Advertisement

Convex bodies with pinched Mahler volume under the centro-affine normal flows

  • Mohammad N. IvakiEmail author
Article

Abstract

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke–Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitably rescaled solutions converge sequentially to the unit ball in the \(\mathcal {C}^{\infty }\) topology modulo \(SL(n+1)\).

Mathematics Subject Classification

Primary 53C44 52A05 Secondary 35K55 

Notes

Acknowledgments

I am indebted to the referees whose comments and suggestions have led to improvements of this article.

References

  1. 1.
    Alessandroni, R., Sinestrari, C.: Evolution of hypersurfaces by powers of the scalar curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. 9, 541–571 (2010)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Andrews, B.: Harnack inequalities for evolving hypersrufaces. Math. Z. 217, 179–197 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Andrews, B.: Contraction of convex hypersurfaces by their affine normal. J. Differ. Geom. 43, 207–230 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Andrews, B.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138(1), 151–161 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pacif. J. Math. 195(1), 1–34 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Andrews, B., Chen, X.: Surfaces moving by powers of Gauss curvature. Pure Appl. Math. Q. 8(4), 825–834 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Andrews, B., McCoy, J.: Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. Trans. Am. Math. Soc. 364(7), 3427–3447 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Andrews, B., McCoy, J., Zheng, Y.: Contracting convex hypersurfaces by curvature. Calc. Var. Partial Differ. Equ. 47(3–4), 611–665 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Angenent, S., Sapiro, G., Tannenbaum, A.: On the heat equation for non-convex curves. J. Am. Math. Soc. 11(3), 601–634 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ball, B., Böröczky, K.J.: Stability of some versions of the Prékopa–Leindler inequality. Monatsh. Math. 163, 1–14 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Blaschke, W.: Über affine Geometrie I. Isoperimetrische Eigenschaften von Ellipse und Ellipsoid. Leipz. Ber. 68, 217–239 (1916)Google Scholar
  12. 12.
    Cabezas-Rivas, E., Sinestrari, C.: Volume-preserving flow by powers of the \(m\)-th mean curvature. Calc. Var. Partial Differ. Equ. 38, 441–469 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Chen, S.: Classifying convex compact ancient solutions to the affine curve shortening flow, J. Geom. Anal. (2013). doi:  10.1007/s12220-013-9456-z
  14. 14.
    Chow, B.: Deforming convex hypersurfaces by the \(n\)-th root of the Gaussian curvature. J. Differ. Geom. 22(1), 117–138 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Guan, P., Ni, L.: Entropy and a convergence theorem for Gauss curvature flow in high dimension, preprint (2013). http://arxiv.org/abs/1306.0625
  16. 16.
    Ivaki, M.N.: Centro-affine curvature flows on centrally symmetric convex curves. Trans. Am. Math. Soc. 366(11), 5671–5692 (2014)Google Scholar
  17. 17.
    Ivaki, M.N.: Stability of the \(p\)-affine isoperimetric inequality. J. Geom. Anal. 24(4), 1898–1911 (2014). doi:  10.1007/s12220-013-9401-1 CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ivaki, M.N.: The planar Busemann-Petty centroid inequality and its stability. Trans. Am. Math. Soc. (2014). http://arxiv.org/abs/1312.4834v6 (to appear)
  19. 19.
    Ivaki, M.N.: Centro-affine normal flows on curves: Harnack estimates and ancient solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire (2014). doi: 10.1016/j.anihpc.2014.07.001
  20. 20.
    Ivaki, M.N.: Stability of the Blaschke-Santaló inequality in the plane. Monatsh. Math. (2014). doi: 10.1007/s00605-014-0651-1
  21. 21.
    Ivaki, M.N.: A note on the Gauss curvature flow, preprint (2014). http://arxiv.org/abs/1409.2629v2
  22. 22.
    Ivaki, M.N.: Classification of compact convex ancient solutions of the planar affine normal flow. J. Geom. Anal. (2014) (to appear)Google Scholar
  23. 23.
    Ivaki, M.N., Stancu A.: Volume preserving centro-affine normal flows, Commun. Anal. Geom. (2013) doi: 10.4310/CAG.2013.v21.n3.a9
  24. 24.
    Krylov, N.V., Safonov, V.M.: A certain property of solutions of parabolic equations with measurable coefficients, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 44(1), 161–175 (1980)MathSciNetGoogle Scholar
  25. 25.
    Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in domains. Izvestiya: Math. 20(3), 459–492 (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. D. Reidel Publishing Co., Dordrecht (1987)CrossRefzbMATHGoogle Scholar
  27. 27.
    Loftin, J., Tsui, M.P.: Ancient solutions of the affine normal flow. J. Differ. Geom. 78, 113–162 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Lutwak, E.: The Brunn–Minkowski–Fiery theory II:affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Sapiro, G., Tannenbaum, A.: On affine plane curve evolution. J. Funct. Anal. 119, 79–120 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Schnürer, O.C.: Surfaces contracting with speed \(|A|^2\). J. Differ. Geom. 71(3), 347–363 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Schulze, F.: Convexity estimates for flows by powers of the mean curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(2), 261–277 (2006)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Smoczyk, K.: Starshaped hypersurfaces and the mean curvature flow. Manuscr. Math. 95(2), 225–236 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Stancu, A.: Centro-affine invariants for smooth convex bodies. Int. Math. Res. Not. IMRN (2011). doi: 10.1093/imrn/rnr110
  34. 34.
    Tsai, H.D.: \(C^{2,\alpha }\) estimate of a parabolic Monge–Ampère equation on \(\mathbb{S}^n\). Proc. Am. Math. Soc. 131(10), 3067–3074 (2003)CrossRefzbMATHGoogle Scholar
  35. 35.
    Tso, K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math. 38, 867–882 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Wu, C., Tian, D., Li, G.: Forced flows by powers of the \(m\)-th mean curvature. Armen. J. Math. 3, 61–91 (2010)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWienAustria

Personalised recommendations