Complete λ-Hypersurfaces in Euclidean Spaces

Abstract

In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.

Keywords

Self-shrinker / λ-Hypersurface / Mean curvature flow / Weighted volume / Rigidity theorem

References

[1] Abresch U, Langer J. The normalized curve shortening flow and homothetic solutions. J. Differential Geom., 1986, 23: 175-196

[2] Ancari S, Igor M. Rigidity theorems for complete λ-hypersurfaces. Arch. Math., 2021, 117: 105-120

[3] Ancari S, Igor M. Volume estimates and classification theorem for constant weighted mean curvature hypersurfaces. J. Geom. Anal., 2021, 31: 3764-3782

[4] Andrews B, Li H, Wei Y. F-stability for self-shrinking solutions to mean curvature flow. Asian J. Math., 2014, 18: 757-777

[5] Angenent S. Shrinking Doughnuts, in Nonlinear Diffusion Equations and Their Equilibrium States, Birkhaüser, Boston-Basel-Berlin, 1992, 7: 21-38

[6] Barbosa E, Santana F, Upadhyay A. λ-Hypersurfaces on shrinking gradient Ricci solitons. J. Math. Anal. Appl., 2020, 492: 124470 18 pp

[7] Borell C. The Brunn-Minkowski inequality in Gauss space. Invent. Math., 1975, 30: 207-216

[8] Brakke K. The motion of a surface by its mean curvature, 1978, Princeton, N.J.: Princeton University Press

[9] Brendle S. Embedded self-similar shrinkers of genus 0. Ann. of Math., 2016, 183: 715-728

[10] Cao H-D, Li H. A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Cale. Var. Partial Differential Equations, 2013, 46: 879-889

[11] Chang J-E. 1-Dimensional solutions of λ-self shrinkers. Geom. Dedicata, 2017, 189: 97-112

[12] Cheng Q -M, Li Z, Wei G. Complete self-shrinkers with constant norm of the second fundamental form. Math. Z., 2022, 300: 995-1018

[13] Cheng Q -M, Ogata S. 2-Dimensional complete self-shrinkers in ℝ³. Math. Z., 2016, 284: 537-542

[14] Cheng Q -M, Ogata S, Wei G. Rigidity theorems of λ-hypersurfaces. Comm. Anal. Geom., 2016, 24: 45-58

[15] Cheng Q -M, Peng Y. Complete self-shrinkers of the mean curvature flow. Calc. Var. Partial Differential Equations, 2015, 52: 497-506

[16] Cheng Q -M, Wei G. A gap theorem for self-shrinkers. Trans. Amer. Math. Soc., 2015, 367: 4895-4915

[17] Cheng Q -M, Wei G. Complete λ-hypersurfaces of weighted volume-preserving mean curvature flow. Calc. Var. Partial Differential Equations, 2018, 57: 32

[18] Cheng, Q. -M. and Wei, G., The Gauss image of λ-hypersurfaces and a Bernstein type problem, arXiv:1410.5302.

[19] Cheng Q -M, Wei G. Geometry of complete λ-hypersurfaces. Sci. Sin. Math., 2018, 48: 699-710 in Chinese)

[20] Cheng Q -M, Wei G. Examples of compact λ-hypersurfaces in Euclidean spaces. Sci. China Math., 2021, 64: 155-166

[21] Cheng Q -M, Wei G. Complete λ-surfaces in R ³. Calc. Var. Partial Differential Equations, 2021, 60: 46

[22] Cheng, Q. -M. and Wei, G., Stability and area growth of λ-hypersurfaces, to appear in Comm. Anal. Geom., arXiv:1911.00631.

[23] Cheng X, Zhou D. Volume estimate about shrinkers. Proc. Amer. Math. Soc., 2013, 141: 687-696

[24] Chopp D L. Computation of self-similar solutions for mean curvature flow. Experiment. Math., 1994, 3: 1-15

[25] Colding T H, Ilmanen T, Minicozzi W P II Rigidity of generic singularities of mean curvature flow. Publ. Math. Inst. Hautes Etudes Sci., 2015, 121: 363-382

[26] Colding T H, Minicozzi W P II Generic mean curvature flow I; Generic singularities. Ann. of Math., 2012, 175: 755-833

[27] Ding Q. A rigidity theorem on the second fundamental form for self-shrinkers. Trans. Amer. Math. Soc., 2018, 370: 8311-8329

[28] Ding, Q. and Wang, Z. Z., On the self-shrinking systems in arbitrary codimensional space, 2010, arXiv:1012.0429.

[29] Ding Q, Xin Y L. Volume growth, eigenvalue and compactness for self shrinkers. Asian J. Math., 2013, 17: 443-456

[30] Ding Q, Xin Y L. The rigidity theorems of self-shrinkers. Trans. Amer. Math. Soc., 2014, 366: 5067-5085

[31] Drugan, G., Lee, H. and Nguyen, X. H., A survey of closed self-shrinkers with symmetry, Results Math., 73, 2018, Art. 32, 32 pp.

