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Critical point theorems of Finsler manifolds
by
Laszlo Kozma
University of Debrecen, Hungary
Coauthors: Csaba Varga (Cluj-Napoca, Romania), Alexandru Kristaly (Cluj-Napoca, Romania)
Poster
In this paper we consider a dominating Finsler metric on a complete Riemannian manifold. First we prove that the energy integral of the Finsler metric satisfies the Palais-Smale condition, and ask for the number of geodesics with endpoints in two given submanifolds. Using Lusternik-Schnirelman theory of critical points we obtain some multiplicity results for the number of Finsler-geodesics between two submanifolds.
Let M be a finite dimensional noncompact or a compact manifold and let M1 respectively M2 be two submanifolds of M. Many authors studied the problem in the Riemannian case (see [2], [10], [5], [12], [8], [11], ):
What is the number of geodesics with endpoints in M1 and M2 and which are orthogonal to M1 and M2 ?The purpose of our study is to examine the existence and the number of Finsler-geodesics joining orthogonally M1 and M2 when a Finsler metric is given on a complete Riemannian manifold. The existence of closed geodesics in the case of Finsler space has been studied by F. Mercuri see [4]. Following its considerations we shall extend some of the Riemannian results of K. Grove [2], J.P. Serre [10] and J.T. Schwartz [8] for geodesics of Finsler spaces with endpoints in two given submanifolds. Using the methods of D. Motreanu [5], T. Wang [12] and Cs. Varga - G. Farkas [11] it is possible to extend these results for locally convex cases.
In the first section, following [4] we describe the Riemann-Hilbert manifold \LambdaNM of absolutely continuous maps from the unit interval I=[0, 1] to M with endpoints in N subset M×M. The second section is devoted to the study of energy integral of a Finsler metric. We consider only such a Finsler metric which dominates the underlying Riemannian structure of the manifold. We show that the energy integral of class C-2, and the geodesics of the Finsler metric F joining orthogonally M1 and M2 are just the critical points of the energy integral. In the third section we prove that the energy functional of a Finsler metric satisfies the Palais-Smale condition on a complete manifold. This generalizes the analogous result of [2] for Finsler metrics. In the last section, applying the results of [[10] and [8] we deduce some multiplicity results for geodesics of Finsler spaces joining M1 and M2.
Bibliography
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[8] J.T. Schwartz, Generalizing the Lusternik-Schnirelmann theory of critical points, Comm. on Pure and Applied Math., 17(1964), 307-315.
[9] A. Szulkin, Ljusternik-Schnirelmann theory on C1 manifold, Analyse non linéare, Ann. Inst. Henri Poincaré, vol 5, no 2, 1988, 119-139.
[10] J.P. Serre, Homologie singulier des espaces fibrés, Ann. of Math., 54(1951), 425-505.
[11] Cs. Varga, G. Farkas, Lusternik-Schnirelmann theory on closed subsets of C1-manifolds, Studia Univ. Babe s-Bolyai, Mathematica, 38, 2(1993).
[12] T. Wang, Lusternik-Schnirelmann category theory on closed subsets of C2-Banach manifold, J. Math. Anal. Appl., 149(1990), 412-423.
Date received: June 15, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-25.