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April  2020, 40(4): 2017-2035. doi: 10.3934/dcds.2020104

A class of anisotropic expanding curvature flows

Weimin Sheng , and  Caihong Yi

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: Weimin Sheng

Received  October 2018 Revised  October 2019 Published  January 2020

Fund Project: The authors were supported by NSFC, grant nos. 11971424 and 11571304.

In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean $ R^{n+1} $ with speed $ u^\alpha\sigma_k^\beta $ firstly, where $ u $ is support function of the hypersurface, $ \alpha, \beta \in R^1 $, and $ \beta>0 $, $ \sigma_k $ is the $ k $-th symmetric polynomial of the principal curvature radii of the hypersurface, $ k $ is an integer and $ 1\le k\le n $. For $ \alpha\le1-k\beta $, $ \beta>\frac{1}{k} $ we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $ \alpha\le1-k\beta $, $ \beta>\frac{1}{k} $, we prove that the flow with the speed $ fu^\alpha\sigma_k^\beta $ exists for all time and converges smoothly after normalisation to a soliton which is a solution of $ fu^{\alpha-1}\sigma_k^{\beta} = c $ provided that $ f $ is a smooth positive function on $ S^n $ and satisfies that $ (\nabla_i\nabla_jf^{\frac{1}{1+k\beta-\alpha}}+\delta_{ij}f^{\frac{1}{1+k\beta-\alpha}}) $ is positive definite. When $ \beta = 1 $, our argument provides a proof to the well-known $ L_p $ Christoffel-Minkowski problem for the case $ p\ge k+1 $ where $ p = 2-\alpha $, which is identify with Ivaki's recent result. Especially, we obtain the same result for $ k = n $ without any constraint on smooth positive function $ f $. Finally, we also give a counterexample for the two anisotropic expanding flows when $ \alpha>1-k\beta $.

Keywords: Expanding flow, asymptotic behaviour, uniformly convex hypersuface, $ L_p $ Christoffel-Minkowski problem, smooth solution.
Mathematics Subject Classification: 35K96, 53C44.
Citation: Weimin Sheng, Caihong Yi. A class of anisotropic expanding curvature flows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2017-2035. doi: 10.3934/dcds.2020104
References:
[1]

R. Alessandroni and C. Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 541-571.   Google Scholar

[2]

B. Andrews, Entropy estimates for evolving hypersurfaces, Communications in Analysis and Geometry, 2 (1994), 267-275.   Google Scholar

[3]

B. Andrews, Gauss curvature flow: The fate of the rolling stones, Invent. Math., 138 (1999), 151-161.  doi: 10.1007/s002220050344.  Google Scholar

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B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171.  doi: 10.1007/BF01191340.  Google Scholar

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B. Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, International Mathematics Research Notices, 20 (1997), 1001-1031.  doi: 10.1155/S1073792897000640.  Google Scholar

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B. Andrews and J. McCoy, Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc., 364 (2012), 3427-3447.  doi: 10.1090/S0002-9947-2012-05375-X.  Google Scholar

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B. Andrews, J. McCoy and Y. Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations, 47 (2013), 611-665.  doi: 10.1007/s00526-012-0530-3.  Google Scholar

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S. Brendle, K. Choi and P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math., 219 (2017), 1-16.  doi: 10.4310/ACTA.2017.v219.n1.a1.  Google Scholar

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K. Choi and P. Daskalopoulos, Uniqueness of closed self-similar solutions to the Gauss curvature flow, arXiv: 1609.05487. Google Scholar

[10]

K.-S. Chou and X.-J. Wang, A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 733-751.  doi: 10.1016/S0294-1449(00)00053-6.  Google Scholar

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K.-S. Chou and X.-J. Wang, The $L_p$ Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. in Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004.  Google Scholar

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B. Chow, Deforming convex hypersurfaces by the $n$-th root of the Gaussian curvature, J. Differential Geom., 22 (1985), 117-138.  doi: 10.4310/jdg/1214439724.  Google Scholar

[13]

B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math., 87 (1987), 63-82.  doi: 10.1007/BF01389153.  Google Scholar

[14]

B. Chow and D.-H. Tsai, Expansion of convex hypersurfaces by nonhomogeneous functions of curvature, Asian J. Math., 1 (1997), 769-784.  doi: 10.4310/AJM.1997.v1.n4.a7.  Google Scholar