[32] Drugan G, Kleene S J. Immersed self shrinkers. Trans. Amer. Math. Soc., 2017, 369: 7213-7250

[33] Ecker K, Huisken G. Mean curvature evolution of entire graphs. Ann. of Math., 1989, 130: 453-471

[34] Guang, Q., Self-shrinkers and translating solitons of mean curvature flow, Thesis (Ph.D.)-Massachusetts Institute of Technology, 2016.

[35] Guang Q. Gap and rigidity theorems of λ-hypersurfaces. Proc. Am. Math. Soc., 2018, 146: 4459-4471

[36] Guang, Q., Gap and rigidity theorems of λ-hypersurfaces, 2014, arXiv:1405.4871v1.

[37] Guang, Q. and Zhu, J. J., Rigidity and curvature estimates for graphical self-shrinkers, Calc. Var. Partial Differential Equations, 56, 2017, Art. 176, 18 pp.

[38] Guang Q, Zhu J J. On the rigidity of mean convex self-shrinkers. Int. Math. Res. Not., 2018, 20: 6406-6425

[39] Halldorsson H. Self-similar solutions to the curve shortening flow. Trans. Amer. Math. Soc., 2012, 364: 5285-5309

[40] Hartman P, Wintner A. Umbilical points and W-surfaces. Amer. J. Math., 1954, 76: 502-508

[41] Heilman S. Symmetric convex sets with minimal Gaussian Surface area. Amer. J. Math., 2021, 143: 53-94

[42] Heilman S. The structure of Gaussian minimal bubbles. J. Geom. Anal., 2021, 31: 6307-6348

[43] Huisken G. Flow by mean curvature convex surfaces into spheres. J. Differential Geom., 1984, 20: 237-266

[44] Huisken G. Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom., 1990, 31: 285-299

[45] Huisken G. Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds, Los Angeles, CA, 1990, 1993, Providence, RI: Amer. Math. Soc. 175-191 Part 1

[46] Huisken G. The volume preserving mean curvature flow. J. reine angew. Math., 1987, 382: 35-48

[47] Kapouleas N, Kleene S J, Møller N M. Mean curvature self-shrinkers of high genus: non-compact examples. J. reine angew. Math., 2018, 739: 1-39

[48] Kleene S, Møller N M. Self-shrinkers with a rotation symmetry. Trans. Amer. Math. Soc., 2014, 366: 3943-3963

[49] Lawson H B. Local rigidity theorems for minimal hypersurfaces. Ann. of Math., 1969, 89: 187-197

[50] Le N Q, Sesum N. Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Comm. Anal. Geom., 2011, 19: 1-27

[51] Lee T. Convexity of λ-hypersurfaces. Proc. Amer. Math. Soc., 2022, 150: 1735-1744

[52] Lei L, Xu H, Xu Z. A new pinching theorem for complete self-shrinkers and its generalization. Sci. China Math., 2020, 63: 1139-1152

[53] Li H, Wei Y. Lower volume growth estimates for self-shrinkers of mean curvature flow. Proc. Amer. Math. Soc., 2014, 142: 3237-3248

[54] Li, Z. and Wei, G., An immersed S ⁿ λ-hypersurface, arXiv:2204.11390.

[55] McGonagle M, Ross J. The hyperplane is the only stable, smooth solution to the isoperimetric problem in gaussian space. Geom. Dedicata, 2015, 178: 277-296

[56] Møller, N. M., Closed self-shrinking surfaces in ℝ³ via the torus, 2011, arXiv:1111.7318.

[57] Nguyen X H. Construction of complete embedded self-similar surfaces under mean curvature flow. I. Trans. Amer. Math. Soc., 2009, 361: 1683-1701

[58] Nguyen X H. Construction of complete embedded self-similar surfaces under mean curvature flow. II. Adv. Differential Equations, 2010, 15: 503-530

[59] Nguyen X H. Construction of complete embedded self-similar surfaces under mean curvature flow. III. Duke Math. J., 2014, 163: 2023-2056

[60] Ross J. On the existence of a closed embedded rotational λ-hypersurface. J. Geom., 2019, 110: 26

[61] Stone A. A density function and the structure of singularities of the mean curvature flow. Calc. Var. Partial Differential Equations, 1994, 2: 443-480

[62] Wang H, Xu H, Zhao E. Gap theorems for complete λ-hypersurfaces. Pacific J. Math., 2017, 288: 453-474

[63] Wang L. A Bernstein type theorem for self-similar shrinkers. Geom. Dedicata, 2011, 151: 297-303

[64] Wei G, Peng Y. A note on rigidity theorem of λ-hypersurfaces. Proc. Roy. Soc. Edinburgh Sect. A, 2019, 149: 1595-1601

[65] Xu H, Lei L, Xu Z. The second pinching theorem for complete λ-hypersurfaces. Sci. Sin. Math., 2018, 48: 245-254 in Chinese)

[66] Zhu Y, Chen Q. Classification and rigidity of λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Houston J. Math., 2019, 45: 1037-1053

[67] Zhu Y, Fang Y, Chen Q. Complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Sci. China Math., 2018, 61: 929-942