Abstract
We consider compact, strictly convex, origin-symmetric, smooth hypersurfaces inℝn+1 shrinking with speed given by powers of their centro-affine curvature. We show that, as long as the support function of the evolving convex bodies is bounded from both sides, the centro-affine curvature is also bounded above and below. We prove that the flow’s singularity which appears when the support function goes to zero is a compact contained in a hyperplane of dimension (n − 1). This information is exploited in ℝ3 to show that these flows shrink any admissible surface to a point and that, up to SL(3) transformations, the rescaled images of the evolving surface converge, in the Hausdorff metric, to a ball.
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References
B. Andrews, Contraction of convex hypersurfaces by their affine normal, J. Diff. Geom. 43 (1996), 207–230.
K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), 351–359.
A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999), 1–13.
D. Hug, Curvature Relations and Affine Surface Area for a General Convex Body and its Polar, Results in Math. 29 (1996), 233–248.
M. N. Ivaki, Centro-affine curvature flows on centrally symmetric convex curves, Trans. Amer. Math. Soc, to appear.
M. N. Ivaki, A flow approach to the L−2 Minkowski problem, Adv. in Appl. Math. 50 (2013), 445–464.
M. N. Ivaki and A. Stancu, Volume preserving centro affine normal flows, Comm. Anal. Geom. 21 (2013), 1–15.
K. Leichtweiss, “Affine Geometry of Convex bodies”, Johann Ambrosius Barth Verlag, Heidelberg, 1998.
J. Lu and X.-J. Wang, Rotationally symmetric solutions to the L p -Minkowski problem, J. Differential Equations 254 (2013), 983–1005.
M. Ludwig and M. Reitzner, A classification of SL(n) invariant valuations, Annals of Math. 172 (2010), 1223–1271.
E. Lutwak, The Brunn-Minkowski-Firey theory. I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150.
E. Lutwak, The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas, Adv. in Math. 118 (1996), 244–294.
D. S. Mitrinović, “Analytic Inequalities”, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
G. Sapiro and A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994), 79–120.
R. Schneider, “Convex Bodies: The Brunn-Minkowski Theory”, Cambridge Univ. Press, New York, 1993.
G. C. Shephard and R. J. Webster, Metrics for sets of convex bodies, Mathematika 12 (1965), 73–88.
A. Stancu, Centro-Affine Invariants for Smooth Convex Bodies, Int. Math. Res. Notices — IMRN (2012), 2289–2320.
A. Stancu, Some affine invariants revisited, 2012, Asymptotic geometric analysis. Proceedings of the Fall 2010 Fields Institute Thematic Program, M. Ludwig et al. (eds.), Springer, 2013, 341–357.
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Stancu, A. (2013). Flows by powers of centro-affine curvature. In: Chambolle, A., Novaga, M., Valdinoci, E. (eds) Geometric Partial Differential Equations proceedings. CRM Series, vol 15. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-473-1_13
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DOI: https://doi.org/10.1007/978-88-7642-473-1_13
Publisher Name: Edizioni della Normale, Pisa
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