ScholarWorks@숙명여자대학교

초록

Let Sigma be a k-dimensional complete proper minimal submanifold in the Poincare ball model B-n of hyperbolic geometry. If we consider Sigma as a subset of the unit ball B-n in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold Sigma and the ideal boundary partial derivative(infinity)Sigma, say Vol(R)(Sigma) and Vol(R)(partial derivative(infinity)Sigma), respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if Vol(R)(partial derivative(infinity)Sigma) >= Vol(R)(Sk-1), then Sigma satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such Sigma, we further obtain a sharp lower bound for the Euclidean volume Vol(R)(Sigma), which is an extension of Fraser-Schoen and Brendle's recent results to hyperbolic space. Moreover we introduce the Mobius volume of Sigma in B-n to prove an isoperimetric inequality via the M bius volume for Sigma.

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