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W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11.  doi: 10.1112/S0025579300005714.  Google Scholar

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C. Gerhardt, Non-scale-invariant inverse curvature flows in Euclidean space, Car. Var. Partial Differential Equations, 49 (2014), 471-489.  doi: 10.1007/s00526-012-0589-x.  Google Scholar

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P. F. Guan and C. S. Lin, On Equation $\det(u_ij+u\delta_ij) = u^{p}f $ on $S^n$, Preprint No 2000-7, NCTS in Tsing-Hua University, 2000. Google Scholar

[18]

P. F. Guan and X.-N. Ma, Christoffel-Minkowski problem I: Convexity of solutions of a hessian equation, Invent. Math., 151 (2003), 553-577.  doi: 10.1007/s00222-002-0259-2.  Google Scholar

[19]

P. F. Guan and L. Ni, Entropy and a convergence theorem for Gauss curvature flow in high dimensions, J. Eur. Math. Soc., 19 (2017), 3735-3761.  doi: 10.4171/JEMS/752.  Google Scholar

[20]

P. F. Guan and C. Xia, $L^p$ Christoffel-Minkowski problem: The case $1 < p < k+1$, Cal. Var. Partial Differential Equations, 57 (2018), Art. 69, 23 pp. doi: 10.1007/s00526-018-1341-y.  Google Scholar

[21]

C. Q. Hu, X.-N. Ma and C. L. Shen, On the Christoffel-Minkowski problem of Firey's $p$-sum, Cal. Var. Partial Differential Equations, 21 (2004), 137-155.  doi: 10.1007/s00526-003-0250-9.  Google Scholar

[22]

Y. Huang, E. Lutwak, D. Yang and G. Y. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math., 216 (2016), 325-388.  doi: 10.1007/s11511-016-0140-6.  Google Scholar

[23]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998.  Google Scholar

[24]

M. N. Ivaki, Deforming a hypersurface by principal radii of curvature and support function, Calc. Var. Partial Differential Equations, 58 (2019), Art. 1, 18 pp. doi: 10.1007/s00526-018-1462-3.  Google Scholar

[25]

N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-010-9557-0.  Google Scholar

[26]

Q.-R. Li, W. M. Sheng and X.-J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc., (2019). doi: 10.4171/JEMS/936.  Google Scholar

[27]

Q.-R. Li, W. M.Sheng and X.-J. Wang, Asymptotic convergence for a class of fully nonlinear curvature flows, J. Geom. Anal., 3 (2019), 1-27.   Google Scholar

[28] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511526282.  Google Scholar
[29]

J. I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom., 33 (1991), 91-125.  doi: 10.4310/jdg/1214446031.  Google Scholar

[30]

X.-J. Wang, Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature, Trans. Amer. Math. Soc., 348 (1996), 4501-4524.  doi: 10.1090/S0002-9947-96-01650-9.  Google Scholar

[31]

C. Xia, Inverse anisotropic curvature flow from convex hypersurfaces, J. Geom. Anal., 27 (2017), 2131-2154.  doi: 10.1007/s12220-016-9755-2.  Google Scholar

show all references

References:
[1]

R. Alessandroni and C. Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 541-571.   Google Scholar

[2]

B. Andrews, Entropy estimates for evolving hypersurfaces, Communications in Analysis and Geometry, 2 (1994), 267-275.   Google Scholar

[3]

B. Andrews, Gauss curvature flow: The fate of the rolling stones, Invent. Math., 138 (1999), 151-161.  doi: 10.1007/s002220050344.  Google Scholar

[4]

B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171.  doi: 10.1007/BF01191340.  Google Scholar

[5]

B. Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, International Mathematics Research Notices, 20 (1997), 1001-1031.  doi: 10.1155/S1073792897000640.  Google Scholar

[6]

B. Andrews and J. McCoy, Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature, Trans. Amer. Math. Soc., 364 (2012), 3427-3447.  doi: 10.1090/S0002-9947-2012-05375-X.  Google Scholar

[7]

B. Andrews, J. McCoy and Y. Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations, 47 (2013), 611-665.  doi: 10.1007/s00526-012-0530-3.  Google Scholar

[8]

S. Brendle, K. Choi and P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math., 219 (2017), 1-16.  doi: 10.4310/ACTA.2017.v219.n1.a1.  Google Scholar

