On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms
Rifaqat Ali, Fatemah Mofarreh, Nadia Alluhaibi, Akram Ali, Iqbal Ahmad
Abstract
In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold N n in Sasakian space forms N ˜ 2 n + 1 ( ϵ ) . We prove that a minimal Legendrian submanifolds in a Sasakian space form is isometric to a standard sphere S n if the Ricci curvature satisfies an extrinsic condition which includes a gradient of a function, the constant holomorphic sectional curvature of the ambient space and a dimension of N n . We also obtain a Simons-type inequality for the same ambient space forms N ˜ 2 n + 1 ( ϵ ) .
Topics & Concepts
SubmanifoldMathematicsMathematical analysisHolomorphic functionMean curvatureAmbient spaceSpace formSpace (punctuation)Pure mathematicsSectional curvatureLaplace operatorDimension (graph theory)Eigenvalues and eigenvectorsCurvatureConstant (computer programming)Scalar curvatureGeometryPhysicsComputer scienceProgramming languageQuantum mechanicsLinguisticsPhilosophyGeometric Analysis and Curvature FlowsGeometry and complex manifoldsNonlinear Partial Differential Equations