Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space (Q741787)

In this paper, the authors obtain some isoperimetric inequalities for \(k\)-dimensional complete proper minimal submanifolds in the Poincaré ball model of the \(n\)-dimensional hyperbolic space. Let \(B^n\subset {\mathbb R}^n\) be the Euclidean unit ball centered at \(0\), and \(\Sigma\subset B^n\) be a complete proper minimal (for the hyperbolic metric) submanifold. Let \(\partial_\infty\Sigma=\overline{\Sigma}\cap\partial B^n\) be the ideal boundary of \(\Sigma\). Then, the authors prove in Theorem~2.3 that \[ \text{Vol}_{\mathbb R}(\Sigma)\leq {1\over k}\,\text{Vol}_{\mathbb R}(\partial_\infty\Sigma), \] where \(\text{Vol}_{\mathbb R}(\Sigma)\) is the \(k\)-dimensional Euclidean volume of \(\Sigma\) and \(\text{Vol}_{\mathbb R}(\partial_\infty\Sigma)\) is the \((k-1)\)-dimensional Euclidean volume of \(\partial_\infty\Sigma\). Equality holds if and only if \(\Sigma\) is a \(k\)-dimensional Euclidean ball \(B^k\). They also prove in Theorem 3.1 that the quotient \[ {\text{Vol}_{\mathbb R}(\Sigma\cap B_r)\over r^k}, \] where \(B_r\) is the Euclidean ball of center \(0\) and radius \(0<r<1\), is non-decreasing. A corollary of this result is that when \(0\in\Sigma\), then \(\text{Vol}_{\mathbb R}(\Sigma)\geq \text{Vol}_{\mathbb R}(B_k)\), with equality precisely when \(\Sigma=B^k\). In the last section, it is shown in Theorem 4.7 that the isoperimetric inequality in \({\mathbb R}^k\) holds for \(\text{Vol}_{M}(\Sigma)\) and \(\text{Vol}_M(\partial_\infty\Sigma)\), where \(\text{Vol}_M\) is the Möbius volume, defined as \(\sup_g (\text{Vol}_{\mathbb R}\circ g)\), where \(g\) belongs to the Möbius transformations of the ball \(B^n\) and the sphere \(\partial B^n\), respectively. Equality holds when \(\Sigma\) is the \(k\)-dimensional ball.