## A class of curvature flows expanded by support function and curvature function

HTML articles powered by AMS MathViewer

- by Shanwei Ding and Guanghan Li
- Proc. Amer. Math. Soc.
**148**(2020), 5331-5341 - DOI: https://doi.org/10.1090/proc/15189
- Published electronically: September 18, 2020
- PDF | Request permission

## Abstract:

In this paper, we consider a class of expanding flows of closed, smooth, uniformly convex hypersurfaces in Euclidean $\mathbb {R}^{n+1}$ with speed $u^\alpha f^\beta$ ($\alpha , \beta \in \mathbb {R}^1$), where $u$ is the support function of the hypersurface, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If $\alpha \leqslant 0<\beta \leqslant 1-\alpha$, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.## References

- Ben Andrews,
*Contraction of convex hypersurfaces in Euclidean space*, Calc. Var. Partial Differential Equations**2**(1994), no. 2, 151–171. MR**1385524**, DOI 10.1007/BF01191340 - Ben Andrews,
*Contraction of convex hypersurfaces by their affine normal*, J. Differential Geom.**43**(1996), no. 2, 207–230. MR**1424425** - Ben Andrews,
*Gauss curvature flow: the fate of the rolling stones*, Invent. Math.**138**(1999), no. 1, 151–161. MR**1714339**, DOI 10.1007/s002220050344 - Ben Andrews,
*Pinching estimates and motion of hypersurfaces by curvature functions*, J. Reine Angew. Math.**608**(2007), 17–33. MR**2339467**, DOI 10.1515/CRELLE.2007.051 - Ben Andrews and James McCoy,
*Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature*, Trans. Amer. Math. Soc.**364**(2012), no. 7, 3427–3447. MR**2901219**, DOI 10.1090/S0002-9947-2012-05375-X - Ben Andrews, James McCoy, and Yu Zheng,
*Contracting convex hypersurfaces by curvature*, Calc. Var. Partial Differential Equations**47**(2013), no. 3-4, 611–665. MR**3070558**, DOI 10.1007/s00526-012-0530-3 - Simon Brendle, Kyeongsu Choi, and Panagiota Daskalopoulos,
*Asymptotic behavior of flows by powers of the Gaussian curvature*, Acta Math.**219**(2017), no. 1, 1–16. MR**3765656**, DOI 10.4310/ACTA.2017.v219.n1.a1 - Bennett Chow,
*Deforming convex hypersurfaces by the $n$th root of the Gaussian curvature*, J. Differential Geom.**22**(1985), no. 1, 117–138. MR**826427** - Bennett Chow,
*Deforming convex hypersurfaces by the square root of the scalar curvature*, Invent. Math.**87**(1987), no. 1, 63–82. MR**862712**, DOI 10.1007/BF01389153 - Bennett Chow and Dong-Ho Tsai,
*Expansion of convex hypersurfaces by nonhomogeneous functions of curvature*, Asian J. Math.**1**(1997), no. 4, 769–784. MR**1621575**, DOI 10.4310/AJM.1997.v1.n4.a7 - William J. Firey,
*Shapes of worn stones*, Mathematika**21**(1974), 1–11. MR**362045**, DOI 10.1112/S0025579300005714 - Claus Gerhardt,
*Non-scale-invariant inverse curvature flows in Euclidean space*, Calc. Var. Partial Differential Equations**49**(2014), no. 1-2, 471–489. MR**3148124**, DOI 10.1007/s00526-012-0589-x - Heinz Otto Cordes,
*Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen*, Math. Ann.**131**(1956), 278–312 (German). MR**91400**, DOI 10.1007/BF01342965 - Gerhard Huisken,
*Flow by mean curvature of convex surfaces into spheres*, J. Differential Geom.**20**(1984), no. 1, 237–266. MR**772132** - Gerhard Huisken and Carlo Sinestrari,
*Convexity estimates for mean curvature flow and singularities of mean convex surfaces*, Acta Math.**183**(1999), no. 1, 45–70. MR**1719551**, DOI 10.1007/BF02392946 - Mohammad N. Ivaki and Alina Stancu,
*Volume preserving centro-affine normal flows*, Comm. Anal. Geom.**21**(2013), no. 3, 671–685. MR**3078952**, DOI 10.4310/CAG.2013.v21.n3.a9 - Mohammad N. Ivaki,
*Deforming a hypersurface by Gauss curvature and support function*, J. Funct. Anal.**271**(2016), no. 8, 2133–2165. MR**3539348**, DOI 10.1016/j.jfa.2016.07.003 - Mohammad N. Ivaki,
*Deforming a hypersurface by principal radii of curvature and support function*, Calc. Var. Partial Differential Equations**58**(2019), no. 1, Paper No. 1, 18. MR**3880311**, DOI 10.1007/s00526-018-1462-3 - N. V. Krylov,
*Nonlinear elliptic and parabolic equations of the second order*, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR**901759**, DOI 10.1007/978-94-010-9557-0 - L. Nirenberg,
*On a generalization of quasi-conformal mappings and its application to elliptic partial differential equations*, Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N.J., 1954, pp. 95–100. MR**0066532** - Qi-Rui Li, Weimin Sheng, and Xu-Jia Wang,
*Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems*, J. Eur. Math. Soc. (JEMS)**22**(2020), no. 3, 893–923. MR**4055992**, DOI 10.4171/jems/936 - W. Sheng and C. Yi,
*A class of anisotropic expanding curvature flows*, Discrete and Continuous Dynamical Systems**40(4)**(2020), 2017-2035. - John I. E. Urbas,
*An expansion of convex hypersurfaces*, J. Differential Geom.**33**(1991), no. 1, 91–125. MR**1085136** - Chao Xia,
*Inverse anisotropic curvature flow from convex hypersurfaces*, J. Geom. Anal.**27**(2017), no. 3, 2131–2154. MR**3667425**, DOI 10.1007/s12220-016-9755-2

## Bibliographic Information

**Shanwei Ding**- Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
- ORCID: 0000-0002-8383-5219
**Guanghan Li**- Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
- Received by editor(s): March 20, 2020
- Received by editor(s) in revised form: May 14, 2020
- Published electronically: September 18, 2020
- Additional Notes: This research was partially supported by NSFC (Nos. 11761080 and 11871053).
- Communicated by: Jia-Ping Wang
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5331-5341 - MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/proc/15189
- MathSciNet review: 4163845