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Gap theorems for complete submanifolds in the hyperbolic space

Jianling Liu, Yong Luo

TL;DR

This work advances rigidity theory for submanifolds in hyperbolic space by establishing gap theorems for complete submanifolds with parallel mean curvature and, separately, for complete hypersurfaces with scalar curvature $R=n(1-n)$. The authors extend prior sphere and Euclidean-space results to the hyperbolic setting, leveraging Simons' formula, Lin’s eigenvalue estimate, and a refined integral-inequality framework with cutoff functions to show that small total curvature forces the trace-free second fundamental form to vanish, yielding total geodesicity. The second result translates Li–Xu–Zhou’s Euclidean/spherical hypersurface gaps to hyperbolic space under an elliptic-condition constraint on the Newton transformation, again concluding total geodesicity under a small $L^n$-norm of the mean curvature. Together, the results deepen the understanding of rigidity phenomena in hyperbolic ambient spaces and illustrate the power of eigenvalue and integral techniques in geometric analysis.

Abstract

Based on the seminal Simons' formula, Shen \cite{Shen} and Lin-Xia \cite{LX} obtained gap theorems for compact minimal submanifolds in the unit sphere in the late 1980's. Then due to the effect of Xu \cite{Xu}, Ni \cite{Ni}, Yun \cite{Yun} and Xu-Gu \cite{XuG}, we achieved a comprehensive understanding of gap phenomena of complete submanifolds with parallel mean curvature vector field in the sphere or in the Euclidean space. But such kind of results in case of the hyperbolic space were obtained by Wang-Xia \cite{XiaW}, Lin-Wang \cite{LW} and Xu-Xu \cite{XX} until relatively recently and are not quite complete so far. In this paper first we continue to study gap theorems for complete submanifolds with parallel mean curvature vector field in the hyperbolic space, which generalize or extend several results in the literature. Second we prove a gap theorem for complete hypersurfaces with constant scalar curvature $n(1-n)$ in the hyperbolic space, which extends related results due to Bai-Luo \cite{BL2} in cases of the Euclidean space and the unit sphere. Such kind of results in case of the hyperbolic space are more complicated, due to some extra bad terms in the Simons' formula, and one of main ingredients of our proofs is an estimate for the first eigenvalue of complete submanifolds in the hyperbolic space obtained by Lin \cite{Lin}.

Gap theorems for complete submanifolds in the hyperbolic space

TL;DR

This work advances rigidity theory for submanifolds in hyperbolic space by establishing gap theorems for complete submanifolds with parallel mean curvature and, separately, for complete hypersurfaces with scalar curvature . The authors extend prior sphere and Euclidean-space results to the hyperbolic setting, leveraging Simons' formula, Lin’s eigenvalue estimate, and a refined integral-inequality framework with cutoff functions to show that small total curvature forces the trace-free second fundamental form to vanish, yielding total geodesicity. The second result translates Li–Xu–Zhou’s Euclidean/spherical hypersurface gaps to hyperbolic space under an elliptic-condition constraint on the Newton transformation, again concluding total geodesicity under a small -norm of the mean curvature. Together, the results deepen the understanding of rigidity phenomena in hyperbolic ambient spaces and illustrate the power of eigenvalue and integral techniques in geometric analysis.

Abstract

Based on the seminal Simons' formula, Shen \cite{Shen} and Lin-Xia \cite{LX} obtained gap theorems for compact minimal submanifolds in the unit sphere in the late 1980's. Then due to the effect of Xu \cite{Xu}, Ni \cite{Ni}, Yun \cite{Yun} and Xu-Gu \cite{XuG}, we achieved a comprehensive understanding of gap phenomena of complete submanifolds with parallel mean curvature vector field in the sphere or in the Euclidean space. But such kind of results in case of the hyperbolic space were obtained by Wang-Xia \cite{XiaW}, Lin-Wang \cite{LW} and Xu-Xu \cite{XX} until relatively recently and are not quite complete so far. In this paper first we continue to study gap theorems for complete submanifolds with parallel mean curvature vector field in the hyperbolic space, which generalize or extend several results in the literature. Second we prove a gap theorem for complete hypersurfaces with constant scalar curvature in the hyperbolic space, which extends related results due to Bai-Luo \cite{BL2} in cases of the Euclidean space and the unit sphere. Such kind of results in case of the hyperbolic space are more complicated, due to some extra bad terms in the Simons' formula, and one of main ingredients of our proofs is an estimate for the first eigenvalue of complete submanifolds in the hyperbolic space obtained by Lin \cite{Lin}.
Paper Structure (7 sections, 17 theorems, 106 equations)

This paper contains 7 sections, 17 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Basic notations
  4. Submanifolds
  5. Hypersurfaces
  6. Proofs of Theorem \ref{['th:1.3']} and Theorem \ref{['main thm2']}
  7. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1.1

Let $M^n(n\geq 3)$ be a complete submanifold with parallel mean curvature vector field in the unit sphere or the Euclidean space. Denote by $Å$ the trace free second fundamental form of $M$. There exists an explicit positive constant $C_1(n)$ depending only on $n$, such that if then $Å=0$, i.e. $M$ is totally umbilical. In particular, if $M$ is a minimal submanifold in the Euclidean space, then i

Theorems & Definitions (19)

  • Theorem 1.1: XuXuG
  • Theorem 1.2: XiaW
  • Theorem 1.3: XHGH
  • Theorem 1.4: XX
  • Theorem 1.5: LW
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: LXZ
  • Theorem 1.9: BL2
  • Theorem 1.10: BL2
  • ...and 9 more