Lower bounds for the mean curvature of hollow tubes around complex hypersurfaces and totally real submanifolds
type: Item
instance of: scholarly article
title: Lower bounds for the mean curvature of hollow tubes around complex hypersurfaces and totally real submanifolds
zbMATH Open document ID: 0843.53044
author: Vicente Miquel
author: Vicente Palmer
published in: Illinois Journal of Mathematics
publication date: 1996-08-25T00:00:00Z
review text: In [Compos. Math. 86, No. 3, 317-335 (1993; Zbl 0782.53054)]\ the authors derived comparison theorems for the mean curvature of tubular hypersurfaces in complete Kähler manifolds, taking as models the tubular hypersurfaces around a complex projective space embedded as a totally geodesic submanifold in a complex projective space \(\mathbb{C} P^n\). The paper under review continues this work. Here the authors derive comparison theorems for the mean curvature of a tubular hypersurface about a compact commplex hypersurface \(P\) and about a compact totally real submanifold \(P\) of dimension \(n\) in a complex Kähler manifold \(M^{2n}\), with some specific curvature bounds, when the models for the comparison are the tubular hypersurfaces about the complex hyperquadric \(Q\) and the real projective space \(\mathbb{R} P^n\) embedded in \(\mathbb{C} P^n\), in particular when \(P\) is totally geodesic. In the above mentioned paper they used the Jacobi equation technique while now they make use of the Riccati equation. Moreover, they derive comparisons for the relative volume, the first Dirichlet eigenvalue and the mean exit time. They also obtain several necessary geometrical properties when the equality is obtained in their comparison formulas but they are still unable to decide whether these equalities characterize the model pairs \((Q, \mathbb{C} P^n)\), \((\mathbb{R} P^n, \mathbb{C} P^n)\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 856556
zbMATH Keywords: comparison theorems
zbMATH Keywords: mean curvature
zbMATH Keywords: tubular hypersurface
zbMATH Keywords: Riccati equation
zbMATH Keywords: relative volume
zbMATH Keywords: first Dirichlet eigenvalue
zbMATH Keywords: mean exit time
reviewed by: Lieven Vanhecke
MaRDI profile type: Publication
Recommended article: Q4694933
Recommended article: Q4816558
Recommended article: CHARACTERIZATIONS OF HYPERSURFACES OF TYPE A2 IN A COMPLEX PROJECTIVE SPACE
Recommended article: Immersions of compact Riemann manifolds into a ball of a complex space form
Recommended article: Comparison theorems for the volume of a complex submanifold of a Kaehler manifold
Recommended article: Q4523983
Recommended article: Q4938704
Recommended article: Characterizations of real hypersurfaces in complex space forms in terms of curvature tensors
Recommended article: A rigidity theorem for the pair \(2, \mathbb{C} P^n\) (complex hyperquadric, complex projective space)
Recommended article: Volumes of Tubes About Complex Submanifolds of Complex Projective Space
description: scientific article; zbMATH DE number 856556