## Ancient solution of mean curvature flow in space forms

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- by Li Lei, Hongwei Xu and Entao Zhao PDF
- Trans. Amer. Math. Soc.
**374**(2021), 2359-2381 Request permission

## Abstract:

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.## References

- Ben Andrews and Charles Baker,
*Mean curvature flow of pinched submanifolds to spheres*, J. Differential Geom.**85**(2010), no. 3, 357–395. MR**2739807** - Sigurd Angenent, Panagiota Daskalopoulos, and Natasa Sesum,
*Unique asymptotics of ancient convex mean curvature flow solutions*, J. Differential Geom.**111**(2019), no. 3, 381–455. MR**3934596**, DOI 10.4310/jdg/1552442605 - C. Baker,
*The mean curvature flow of submanifolds of high codimension*, arXiv:1104.4409v1, 2011. - Simon Brendle and Kyeongsu Choi,
*Uniqueness of convex ancient solutions to mean curvature flow in $\Bbb R^3$*, Invent. Math.**217**(2019), no. 1, 35–76. MR**3958790**, DOI 10.1007/s00222-019-00859-4 - S. Brendle and K. Choi,
*Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions*, arXiv:1804.00018, 2018. - Simon Brendle, Gerhard Huisken, and Carlo Sinestrari,
*Ancient solutions to the Ricci flow with pinched curvature*, Duke Math. J.**158**(2011), no. 3, 537–551. MR**2805067**, DOI 10.1215/00127094-1345672 - P. Bryan, M. N. Ivaki, and J. Scheuer,
*On the classification of ancient solutions to curvature flows on the sphere*, arXiv:1604.01694v2, 2016. - Paul Bryan and Janelle Louie,
*Classification of convex ancient solutions to curve shortening flow on the sphere*, J. Geom. Anal.**26**(2016), no. 2, 858–872. MR**3472819**, DOI 10.1007/s12220-015-9574-x - Shiu Yuen Cheng, Peter Li, and Shing-Tung Yau,
*Heat equations on minimal submanifolds and their applications*, Amer. J. Math.**106**(1984), no. 5, 1033–1065. MR**761578**, DOI 10.2307/2374272 - S. S. Chern, M. do Carmo, and S. Kobayashi,
*Minimal submanifolds of a sphere with second fundamental form of constant length*, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59–75. MR**0273546** - K. Choi and C. Mantoulidis,
*Ancient gradient flows of elliptic functionals and Morse index*, arXiv:1902.07697v2, 2019. - Panagiota Daskalopoulos, Richard Hamilton, and Natasa Sesum,
*Classification of compact ancient solutions to the curve shortening flow*, J. Differential Geom.**84**(2010), no. 3, 455–464. MR**2669361** - Panagiota Daskalopoulos,
*Ancient solutions to geometric flows*, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III, Kyung Moon Sa, Seoul, 2014, pp. 773–788. MR**3729051** - Robert Haslhofer and Or Hershkovits,
*Ancient solutions of the mean curvature flow*, Comm. Anal. Geom.**24**(2016), no. 3, 593–604. MR**3521319**, DOI 10.4310/CAG.2016.v24.n3.a6 - Gerhard Huisken,
*Flow by mean curvature of convex surfaces into spheres*, J. Differential Geom.**20**(1984), no. 1, 237–266. MR**772132** - Gerhard Huisken,
*Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature*, Invent. Math.**84**(1986), no. 3, 463–480. MR**837523**, DOI 10.1007/BF01388742 - Gerhard Huisken,
*Deforming hypersurfaces of the sphere by their mean curvature*, Math. Z.**195**(1987), no. 2, 205–219. MR**892052**, DOI 10.1007/BF01166458 - Gerhard Huisken and Carlo Sinestrari,
*Convex ancient solutions of the mean curvature flow*, J. Differential Geom.**101**(2015), no. 2, 267–287. MR**3399098** - H. Blaine Lawson Jr.,
*Local rigidity theorems for minimal hypersurfaces*, Ann. of Math. (2)**89**(1969), 187–197. MR**238229**, DOI 10.2307/1970816 - L. Lei and H. W. Xu,
*An optimal convergence theorem for mean curvature flow of arbitrary codimension in hyperbolic spaces*, arXiv:1503.06747, 2015. - L. Lei and H. W. Xu,
*Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theorem*, arXiv:1506.06371, 2015. - Li Lei and Hong-Wei Xu,
*New developments in mean curvature flow of arbitrary codimension inspired by Yau rigidity theory*, Proceedings of the Seventh International Congress of Chinese Mathematicians, Vol. I, Adv. Lect. Math. (ALM), vol. 43, Int. Press, Somerville, MA, 2019, pp. 327–348. MR**3971877** - Li An-Min and Li Jimin,
