Separable calibrations and minimal surfaces (Q1376671)

As is well-known, if \(\mathbb{R}^n =V\oplus W\) is an orthogonal decomposition of \(\mathbb{R}^n\), and if \(\varphi\in \wedge^r(V)\) and \(\eta\in \wedge^s(W)\), then \(|\varphi \wedge \eta |^*= |\varphi |^*|\eta|^*\) (cf. \textit{R. Harvey} and \textit{H. B. Lawson} [Acta Math. 148, 47-157 (1982; Zbl 0584.53021)]). In this paper, the author studies the \(p\)-forms in \(\mathbb{R}^n\), which can be expressed in the form: \(\Omega= \Omega_1+ e^*_{V_1} \wedge \Omega_2+ \cdots+ e^*_{V_1} \wedge\cdots \wedge e^*_{V_k} \wedge \Omega_{k+1}\), where \(\mathbb{R}^n= V_1\oplus \cdots \oplus V_{k+1}\) is an orthogonal decomposition of \(\mathbb{R}^n\), \(\dim V_t =p_t\geq 2\) for \(t\leq k\), \(\Omega_i\in \wedge^{q_i} (W_i) \), \(q_i=p -\sum_{t<i} p_t\), \(e^*_{V_i}\) is the unit \(p_i\)-form of \(V_i\), \(W_i= \oplus_{t>i} V_t\) for \(i<k\), and \(W_{k+1} =V_{k+1}\). In such a case, the comass of \(\Omega\) can be calculated by the formula \(|\Omega |^* =\max|\Omega_i |^*\), and the maximal direction of \(\Omega\) can be defined by the maximal directions of the terms whose comass equals \(\Omega\). It is then proved that \(|\Omega \wedge\eta |^*= |\Omega |^*|\eta|^*\) for \(\Omega\) expressed as above. Some examples to illustrate the set of such separable forms are given. Finally, as a geometric application, minimal surfaces in almost product manifolds are considered.