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”.
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.
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