Critical point theorems on Finsler manifolds
Given two submanifolds \(M_1\) and \(M_2\) of a complete Riemannian manifold \(M\), an intensively studied problem is that of the number of geodesics with endpoints in \(M_1\) and \(M_2\) which are orthogonal to \(M_1\) and \(M_2\). In this paper the same problem is studied when the geodesics are replaced with Finsler-geodesics of a Finsler metric \(F\) that dominates the Riemannian metric, i.e., \(F(u)\geq H_0 |u| \) for some \(H_0 > 0\), where \(|.|\) denotes the Riemannian norm. First, the authors describe the Riemann-Hilbert manifold \(\Lambda _N M\) of absolutely continuous maps from \(I= [0,1]\) to \(M\) with endpoints in \(N\subset M\times M\). Secondly, they show that the energy integral \(\widetilde{L} :\Lambda_NM \rightarrow \mathbb{R}\) of \(F\) given by \(\widetilde{L}(c) = \int_I F^2(\dot{c}(t))\,dt \) is of class \(C^{2-}\) on \(\Lambda_NM\) and the geodesics of \(F\) joining orthogonally \(M_1\) and \(M_2\) are just the critical points of the energy integral \(\widetilde{L}: \Lambda_{M_1 \times M_2}\rightarrow \mathbb{R}.\) Thirdly, they prove that the energy functional \(\widetilde{L} :\Lambda_N M \rightarrow \mathbb{R}\) satisfies the Palais-Smale condition. Finaly, using these results they prove Theorem 4: Let \(M\) be a smooth, complete, finite-dimensional Riemannian manifold with a dominating Finsler metric \(F\) and let \(M_1\) and \(M_2\) be closed submanifolds of \(M\) with say \(M_1\) compact. Then, in any homotopy class of curves from \(M_1\) to \(M_2\) there exists a Finsler-geodesic joining \(M_1\) and \(M_2\) orthogonally with length smaller than that of any other curve in this class. Furthermore, there are at least cat\(\Lambda_ {M_1\times M_2} M\) geodesics joining \(M_1\) and \(M_2\) orthogonally. Theorem 5: Let \(M\) be a smooth, compact, connected, finite dimensional Finsler manifold. Suppose that \(M\) is simply connected and let \(M_1,M_2\) be two closed submanifolds of \(M\) such that \(M_1\bigcap M_2 = \emptyset, M_1\) is contractible. Then there are infinitely many Finsler-geodesics joining orthogonally \(M_1\) and \(M_2\). Theorem 6: Let \(M\) be a smooth, complete, non-contractible, finite dimensional Riemannian manifold endowed with a dominating Finsler metric \(F\) and let \(M_1\) and \(M_2\) be two closed and contractible submanifolds of \(M\) such that \(M_1\) or \(M_2\) is compact. Then there exist infinitely many Finsler-geodesics joining \(M_1\) and \(M_2\) orthogonally.
scientific article; zbMATH DE number 2096230