• Seo, Keomkyo (Department of Mathematics Sookmyung Women's University)
  • Received : 2010.05.13
  • Accepted : 2010.06.07
  • Published : 2010.06.30


Let C be a closed convex set in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$. Assume that ${\Sigma}$ is an n-dimensional compact minimal submanifold outside C such that ${\Sigma}$ is orthogonal to ${\partial}C$ along ${\partial}{\Sigma}{\cap}{\partial}C$ and ${\partial}{\Sigma}$ lies on a geodesic sphere centered at a fixed point $p{\in}{\partial}{\Sigma}{\cap}{\partial}C$ and that r is the distance in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$ from p. We make use of a modified volume $M_p({\Sigma})$ of ${\Sigma}$ and obtain a sharp relative isoperimetric inequality $$\frac{1}{2}n^n{\omega}_nM_p({\Sigma})^{n-1}{\leq}Vol({\partial}{\Sigma}{\sim}{\partial}C)^n$$, where ${\omega}_n$ is the volume of a unit ball in ${\mathbb{R}}^n$ Equality holds if and only if ${\Sigma}$ is a totally geodesic half ball centered at p.


  1. C. Bandle, Isoperimetric inequalities and applications, Pitman, 1980.
  2. J. Choe, Relative isoperimetric inequality for domains outside a convex set, Arch. Inequal. Appl. 1(2003), 241-250.
  3. J. Choe, The double cover relative to a convex set and the relative isoperimetric inequality, J. Aust. Math. Soc. 80(2006), 375-382.
  4. J. Choe, M. Ghomi and M. Ritore, The relative isoperimetric inequality outside convex domains in $\mathbb{B}^n$, Calc. Var. Partial Differential Equations 29(2007), no. 4, 421-429.
  5. J. Choe and R. Gulliver, Isoperimetric inequalities on minimal submanifolds of space forms, Manuscripta Math. 77(1992), 169-189.
  6. J. Choe and M. Ritore, The relative isoperimetric inequality in Cartan- Hadamard 3-manifolds, J. reine angew. Math. 605(2007), 179-191.
  7. I. Kim, Relative isoperimetric inequality and linear isoperimetric inequality for minimal submanifolds, Manuscripta Math. 97(1998), 343-352.
  8. K. Seo, Relative isoperimetric inequality on a curved surface, J. Math. Kyoto Univ. 46(2006), 525-533.
  9. K. Seo, Relative isoperimetric inequality for minimal surfaces outside a convex set, Arch. Math. (Basel) 90(2008), 173-180.