Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 26, No. 2, pp. 209-219 (2010)

Geodesics on non-complete Finsler manifolds

Rossella Bartolo

Politecnico di Bari

Abstract: In this note we deal with domains $D$ (i.e. connected open subsets) of a Finsler manifold $(M,F)$. At first we carry out a comparison between different notions of convexity for their boundaries. Then a careful application of variational methods to the geodesic problem yields that the convexity of $\partial D$ is equivalent to the existence of a minimal geodesic for each pair of points of $D$. Furthermore multiplicity of connecting geodesics can be obtained if $D$ is not contractible.

Keywords: Finsler manifold, minimizing geodesic, convex boundary, penalization technique

Classification (MSC2000): 53C60; 58B20, 53C22

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