On complete submanifolds with parallel mean curvature in product spaces

Abstract

We prove a Simons type formula for submanifolds with parallel mean curvature vector field in product spaces of type M n (c)×R, where M n (c) is a space form with constant sectional curvature c, and then we use it to characterize some of these submanifolds.2000 Mathematics Subject Classification. 53A10, 53C42.

On complete submanifolds with parallel normalized mean curvature in product spaces

“…Working with Cheng-Yau's differential operator, we use an alternative approach to find a Simons-type formula for pnmc submanifolds in product spaces M n (c) × R (cf. proposition 2.2) which generalize the formula obtained by Fetcu and Rosenberg [18]. Next, we present some auxiliary results and we establish a maximum principle for an arbitrary semielliptic operator (cf.…”

“…An interesting question is to describe submanifolds immersed in a product space using Simons' formula. In this direction, Batista [6] found a Simons' type formula for surfaces with cmc immersed in the product spaces M 2 (c) × R. Allied to the generalized maximum principle of Omori-Yau, he used this formula in order to characterize complete surfaces with cmc in the spaces H 2 × R and S 2 × R. Passing to higher dimension and codimension, we highlight the studies of Fetcu, Oniciuc and Rosenberg in [17,18], concerning to submanifolds with parallel mean curvature vector field (pmc submanifolds) in the product spaces M n (c) × R. Indeed, they computed a Simons type formula for pmc submanifolds in M n (c) × R. More precisely, in [17], they showed that, under suitable restrictions on the square of the norm of the second fundamental form of Σ m , it should be a cmc totally umbilical hypersurfaces in M m+1 (c) → M n (c) for n > m 3. In [18] they obtained gap results for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and also, classify proper-biharmonic pmc surfaces in S n × R.…”

A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds

“…During the last 50 years, there are many research done on submanifolds with parallel mean curvature vector. Among others, for submanifolds with parallel mean curvature vector in real space forms, see [142,143,144,145,146,147,148,149]; for surfaces with parallel mean curvature vector in complex space forms, see [150,151,152,154,155,156,153]; for surfaces with parallel mean curvature vector in indefinite space forms, see [157,158,159,160,161,162]; for surfaces with parallel mean curvature vector in homogeneous spaces or symmetric spaces, see [163,164]; for surfaces with parallel mean curvature vector in Sasakian space forms, see [165]; and for surfaces with parallel mean curvature vector in reducible manifolds, see [166,167,168,169]. For general references of submanifolds with parallel mean curvature vector, see [170].…”

Submanifolds with Parallel Mean Curvature Vector Field in Product Spaces

“…|Φ| 2 = 2(H 2 + c) − 3c|T | 2 . Using the compatibility (11) to the third equation of (43), we obtain 0 = λ∇ ⊥ k N 3 + μ∇ ⊥ k N 4 = T k λh 3 kk + μh 4 kk for all k, i.e.,…”