Prescribing mean curvature vectors for foliations (Q1425765)

Let \((M,{\mathcal F})\) be an oriented closed manifold \(M\) with an oriented foliation \({\mathcal F}\), \(m= \dim M\), \(p= \dim {\mathcal F}\). Let \(H_{{\mathcal F},g}= \sum^p_{i=1} (\nabla_{E_i}E_i)^\perp\) be the mean curvature of \({\mathcal F}\) with respect to a Riemannian metric \(g\) with its Levi-Cività connection \(\nabla\), \(E_1,\dots, E_p\) being a local orthonormal frame of vector fields tangent to \({\mathcal F}\). Let \(A\) denote a closed subset of \(M\) contained in \(M\setminus\Sigma\), and \(P_{{\mathcal F},X}(A)\) the closed linear space generated by all Dirac currents \(X(x)\wedge v_1\wedge\cdots\wedge v_{p-1}\), where \(v_1,\dots, v_{p-1}\in T_x{\mathcal F}\), \(x\in A\), \(X\) is a vector field, \(\Sigma\) the singular set of \(X\). Denote by \(C_{{\mathcal F},X}(A)\) the closed convex cone generated by the union \(C_{\mathcal F}(A)\cup \widetilde C_{{\mathcal F},X}(A)\), \(C_{\mathcal F}(A)=\) the closed convex cone generated by all Dirac currents of the form \(e_1\wedge\cdots\wedge e_p\), where \((e_1,e_2,\dots,e_p)\) is a positive oriented frame of the space tangent to the leaf through \(x\), \(T_x{\mathcal F}\). \(\widetilde C_{{\mathcal F},X}(A)\) denotes the closed convex cone generated by all boundaries \(\partial(-X(x)\wedge e_1\wedge\cdots\wedge e_p)\) for \(x\in A\). The following problem is studied: Given \((M,{\mathcal F})\) and \(X\), find conditions for \(X\) to coincide with \(H_{{\mathcal F},g}\) for some \(g\). The authors prove two theorems. Theorem 1: Suppose that \(g_0\) is a Riemannian metric on a neighbourhood \(U\) of \(\Sigma\) on which \(X= H_{{\mathcal F},g_0}\). Let \(B\) be a smooth compact submanifold of \(M\) with \(\Sigma\subset \text{Int}(B)\subset B\subset U\) and let \(A\) be the closure of \(M\setminus B\), so that \(\partial A=\partial B= A\cap B\). Then there exists a Riemannian metric \(g\) on \(M\) extending \(g_0| B\) and such that \(X= H_{{\mathcal F},g}\) on \(M\) if and only if the cone \(C_{{\mathcal F},X}(A)\) has a compact base and intersects the space \(P_{{\mathcal F},X}(A)\) trivially. Let \(B_{\mathcal F}(\Sigma)\) be the space generated by all boundaries of the form \(\partial(w\wedge v_1\wedge\cdots\wedge v_p)\), \(w\in T_2M\), \(v_i\in T_xM\), \(x\in\Sigma\) and \(P_{{\mathcal F},E}\) is the space generated by all Dirac currents of the form \(w\wedge v_1\wedge\cdots\wedge v_{p-1}\), where \(w\in E_x\), \(v_i\in T_x{\mathcal F}\), \(x\in M\). Theorem 2: If \(X\) is a vector field on \((M,{\mathcal F})\) which takes values in a subbundle \(E\subset TM\) complementary to \(T{\mathcal F}\), \(E\) and \(\Sigma= \{x\in M; X(x)= 0\}\) satisfy \(C_{\mathcal F}\cap (P_{{\mathcal F},E}+ B_{\mathcal F}(\Sigma))= \{0\}\), \(X/U\) is a gradient field for some open neighbourhood \(U\) of \(\Sigma\), and the cone \(C_{{\mathcal F},X}(A)\) has a compact base and intersects the subspace \(P_{{\mathcal F},X}(A)\) trivially for every closed set \(A\subset M\setminus\Sigma\), then there exists a Riemannian metric \(g\) on \(M\) for which \(X= H_{{\mathcal F},g}\), the mean curvature of the foliation \({\mathcal F}\) on \((M,g)\). Some examples are presented. The methods are expressed in terms of currents.