Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 19, 2015

Allard-type boundary regularity for C1,α boundaries

Theodora Bourni

Abstract

In this paper, we show boundary monotonicity formulae for rectifiable varifolds having a C1,α “boundary”. In particular, we show that the area ratios of balls centered at this “boundary” satisfy a nice monotonicity formula, similar to that for interior balls proved by Allard in [Ann. of Math. (2) 95 (1972), 417–491]. This extends the boundary monotonicity formulae of Allard [Ann. of Math. (2) 101 (1975), 418–446], which require that the boundary is C1,1. As a corollary, the regularity results of [Ann. of Math. (2) 101 (1975), 418–446] extend to this case and provide a regularity result for rectifiable varifolds with a C1,α “boundary”.

MSC: 49Q15

I would like to thank Professor Leon Simon for suggesting this problem to me and for all his helpful advice. Furthermore, I would like to thank Ulrich Menne and Alexander Volkmann for fruitful discussions on Allard's papers [Ann. of Math. (2) 95 (1972), 417–491; Ann. of Math. (2) 101 (1975), 418–446] and I would further like to thank Ulrich Menne for all his useful suggestions in writing this paper. Finally, I would like to thank Mat Langford for his time in proofreading this paper.

Received: 2014-9-10
Revised: 2014-12-9
Accepted: 2015-1-21
Published Online: 2015-2-19
Published in Print: 2016-4-1

© 2016 by De Gruyter

Scroll Up Arrow