Abstract
In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space \({\mathbb {R}}^{n+1}\) with speed \(r^{\alpha } \sigma _k\), where \(\sigma _k\) is the k-th elementary symmetric polynomial of the principal curvatures, \(\alpha \in {\mathbb {R}}^1\), and r is the distance from the hypersurface to the origin. If \(\alpha \ge k+1\), we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If \(\alpha <k+1\), a counterexample is given for the above convergence. In the case \(k=1\) and \(\alpha \ge 2\), we also prove that the flow converges to a round point if the initial hypersurface is weakly mean-convex and star-shaped.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alessandroni, R., Sinestrari, C.: Evolution of hypersurfaces by powers of the scalar curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. 9(3), 541–571 (2010)
Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. PDEs 2(2), 151–171 (1994)
Andrews, B.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138(1), 151–161 (1999)
Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pac. J. Math. 195(1), 1–34 (2000)
Andrews, B., McCoy, J.: Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. Trans. Amer. Math. Soc. 364(7), 3427–3447 (2012)
Andrews, B., Guan, P.F., Ni, L.: Flow by power of the Gauss curvature. Adv. Math. 299, 174–201 (2016)
Andrews, B; Wei, Y. Volume preserving flow by powers of \(k\)-th mean curvature. To appear in J. Differ. Geom. arXiv:1708.03982
Brendle, S., Choi, K., Daskalopoulos, P.: Asymptotic behavior of flows by powers of the Gauss curvature. Acta Math. 219, 1–16 (2017)
Bryan, P., Ivaki, M., Scheuer, J.: A unified flow approach to smooth, even \(L_p\)-Minkowski problems. Anal. PDE 12, 259–280 (2019)
Caffarelli, L., Cabré, X.: Fully nonlinear elliptic equations, p. vi+104. American Mathematical Society, Providence (1995)
Cabezas-Rivas, E., Sinestrari, C.: Volume-preserving flow by powers of the mth mean curvature. Calc. Var. PDEs 38(3–4), 441–469 (2010)
Chou, K.-S.: (Tso, Kaising): Deforming a hypersurface by its Gauss–Kronecker curvature. Commun. Pure Appl. Math. 38(6), 867–882 (1985)
Chow, B.: Deforming convex hypersurfaces by the \(n\)-th root of the Gaussian curvature. J. Differ. Geom. 22(1), 117–138 (1985)
Chow, B.: Deforming convex hypersurfaces by the square root of the scalar curvature. Invent. Math. 87(1), 63–82 (1987)
Chow, B., Tsai, D.-H.: Expansion of convex hypersurfaces by nonhomogeneous functions of curvature. Asian J. Math. 1(4), 769–784 (1997)
Firey, W.J.: Shapes of worn stones. Mathematika 21, 1–11 (1974)
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)
Gerhardt, C.: Non-scale-invariant inverse curvature flows in Euclidean space. Calc. Var. PDEs 49(1–2), 471–489 (2014)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, p. xiv+517. Springer, Berlin (2001)
Guan, P., Li, J.: A mean curvature type flow in space forms. Int. Math. Res. Not. IMRN 13, 4716–4740 (2015)
Guan, P., Ni, L.: Entropy and a convergence theorem for Gauss curvature flow in high dimensions. J. Eur. Math. Soc. 19(12), 3735–3761 (2017)
Guan, P., Xia, C.: \(L_p\) Christoffel–Minkowski problem: the case \(1\,<\,p\,<\,k+1\). Calc. Var. PDEs 57(2), 23 (2018)
Hu, C., Ma, X., Shen, C.: On the Christoffel–Minkowski problem of Firey’s \(p\)-sum. Calc. Var. PDEs 21(2), 137–155 (2004)
Huang, Y., Lutwak, E., Yang, D., Zhang, G.: Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems. Acta Math. 216(2), 325–388 (2016)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)
Ivaki, M.: Deforming a hypersurface by principal radii of curvature and support function. Calc. Var. PDEs 58(1), 58:1 (2019)
Krylov, N.V., Safonov, M.V.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Izv. 16(1), 151–164 (1981)
Li, Q.-R., Sheng, W.M., Wang, X.-J.: Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems. To appear in J. Eur. Math. Soc. arXiv:1712.07774
Lieberman, G.: Second Order Parabolic Differential Equations, p. xii+439. World Scientific Publishing Co., Singapore (1996)
Lutwak, E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–50 (1993)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151. Cambridge University Press, Cambridge (2014)
Schnürer, O.C.: Surfaces contracting with speed \(|A|^2\). J. Differ. Geom. 71(3), 347–363 (2005)
Schulze, F.: Convexity estimates for flows by powers of the mean curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci 5(2), 261–277 (2006)
Tian, G., Wang, X.-J.: A priori estimates for fully nonlinear parabolic equations. Int. Math. Res. Not. IMRN 17, 3857–3877 (2013)
Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(3), 355–372 (1990)
Urbas, J.: An expansion of convex hypersurfaces. J. Differ. Geom. 33(1), 91–125 (1991)
Funding
Funding provided by Australian Research Council (Grant No. FL 130100118) and NSF in China (Grant Nos. 11131007 and 11571304).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, QR., Sheng, W. & Wang, XJ. Asymptotic Convergence for a Class of Fully Nonlinear Curvature Flows. J Geom Anal 30, 834–860 (2020). https://doi.org/10.1007/s12220-019-00169-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-019-00169-4