Abstract
We prove a generalization of Reifenberg’s isoperimetric inequality. The main result of this paper is used in Harrison and Pugh (General methods of elliptic minimization, Available on arxiv, 2016) to establish existence of a minimizer for an anisotropically-weighted area functional among a collection of surfaces which satisfies a set of axioms, namely being closed under certain deformations and Hausdorff limits. This problem is known as the axiomatic Plateau problem.
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Notes
If \( \mathcal {H}^{m-1}(A)=0 \), we understand the neighborhood of radius zero of \( A \) to mean the set \( A \) itself.
When \( \mathcal {H}^{m-1}(A)=0 \), Theorem 1 follows from dimension theory (see [2] VII 3, VIII 3’,) whereby such a set \( A \) has no \( m \)-dimensional holes, so the set \( X=A \) satisfies the requirements of Theorem 1. In this case, Theorem 2 gives a different set \( \tilde{Y} \) for each choice of \( L>0 \).
See definition below.
Such a vector field \( V \) is complete, see e.g. [7] Ch. 2 Prop. 1.6, so \( V \) generates a \( 1 \)-parameter group of diffeomorphisms \( \{(\phi _t: \mathbb {R}^n\rightarrow \mathbb {R}^n) : t\in \mathbb {R}\} \). We restrict \( t \) to \( \mathbb {N}\) to generate our sequence of diffeomorphisms \( \{\phi _i \} \).
In the case that \( m=2 \) we encounter the quantity \( \mathcal {H}^{-1}(S\cap W\cap \Sigma ) \), which is zero if and only if \( S\cap W\cap \Sigma \) is empty, and otherwise is infinite.
There is a small error at the corresponding point of Reifenberg’s original proof of Theorem 1. On page 17, his set \( X_i+X_{i+1}+A_i \), which corresponds to our set \( X_i\cup X_{i+1}\cup (A\cap W_i) \) does not necessarily satisfy the hypotheses of his proposition \( P(m_0,N,k_0) \), since the quantity \( \Lambda ^{m-1}(X_i+X_{i+1}+A_i)^{1/(m-1)} \) may be too small. We fix this issue by introducing the quantity \( \mathcal {L}_m \) in our proof.
References
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Communicated by C. De Lellis.
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Pugh, H. Reifenberg’s isoperimetric inequality revisited. Calc. Var. 58, 159 (2019). https://doi.org/10.1007/s00526-019-1602-4
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DOI: https://doi.org/10.1007/s00526-019-1602-4