${\mathbb R}^5$中具有平行Fubini-Pick形式的Calabi超曲面的分类

Abstract

Abstract:

The classifications of locally strongly convex equiaffine hypersurfaces (centroaffine hypersurfaces) with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Blaschke-Berwald metric (centroaffine metric) have been completed in the last decades. In [20], the authors studied Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric and classified 2 and 3-dimensional cases. In this paper, we extend such calssification results to 4-dimensional Calabi hypersurfaces in the affine space ${\mathbb R}^5$.

Key words: Parallel Fubini-Pick form, Calabi geometry, Graph immersion, Tchebychev affine Kähler hypersurface, Affine extremal hypersurface

CLC Number:

O186.1

Ruiwei Xu,Miaoxin Lei. Classification of Calabi Hypersurfaces in ${\mathbb R}^5$ with Parallel Fubini-Pick Form[J].Acta mathematica scientia,Series A, 2022, 42(2): 321-337.

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References 20

1	Bokan N , Nomizu K , Simon U . Affine hypersurfaces with parallel cubic forms. Tôhoku Math J, 1990, 42: 101- 108
2	Calabi E . Improper affine hyperspheres of convex type and a generalization of a theorem of Jörgens. Michgan J Math, 1958, 5: 105- 126
3	Cheng X , Hu Z . An optimal inequality on locally strongly convex centroaffine hypersurfaces. J Geom Anal, 2018, 28: 643- 655 doi: 10.1007/s12220-017-9836-x
4	Cheng X , Hu Z , Moruz M . Classification of the locally strongly convex centroaffine hypersurfaces with parallel cubic form. Results Math, 2017, 72: 419- 469 doi: 10.1007/s00025-017-0651-2
5	Dillen F , Vrancken L . 3-dimensional affine hypersurfaces in ${\mathbb R}^4$ with parallel cubic form. Nagoya Math J, 1991, 124: 41- 53 doi: 10.1017/S0027763000003767
6	Dillen F , Vrancken L , Yaprak L . Affine hypersurfaces with parallel cubic form. Nagoya Math J, 1994, 135: 153- 164 doi: 10.1017/S0027763000005006
7	Hu Z , Li H , Simon U , Vrancken L . On locally strongly convex affine hypersurfaces with parallel cubic form. Part Ⅰ. Diff Geom Appl, 2009, 27: 188- 205 doi: 10.1016/j.difgeo.2008.10.005
8	Hu Z , Li H , Vrancken L . Locally strongly convex affine hypersurfaces with parallel cubic form. J Diff Geom, 2011, 87: 239- 307
9	Kobayashi S, Nomizu K. Foundations of Differential Geometry. Vol Ⅰ. New York: Interscience Publishers, 1963
10	Li A M. Affine Kähler manifolds[R]. Lodz: International Conference in Banach Center of Poland, 2005
11	Li A M, Simon U, Zhao G, Hu Z. Global Affine Differential Geometry of Hypersurfaces. Berlin: De Gruyter, 2015
12	Li A M, Xu R W, Simon U, Jia F. Affine Bernstein Problems and Monge-Ampère Equations. New Jersey: World Scientific, 2010
13	Li A M , Penn G . Uniqueness theorems in affine differential geometry. Part Ⅱ. Results Math, 1988, 13: 308- 317 doi: 10.1007/BF03323247
14	Li H . Variational problems and PDEs in affine differential geometry. Banach Center Publ, 2005, 69 (1): 9- 41
15	Li H , Vrancken L . A basic inequality and new characterization of Whitney spheres in a complex space form. Israel J Math, 2005, 146: 223- 242 doi: 10.1007/BF02773534
16	Liu H L , Wang C P . Centroaffine surfaces with parallel traceless cubic form. Bull Belg Math Soc, 1997, 4: 493- 499
17	Magid M , Nomizu K . On affine surfaces whose cubic forms are parallel relative to the affine metric. Proc Japan Acad Ser A, 1989, 65: 215- 218 doi: 10.2183/pjab.65.215
18	Pogorelov A V. The Minkowski Multidimensional Problem. New York: John Wiley & Sons, 1978
19	Xu R W, Li X X. On the complete solutions to the Tchebychev-Affine-Kähler equation and its geometric significance. 2019(Submmitted)
20	Xu R W , Lei M X . Classification of Calabi hypersurfaces with parallel Fubini-Pick form. Diff Geom Appl, 2021, 74: 101707 doi: 10.1016/j.difgeo.2020.101707

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Classification of Calabi Hypersurfaces in ${\mathbb R}^5$ with Parallel Fubini-Pick Form

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