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PARABOLIC CLASSICAL CURVATURE FLOWS

Published online by Cambridge University Press:  30 October 2017

BRENDAN GUILFOYLE
Affiliation:
School of STEM, Institute of Technology, Tralee, Co. Kerry, Ireland email brendan.guilfoyle@ittralee.ie
WILHELM KLINGENBERG
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK email wilhelm.klingenberg@durham.ac.uk

Abstract

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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References

Andrews, B., ‘Contraction of convex hypersurfaces in Euclidean space’, Calc. Var. Partial Differential Equations 2 (1994), 151171.CrossRefGoogle Scholar
Bloore, F. J., ‘The shape of pebbles’, Math. Geol. 9 (1977), 113122.CrossRefGoogle Scholar
Chern, S. S., ‘Some new characterizations of the Euclidean sphere’, Duke Math. J. 12 (1945), 279290.CrossRefGoogle Scholar
Chern, S. S., ‘On special W-surfaces’, Proc. Amer. Math. Soc. 6 (1955), 783786.Google Scholar
Chow, B., ‘Deforming convex hypersurfaces by the square root of the scalar curvature’, Invent. Math. 87 (1987), 6382.CrossRefGoogle Scholar
Domokos, G. and Gibbons, G. W., ‘The evolution of pebble size and shape in space and time’, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), 30593079.CrossRefGoogle Scholar
Firey, W. J., ‘The shape of worn stones’, Mathematika 21 (1974), 111.CrossRefGoogle Scholar
Gálvez, J. A., Martínez, A. and Milán, F., ‘Linear Weingarten surfaces in ℝ3 ’, Monatsh. Math. 138 (2003), 133144.CrossRefGoogle Scholar
Gerhardt, C., ‘Flow of nonconvex hypersurfaces into spheres’, J. Differential Geom. 32 (1990), 299314.Google Scholar
Guilfoyle, B. and Klingenberg, W., ‘An indefinite Kähler metric on the space of oriented lines’, J. Lond. Math. Soc. 72 (2005), 497509.CrossRefGoogle Scholar
Guilfoyle, B. and Klingenberg, W., ‘On Weingarten surfaces in Euclidean and Lorentzian 3-space’, Differential Geom. Appl. 28 (2010), 454468.CrossRefGoogle Scholar
Guilfoyle, B. and Klingenberg, W., ‘From global to local: an index bound for umbilic points on smooth convex surfaces’, Preprint, 2012, arXiv:1207.5994.Google Scholar
Guilfoyle, B. and Klingenberg, W., ‘A converging Lagrangian curvature flow in the space of oriented lines’, Kyushu J. Math. 70 (2016), 343351.Google Scholar
Hartman, P. and Wintner, A., ‘Umbilical points and W-surfaces’, Amer. J. Math. 76 (1954), 502508.CrossRefGoogle Scholar
Hopf, H., ‘Über Flächen mit einer Relation zwischen den Hauptkrümmungen’, Math. Nachr. 4 (1950–1951), 232249.CrossRefGoogle Scholar
Hopf, H., Differential Geometry in the Large, Lecture Notes in Mathematics, 1000 (Springer, Berlin, 1983).CrossRefGoogle Scholar
Huisken, G., ‘Flow by mean-curvature of convex surfaces into spheres’, J. Differential Geom. 20 (1984), 237266.Google Scholar
Kühnel, W. and Steller, M., ‘On closed Weingarten surfaces’, Monatsh. Math. 146 (2005), 113126.CrossRefGoogle Scholar
Lieberman, G. M., Second Order Parabolic Differential Equations (World Scientific, London, 1996).CrossRefGoogle Scholar
Schnürer, O., ‘Surfaces contracting with speed |A|2 ’, J. Differential Geom. 71 (2005), 347363.Google Scholar
Schulze, F., ‘Evolution of convex hypersurfaces by powers of the mean curvature’, Math. Z. 251 (2005), 721733.CrossRefGoogle Scholar
Smoczyk, K., ‘A representation formula for the inverse harmonic mean curvature flow’, Elem. Math. 60 (2005), 5765.CrossRefGoogle Scholar
Tso, K. S., ‘Deforming a hypersurface by its Gauss–Kronecker curvature’, Comm. Pure Appl. Math. 38 (1985), 867882.CrossRefGoogle Scholar
Voss, K., ‘Über geschlossene Weingartensche Flächen’, Math. Ann. 138 (1959), 4254.CrossRefGoogle Scholar

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