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Register a new userGeometric inequalities for Einstein totally real submanifolds in a complex space form
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Two geometric inequalities are established for Einstein totally real submanifolds in a complex space form. As immediate applications of these inequalities, some non-existence results are obtained.
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We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(kappa ,mu ) $-space forms, Sasakian space forms and non-Sasakian $(kappa ,mu) $-manifolds. Then we obtain obstructions to the existence of minimal isometric immersions of $C$-totally real warped product submanifolds in $(kappa ,mu) $-space forms, non-Sasakian $(kappa ,mu) $-manifolds and Sasakian space forms. In the last, we obtain an example of a warped product $C$-totally real submanifold of a non-Sasakian $(kappa ,mu) $-manifold, which satisfies the equality case of the basic inequality.
In this paper we give local and global parametric classifications of a class of Einstein submanifolds of Euclidean space. The highlight is for submanifolds of codimension two since in this case our assumptions are only of intrinsic nature.
Let $Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar{e} ball model $B^n$ of hyperbolic geometry. If we consider $Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Sigma$ and the ideal boundary $partial_infty Sigma$, say $rvol(Sigma)$ and $rvol(partial_infty Sigma)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $rvol(partial_infty Sigma) geq rvol(mathbb{S}^{k-1})$, then $Sigma$ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such $Sigma$, we further obtain a sharp lower bound for the Euclidean volume $rvol(Sigma)$, which is an extension of Fraser and Schoens recent result cite{FS} to hyperbolic space. Moreover we introduce the M{o}bius volume of $Sigma$ in $B^n$ to prove an isoperimetric inequality via the M{o}bius volume for $Sigma$.
We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.
We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, which represent a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion $iota colon M hookrightarrow Q$ is totally umbilic if and only if the extrinsic shape of every geodesic of $M$ is planar. This extends a well-known result about surfaces in $mathbb{R}^{3}$.
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