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Gap results for biharmonic submanifolds in spheres

Stefan Andronic, Simona Nistor

Abstract

In this paper we determine a larger gap of the mean curvature for a class of proper biharmonic submanifolds with parallel mean curvature vector field in Euclidean spheres. When the bounds of the gap are reached, we obtain splitting results of the submanifold.

Gap results for biharmonic submanifolds in spheres

Abstract

In this paper we determine a larger gap of the mean curvature for a class of proper biharmonic submanifolds with parallel mean curvature vector field in Euclidean spheres. When the bounds of the gap are reached, we obtain splitting results of the submanifold.
Paper Structure (6 sections, 33 theorems, 228 equations)

This paper contains 6 sections, 33 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\lambda$-biharmonic submanifolds
  4. Examples of $\lambda$-biharmonic hypersurfaces
  5. Properties of $PMC$ $\lambda$-biharmonic submanifolds
  6. Biharmonic product submanifolds

Key Result

Theorem \ref{th:SummarizeTheorem}

Let $\varphi_1 : M_1 ^{m_1} \to \mathbb S^{n_1} (r_1)$, $m_1 \geq 2$, and $\varphi_2 : M_2^{m_2} \to \mathbb S^{n_2} (r_2)$, $m_2 \geq 2$, be two non-minimal and $PMC$ submanifolds such that $M^m = M_1^{m_1} \times M_2^{m_2}$ is a proper biharmonic in $\mathbb S^n$, $r_1 ^2 + r_2^2 = 1$ and $n_1 + n

Theorems & Definitions (56)

  • Conjecture
  • Theorem \ref{th:SummarizeTheorem}
  • Proposition 2.1: CaddeoMontaldoOniciuc2002
  • Proposition 2.2: CaddeoMontaldoOniciuc2002
  • Proposition 2.3: OniciucPHD
  • Remark 2.4
  • Proposition 2.5: BalmusOniciuc2012
  • Remark 2.6
  • Theorem 2.7: BalmusOniciuc2012
  • Remark 2.8
  • ...and 46 more