*An intrinsic rigidity theorem for minimal submanifolds in a sphere*, Arch. Math. (Basel)**58**(1992), no. 6, 582–594. MR**1161925**, DOI 10.1007/BF01193528 - Kefeng Liu, Hongwei Xu, Fei Ye, and Entao Zhao,
*Mean curvature flow of higher codimension in hyperbolic spaces*, Comm. Anal. Geom.**21**(2013), no. 3, 651–669. MR**3078951**, DOI 10.4310/CAG.2013.v21.n3.a8 - Kefeng Liu, Hongwei Xu, Fei Ye, and Entao Zhao,
*The extension and convergence of mean curvature flow in higher codimension*, Trans. Amer. Math. Soc.**370**(2018), no. 3, 2231–2262. MR**3739208**, DOI 10.1090/tran/7281 - K. F. Liu, H. W, Xu and E. T. Zhao,
*Mean curvature flow of higher codimension in Riemannian manifolds*, arXiv:1204.0107, 2012. - K. F. Liu, H. W Xu, and E. T. Zhao,
*Deforming submanifolds of arbitrary codimension in a sphere*, arXiv:1204.0106, 2012. - K. F. Liu, H. W Xu, and E. T. Zhao,
*Some recent results on mean curvature flow of arbitrary codimension*, In Proceedings of the Sixth International Congress of Chinese Mathematicians, Taipei, July 14-19, 2013, AMS/IP, Studies in Advanced Math. - S. Lynch and H. T. Nguyen,
*Pinched ancient solutions to the high codimension mean curvature flow*, arXiv:1709.09697, 2017. - J. H. Michael and L. M. Simon,
*Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$*, Comm. Pure Appl. Math.**26**(1973), 361–379. MR**344978**, DOI 10.1002/cpa.3160260305 - Susanna Risa and Carlo Sinestrari,
*Ancient solutions of geometric flows with curvature pinching*, J. Geom. Anal.**29**(2019), no. 2, 1206–1232. MR**3935256**, DOI 10.1007/s12220-018-0036-0 - Katsuhiro Shiohama and Hongwei Xu,
*The topological sphere theorem for complete submanifolds*, Compositio Math.**107**(1997), no. 2, 221–232. MR**1458750**, DOI 10.1023/A:1000189116072 - Katsuhiro Shiohama and Hong Wei Xu,
*Rigidity and sphere theorems for submanifolds*, Kyushu J. Math.**48**(1994), no. 2, 291–306. MR**1294532**, DOI 10.2206/kyushujm.48.291 - James Simons,
*Minimal varieties in riemannian manifolds*, Ann. of Math. (2)**88**(1968), 62–105. MR**233295**, DOI 10.2307/1970556 - Xu-Jia Wang,
*Convex solutions to the mean curvature flow*, Ann. of Math. (2)**173**(2011), no. 3, 1185–1239. MR**2800714**, DOI 10.4007/annals.2011.173.3.1 - H. W. Xu,
*Pinching theorems, global pinching theorems, and eigenvalues for Riemannian submanifolds*, Ph.D. dissertation, Fudan University, 1990. - Hong Wei Xu,
*A rigidity theorem for submanifolds with parallel mean curvature in a sphere*, Arch. Math. (Basel)**61**(1993), no. 5, 489–496. MR**1241055**, DOI 10.1007/BF01207549 - Hong Wei Xu,
*$L_{n/2}$-pinching theorems for submanifolds with parallel mean curvature in a sphere*, J. Math. Soc. Japan**46**(1994), no. 3, 503–515. MR**1276835**, DOI 10.2969/jmsj/04630503 - Hong-wei Xu and Juan-ru Gu,
*A general gap theorem for submanifolds with parallel mean curvature in $\mathbf R^{n+p}$*, Comm. Anal. Geom.**15**(2007), no. 1, 175–193. MR**2301252**

## Additional Information

**Li Lei**- Affiliation: Center of Mathematical Sciences, Zhejiang University, Zhejiang Hangzhou, 310027, People’s Republic of China
- ORCID: 0000-0002-7143-8101
- Email: lei-li@zju.edu.cn
**Hongwei Xu**- Affiliation: Center of Mathematical Sciences, Zhejiang University, Zhejiang Hangzhou, 310027, People’s Republic of China
- MR Author ID: 245171
- Email: xuhw@zju.edu.cn
**Entao Zhao**- Affiliation: Center of Mathematical Sciences, Zhejiang University, Zhejiang Hangzhou, 310027, People’s Republic of China
- MR Author ID: 884026
- Email: zhaoet@zju.edu.cn
- Received by editor(s): December 12, 2019
- Published electronically: January 27, 2021
- Additional Notes: This research was supported by the National Natural Science Foundation of China, Grant Nos. 11531012, 12071424; and the China Postdoctoral Science Foundation, Grant No. BX20180274.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 2359-2381 - MSC (2020): Primary 53C40
- DOI: https://doi.org/10.1090/tran/8267
- MathSciNet review: 4223019

Dedicated: Dedicated to Professor Katsuhiro Shiohama on the occasion of his 80th birthday