathSciNetGoogle Scholar
- 5.H. Fédérer and W. H. Fleming, Normal and integral currents.Ann. of Math. 72 (1960), 458–520.CrossRefMATHMathSciNetGoogle Scholar
- 6.M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves;J. Differential Geom. 23 (1986), 69–96.MATHMathSciNetGoogle Scholar
- 7.M. Grayson, The heat equation shrinks embedded plane curves to round points.J. Differential Geom. 26 (1987), 285–314.MATHMathSciNetGoogle Scholar
- 8.H. Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen,Math. Nachr. 4 (1951), 232–249.CrossRefMATHMathSciNetGoogle Scholar
- 9.H. Hopf, Differential geometry in the large,Lecture Notes in Mathematics No. 1000, (1983) Springer-Verlag.Google Scholar
- 10.G. Huisken, Flow by mean curvature of convex surfaces into spheres.J. Differential Geom. 20 (1984), 237–266.MATHMathSciNetGoogle Scholar
- 11.W. Y. Hsiang, Generalized rotational hypersurfaces of constant mean curvature in the euclidean spaces,J. Differential Geom. 17 (1982), 337–356.MATHMathSciNetGoogle Scholar
- 12.N. Kapouleas, Constant mean curvature surfaces in euclidean three space, (research announcement),Bull. Amer. Math. Soc. 17 (1987), 318–320.CrossRefMATHMathSciNetGoogle Scholar
- 13.N. Kapouleas, Complete constant mean curvature surfaces in euclidean three space, (preprint).Google Scholar
- 14.N. Kapouleas, Compact constant mean curvature surfaces in euclidean three space, (preprint).Google Scholar
- 15.N. Korevaar, R. Kusner, and B. Solomon, The structure of complete embedded surfaces with constant mean curvature,J. Differential Geom. 30 (1989), 465–503.MATHMathSciNetGoogle Scholar
- 16.P. Li, R. Schoen, and S. T. Yau, On the isoperimetric inequality for minimal surfaces,Annali Scuola Norm. Sup. Pisa 21 (1984), 237–244.MathSciNetGoogle Scholar
- 17.W. H. Meeks, III, The topology and geometry of embedded surfaces of constant mean curvature,J. Differential Geom. 27 (1988), 539–552.MATHMathSciNetGoogle Scholar
- 18.J. H. Michael and L. Simon, Sobolev and mean-value inequalities on generalized submanifolds ofRn,Comm. Pure Appl. Math. 26 (1973), 361–379.CrossRefMATHMathSciNetGoogle Scholar
- 19.S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,J. Computational Phys. (1987).Google Scholar
- 20.R. Osserman, The isoperimetric inequality,Bull. Amer. Math. Soc. 84 (1978), 1182–1238.CrossRefMATHMathSciNetGoogle Scholar
- 21.M. H. Protter and H. F. Weinberger,Maximum principles in differential equations, (1984) Springer-Verlag, p. 168.Google Scholar
- 22.J. Sethian, Hypersurfaces moving with curvature-dependent speed: Hamilton-Jacobi equations, conservation laws and numerical algorithms, preprint.Google Scholar
- 23.L. Simon,Lectures in geometric measure theory, Centre for Mathematical Analysis, Australian National University, Canberra, Australia 1983.Google Scholar
- 24.J. Steiner, Sur le maximum et le minimum des figures dans le plan, sur la sphère et dans l’espace en général,J. Reine Angew. Math. 24 (1842), 93–152.CrossRefMATHGoogle Scholar
- 25.H. Wente, Counterexample to a conjecture of H. Hopf,Pacific J. Math. 121 (1986), 193–243.CrossRefMATHMathSciNetGoogle Scholar
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