A Pinching Theorem for Compact Minimal Submanifolds in Warped Products I×fSm(c)

Xin Zhan,Zhonghua Hou

doi:10.3390/math8091445

Abstract

Let Sm(c) be a Euclidean sphere of curvature c>0 and R be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type I×fSm(c), where f:I→R+ is a smooth positive function on an open interval I of R. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products Sm(c)×R to Riemannian warped products I×fSm(c).

Highlights

Let Mn+ p (c) (c 6= 0) be an (n + p)-dimensional real space form with constant sectional curvature c and Mn be an n(≥ 2)-dimensional immersed connected submanifold of Mn+ p (c)
We prove the following result: Main Theorem: Let Mn be an n-dimensional compact minimal submanifold in I × f Sm (c) (m ≥ n ≥ 2) with warping function satisfying c − ( f 02 − f f 00 ) = c1 f 4 for some c1 > 0 at every t ∈ I
From the properties of curvature tensor we find h R(e A, e B )eC, e D i = RCDAB = R ABCD

Summary

Introduction

Let Mn+ p (c) (c 6= 0) be an (n + p)-dimensional real space form with constant sectional curvature c and Mn be an n(≥ 2)-dimensional immersed connected submanifold of Mn+ p (c). [1]) Let Mn be an n(≥ 2)-dimensional immersed submanifold in a real space form. Extended Theorem 3 to the case that the ambient space is a Riemannian warped product I × f Mm (c) by proving a new DDVV type inequality for submanifolds immersed in I × f Mm (c), which is similar to (3) and (6). We prove the following result: Main Theorem: Let Mn be an n-dimensional compact minimal submanifold in I × f Sm (c) (m ≥ n ≥ 2) with warping function satisfying c − ( f 02 − f f 00 ) = c1 f 4 for some c1 > 0 at every t ∈ I. Our theorem can be view as a generalization of Theorem 4

Preliminaries

Proof of Main Theorem