[9]

K. Choi and P. Daskalopoulos, Uniqueness of closed self-similar solutions to the Gauss curvature flow, arXiv: 1609.05487. Google Scholar

[10]

K.-S. Chou and X.-J. Wang, A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 733-751.  doi: 10.1016/S0294-1449(00)00053-6.  Google Scholar

[11]

K.-S. Chou and X.-J. Wang, The $L_p$ Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. in Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004.  Google Scholar

[12]

B. Chow, Deforming convex hypersurfaces by the $n$-th root of the Gaussian curvature, J. Differential Geom., 22 (1985), 117-138.  doi: 10.4310/jdg/1214439724.  Google Scholar

[13]

B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math., 87 (1987), 63-82.  doi: 10.1007/BF01389153.  Google Scholar

[14]

B. Chow and D.-H. Tsai, Expansion of convex hypersurfaces by nonhomogeneous functions of curvature, Asian J. Math., 1 (1997), 769-784.  doi: 10.4310/AJM.1997.v1.n4.a7.  Google Scholar

[15]

W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11.  doi: 10.1112/S0025579300005714.  Google Scholar

[16]

C. Gerhardt, Non-scale-invariant inverse curvature flows in Euclidean space, Car. Var. Partial Differential Equations, 49 (2014), 471-489.  doi: 10.1007/s00526-012-0589-x.  Google Scholar

[17]

P. F. Guan and C. S. Lin, On Equation $\det(u_ij+u\delta_ij) = u^{p}f $ on $S^n$, Preprint No 2000-7, NCTS in Tsing-Hua University, 2000. Google Scholar

[18]

P. F. Guan and X.-N. Ma, Christoffel-Minkowski problem I: Convexity of solutions of a hessian equation, Invent. Math., 151 (2003), 553-577.  doi: 10.1007/s00222-002-0259-2.  Google Scholar

[19]

P. F. Guan and L. Ni, Entropy and a convergence theorem for Gauss curvature flow in high dimensions, J. Eur. Math. Soc., 19 (2017), 3735-3761.  doi: 10.4171/JEMS/752.  Google Scholar

[20]

P. F. Guan and C. Xia, $L^p$ Christoffel-Minkowski problem: The case $1 < p < k+1$, Cal. Var. Partial Differential Equations, 57 (2018), Art. 69, 23 pp. doi: 10.1007/s00526-018-1341-y.  Google Scholar

[21]

C. Q. Hu, X.-N. Ma and C. L. Shen, On the Christoffel-Minkowski problem of Firey's $p$-sum, Cal. Var. Partial Differential Equations, 21 (2004), 137-155.  doi: 10.1007/s00526-003-0250-9.  Google Scholar

[22]

Y. Huang, E. Lutwak, D. Yang and G. Y. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math., 216 (2016), 325-388.  doi: 10.1007/s11511-016-0140-6.  Google Scholar

[23]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998.  Google Scholar

[24]

M. N. Ivaki, Deforming a hypersurface by principal radii of curvature and support function, Calc. Var. Partial Differential Equations, 58 (2019), Art. 1, 18 pp. doi: 10.1007/s00526-018-1462-3.  Google Scholar

[25]

N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Mathematics and its Applications (Soviet Series), 7. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-010-9557-0.  Google Scholar

[26]

Q.-R. Li, W. M. Sheng and X.-J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc., (2019). doi: 10.4171/JEMS/936.  Google Scholar

[27]

Q.-R. Li, W. M.Sheng and X.-J. Wang, Asymptotic convergence for a class of fully nonlinear curvature flows, J. Geom. Anal., 3 (2019), 1-27.   Google Scholar

[28] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511526282.  Google Scholar
[29]

J. I. E. Urbas, An expansion of convex hypersurfaces, J. Differential Geom., 33 (1991), 91-125.  doi: 10.4310/jdg/1214446031.  Google Scholar

[30]

X.-J. Wang, Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature, Trans. Amer. Math. Soc., 348 (1996), 4501-4524.  doi: 10.1090/S0002-9947-96-01650-9.  Google Scholar

[31]

C. Xia, Inverse anisotropic curvature flow from convex hypersurfaces, J. Geom. Anal., 27 (2017), 2131-2154.  doi: 10.1007/s12220-016-9755-2.  Google Scholar

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