On Minimal Hypersurfaces of a Unit Sphere

Amira Ishan, Sharief Deshmukh, Ibrahim Al-Dayel and Cihan Özgür
Additional contact information
Amira Ishan: Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Ibrahim Al-Dayel: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, P.O. Box 65892, Riyadh 11566, Saudi Arabia
Cihan Özgür: Department of Mathematics, İzmir Democracy University, Karabağlar, 35140 İzmir, Turkey

Mathematics, 2021, vol. 9, issue 24, 1-9

Abstract: Minimal compact hypersurface in the unit sphere S n + 1 having squared length of shape operator A 2 < n are totally geodesic and with A 2 = n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in S n + 1 . One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of S n + 1 . It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S 5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in S n + 1 , ( n > 2 ), provided the scalar curvature τ is a constant on integral curves of w . Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in S n + 1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field A w .

Keywords: minimal hypersurfaces; totally geodesic hypersurfaces; sphere; clifford hypersurfaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/9/24/3161/pdf (application/pdf)
https://www.mdpi.com/2227-7390/9/24/3161/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:9:y:2021:i:24:p:3161-:d:697724

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().