Isoperimetric inequalities on minimal submanifolds of space forms (Q1210033)

Let \(D\) be a domain in one of the space forms: \(\mathbb{R}^ n\), \(S^ n\), \(H^ n\), and \(\text{Area}(D)=A\), \(\text{Length}(\partial D)=L\). The well-known isoperimetric inequality tells us \(4\pi A\leq L^ 2+KL^ 2\), where \(K\) is the Gauss curvature of the corresponding space form. Considering that minimal surfaces are viewed as generalized planes, it is tempting to conjecture that these isoperimetric inequalities should still hold for domains on any immersed minimal surfaces in space forms. The truth of this conjecture is not yet known, except in special cases. In this paper, for a domain \(D\) on a \(k\)-dimensional minimal submanifold of a space form the authors introduce a modified volume \(M(D)\) of \(D\) and obtain an isoperimetric inequality \(k^ k\omega_ kM(D)^{k- 1}\leq\text{Volume}(\partial D)^ k\), where \(\omega_ k\) is the volume of the unit ball of \(\mathbb{R}^ k\). This modified isoperimetric inequality is proved under any of the following conditions: (Theorem 4) \(\partial D\) lies on a geodesic sphere of the space form; (Theorem 2) \(k=2\), \(D\subset S^ n_ +\) and \(\partial D\) is weakly connected; (Theorem 1) \(k=2\), \(D\subset S^ n_ +\) and \(\partial D\) is radially connected about a point of \(D\). In the second part of this paper, the authors prove that (Theorem 5) if \(D\) is a domain on a minimal surface in \(S^ n_ +\) (or \(H^ n\) respectively), then \(D\) satisfies the isoperimetric inequality \(2\pi A\leq L^ 2+A^ 2\) \((2\pi A\leq L^ 2-A^ 2\) respectively). Moreover, they show that (Theorem 6) if \(U\) is a \(k\)-dimensional minimal submanifold of \(H^ n\), then \((k-1)\text{Vol}(U)\leq\text{Vol}(\partial U)\).