J. Aust. Math. Soc.  76 (2004), 151-165
An application of Moser iteration to complete minimal submanifolds in a sphere

Leung-Fu Cheung
  Department of Applied Mathematics
  The Hong Kong Polytechnic University
  Hung Hom Kowloon
  Hongkong
  malfcheu@hkpu07.polyu.edu.hk
and
Pui-Fai Leung
  Department of Mathematics
  National University of Singapore
  Lower Kent Ridge Road
  Singapore 119260
  Singapore
  matfredl@nus.edu.sg


Abstract
We apply the Moser iteration method to obtain a pointwise bound on the norm of the second fundamental form from a bound on its $L^{n}$ norm for a complete minimal submanifold in a sphere. As an application we show that a complete minimal submanifold in a sphere with finite total curvature and Ricci curvature bounded away from $-\infty$ must be compact. This complements similar results of Osserman and Oliveira in the case the ambient space is the Euclidean and the hyperbolic space respectively.
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