Isoperimetric inequalities on minimal submanifolds of space forms
- Cite this article as:
- Choe, J. & Gulliver, R. Manuscripta Math (1992) 77: 169. doi:10.1007/BF02567052
For a domainU on a certaink-dimensional minimal submanifold ofSn orHn, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU kk ωkM (D)k-1 ≤Vol(∂D)k, where ωk is the volume of the unit ball ofRk. Also, we prove that ifD is any domain on a minimal surface inS+n (orHn, respectively), thenD satisfies an isoperimetric inequality2π A≤L2+A2 (2π A≤L2−A2 respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofHn, then(k−1) Vol(U)≤Vol(∂U).