Extended Abstracts Fall 2013 by Maria del Mar González Paul C. Yang Nicola Gambino & Joachim Kock

Author:Maria del Mar González, Paul C. Yang, Nicola Gambino & Joachim Kock
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where equality holds if and only if is a k-dimensional unit ball B k in B n .

In order to prove this theorem, we need the monotonicity theorem. By measuring the Euclidean volume rather than the hyperbolic volume in B n , we have the following result.

Theorem 5 (Monotonicity)

Let be a k-dimensional complete minimal submanifold in B n . Then the function is nondecreasing in r for 0 < r < 1. In other words,

which is equivalent to

where ρ is the hyperbolic distance from the origin.

Proof

Use the coarea formula (see [9, 10, 20]). □

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