On curves and surfaces of prescribed mean curvature in Finsler spaces

• Über Kurven und Flächen vorgeschriebener mittlerer Krümmung in Finslerräumen

Feddern, Stephanie; von der Mosel, Heiko (Thesis advisor); Wagner, Alfred (Thesis advisor)

Aachen (2018)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

The thesis discusses curves and surfaces of prescribed curvature in Finsler spaces. We study curves that minimise the Finslerian length functional using a variational approach for parametric Lagrangians that are definite and elliptic due to Buttazzo, Giaquinta and Hildebrandt [BGH98, GH96]. Then we can generalise these results considering the length functional for weak Finsler metrics. Further, the thesis deals with the Plateau problem in Finsler 3-space and generalises the results established by Overath and von der Mosel in [Ove14,OvdM14] to surfaces of prescribed variable Finsler-mean curvature. Therefore, we adapt the anisotropic functional introduced by Clarenz and von der Mosel [CvdM04], whose critical immersions are surfaces of prescribed weighted mean curvature, and prescribe the Finsler-mean curvature as defined by Shen [She98]. The solution of Plateau's problem is obtained using the theory developed by Hildebrandt and von der Mosel [HvdM99, HvdM03b,HvdM03c] for Cartan functionals guaranteeing that the integrand is definite and semi-elliptic. Higher regularity of the minimisers is derived from the existence of dominance functions as defined in [HvdM03a]. We address other boundary value problems such as the partially free and the free boundary value problem as well as the Douglas problem using similar techniques. Finally, we are able to generalise most of the aforementioned results to two-dimensional surfaces of any codimension.