Certain minimal or homologically volume minimizing submanifolds in compact symmetric spaces (Q1067623)

Let M be a simply connected compact symmetric space. Take any point \({\mathfrak O}\) in M and let K be the isotropy subgroup of the group of isometries of M with respect to \({\mathfrak O}\). Then the first conjugate locus \(F_{{\mathfrak O}}(M)\) of M with respect to \({\mathfrak O}\) is expressed as \(F_{{\mathfrak O}}(M)=\cup_{k\in K}k S\) where S is a certain subset of the maximal torus of M through \({\mathfrak O}\). The author defines a certain \(S^ 0\) of S and proves the following theorem : The set \(F^ 0_{{\mathfrak O}}(M)=\cup_{k\in K}k S^ 0\) is a minimal submanifold of M, which is open and dense in \(F_{{\mathfrak O}}(M).\) Next let M be a compact simple Lie group G with a bi-invariant Riemannian metric. Let \(G_ 1\) be a 3-dimensional compact simply connected Lie subgroup of G corresponding to a highest root of the Lie algebra of G. Then constructing a certain calibration on G, the author proves the following theorem : \(G_ 1\) is a homologically volume minimizing submanifold of G. Now let N be a quaternionic Kähler manifold. Then the author obtains the following generalization of a well known fact in Kählerian geometry. Theorem: Let V be a 4r-dimensional quaternionic Kähler submanifold of N. If N is compact, then Vol(V)\(\leq Vol(V')\) for any compact oriented 4r-dimensional submanifold V' such that V and V' define the same class in the homology group \(H_{4r}(N,R)\). If V is noncompact, V is stable under variations of compact supports. Theorem. Let V be a 4r-dimensional oriented compact submanifold of the quaternion projective space \(P^ n(H)\) such that \([V]=[P^ r(H)]\) in \(H_{4r}(M,R)\). Then \(Vol(P^ r(H))\leq Vol(V)\) and the equality holds if V is congruent with \(P^ r(H)\) in \(P^ n(H)\).