Finsler Geometry is used to introduce anisotropic modifications for classical relativity and it also serves as an auxiliary tool for certain standard relativistic spacetimes. The specific issues of regularity of this geometry leads to a variety of problems. Among them, we will discuss: the exclusion of Finsler spacetimes by classic Ehlers-Pirani-Schild axioms [1], the existence of different definitions of Lorentz-Finsler metrics as well as Finsler static or stationary spacetimes [2], the uniqueness of Palatini connections when they are defined on the whole causal cone [3] or the appearance of topological complications in the computation of the causal boundary of spacetimes [4]. We will also discuss a specific Riemannian and Finslerian problem related to the conformal boundary (namely, the convexity of the boundary of a domain implies the convexity of the domain) and show how the improvement of regularity from $C^4$ to $C^{1,1}$ ([5], [6]) permits applications for classic stationary spacetimes [7] as well as for their Finslerian counterparts.
References.
[1] A.N Bernal, M.A. Javaloyes, M. Sánchez. Universe (2020).
[2] M.A. Javaloyes, M. Sánchez. RACSAM (2020).
[3] M.A. Javaloyes, M. Sánchez, F.F. Villaseñor. ATMP (2023), arxiv: 2108.03197.
[4] J.L. Flores, J. Herrera, M. Sánchez. Memoirs AMS (2013).
[5] R.L. Bishop. Indiana M.J. (1974).
[6] R. Bartolo, E. Caponio, A.V. Germinario, M. Sánchez. Calc. Var. PDE (2011).
[7] E. Caponio, A.V. Germinario, M. Sánchez. J. Geom. Anal. 26 (2016).