A short note on rigidity of compact surfaces in space forms

• Autorzy: Costa-Filho Wagner Oliveira

Identyfikatory

DOI

10.4064/ba220613-19-3

• Języki publikacji: EN

Abstrakty

EN

A rigidity result for a compact (without boundary) surface in a space form is obtained under a certain condition involving the Gaussian curvature and mean curvature of the surface. Moreover, as a byproduct of our approach, we shall present an alternative proof of an extension of Li’s theorem (2001) due to Chang (2013), on an integral inequality for Willmore surfaces in space forms.

Słowa kluczowe

EN

compact surfaces; space forms; Gaussian curvature; totally umbilical

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2022
• Tom: Vol. 70, no. 2
• Strony: 159--164
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Costa-Filho Wagner Oliveira

• Campus Arapiraca, Federal University of Alagoas, CEP 57309-005, Arapiraca, Alagoas, Brazil

• https://orcid.org/0000-0002-9536-5322

Bibliografia

• [1] L. J. Alías, J. Meléndez, and O. Palmas, Hypersurfaces with constant scalar curvature in space forms, Differential Geom. Appl. 58 (2018), 65-82.
• [2] H. Alodan and S. Deshmukh, A characterization of spheres in a Euclidean space, New Zealand J. Math. 36 (2007), 93-99.
• [3] L. J. Alías, P. Mastrolia, and M. Rigoli, Maximum Principles and Geometric Applications, Springer Monogr. Math., Springer, Cham, 2016.
• [4] Y.-C. Chang, Willmore surfaces and F-Willmore surfaces in space forms, Taiwanese J. Math. 17 (2013), 109-131.
• [5] S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195-204.
• [6] H. F. de Lima and F. R. dos Santos, Gap theorems for linear Weingarten surfaces, Ann. Univ. Ferrara Sez. VII Sci. Mat. 66 (2020), 89-98.
• [7] A. Gruber, M. Toda, and H. Tran, On the variation of curvature functionals in a space form with application to a generalized Willmore energy, Ann. Global Anal. Geom. 56 (2019), 147-165.
• [8] H. Li, Willmore hypersurfaces in a sphere, Asian J. Math. 5 (2001), 365-377.
• [9] J. Liu and H. Jian, F-Willmore submanifold in space forms, Front. Math. China 6 (2011), 871-886.
• [10] F. C. Marques and A. Neves, The Willmore conjecture, Jahresber. Deutsch. Math.-Verein. 116 (2014), 201-222.
• [11] H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), 211-239.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).