Continuous leafwise harmonic functions on codimension one transversely isometric foliations (Q2814362)

Let \(M\) be a closed \(C^2\) manifold and \({\mathcal F}\) a continuous leafwise foliation on \(M\). Let \(S= (M,{\mathcal I},g)\) be a leafwise \(C^2\) foliation (\(g\) is a certain metric). \(S\) is called Liouville if any continuous leafwise harmonic function is leafwise constant. The author gives a proof for Theorem 1: ``A leafwise \(C^2\) transversally isometric codimension-one function is Liouville.'' NEWLINENEWLINELet \(K\) be a closed \(C^2\) manifold \((\dim K\geq 2)\) equipped with a \(C^2\) Riemannian metric \(g\). Consider \(N=K\times I\), \(I=[0,1]\), \(\pi:N\to K\) the canonical projection and \({\mathcal L}\) a continuous foliation which is transverse to the fibers \(\pi^{-1}(y)\), \(\forall y\in K\). A triplet \((N,{\mathcal L},g)\) is called a leafwise \(C^2\) foliated \(I\)-bundle.NEWLINENEWLINENEWLINE Theorem 2: Assume a leafwise \(C^2\) foliated \(I\)-bundle \((N,{\mathcal L},g)\) does not admit a compact leaf in the interior \(\mathrm{Int}(N)\). Then any continuous leafwise harmonic function is constant on \(N\).NEWLINENEWLINENEWLINE Finally, the author considers the case of discrete group actions.