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Complex sprays and complex curves.

Colum Watt (2001)

Revista Matemática Complutense

After defining what is meant by a complex spray X on a complex manifold M, we introduce the notion of a spray complex curve associated to X. Several equivalent formulations are derived and we give necessary and sufficient conditions for M to admit spray complex curves for X through each point and in each direction. Refinements of this result are then derived for some special cases, for example when X is the horizontal radial vector field associated to a complex Finsler metric.

Conformal ℱ-harmonic maps for Finsler manifolds

Jintang Li (2014)

Colloquium Mathematicae

By introducing the ℱ-stress energy tensor of maps from an n-dimensional Finsler manifold to a Finsler manifold and assuming that (n-2)ℱ(t)'- 2tℱ(t)'' ≠ 0 for any t ∈ [0,∞), we prove that any conformal strongly ℱ-harmonic map must be homothetic. This assertion generalizes the results by He and Shen for harmonics map and by Ara for the Riemannian case.

Cut and singular loci up to codimension 3

Pablo Angulo Ardoy, Luis Guijarro (2011)

Annales de l’institut Fourier

We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension n - 2 is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension n - 3 .

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