Minimal immersions in Finsler spaces Von der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom{Mathematiker Patrick Overath aus Eupen, Belgien Berichter: Univ.-Prof. Dr. Heiko von der Mosel AOR Priv.-Doz. Dr. Alfred Wagner Tag der m¨undlichen Pr¨ufung:18. Juni 2014 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verf¨ugbar. Zusammenfassung Die vorliegende Arbeit verallgemeinert Resultate von [ST03], [SST04] und [CS09] bez¨uglich Finslerminimalfl¨achen in Finslermannigfaltigkeiten, welche mit dem Busemann-Hausdorff- Volumen versehen sind, wovon sich der Finslersche Fl¨acheninhalt ableiten l¨asst. Es wer- den eine Finslermetrik F als auf Rm+1 definiert angenommen und Bedingungen bez¨uglich der m-Symmetrisierung von F formuliert, die die Elliptizit¨atdes Finslerschen Fl¨achenin- halts garantieren. In [ST03] und [SST04] wurden spezielle Randersr¨aumebetrachtet und in [CS09] Finslermannigfaltigkeiten mit sogenannten (α; β)-Metriken, welche Randersmetriken als Spezialfall umfassen. Verallgemeinert werden in der vorliegenden Arbeit die S¨atzevom Bernsteintyp, welche in [SST04] und [CS09] bewiesen wurden, allerdings umfasst diese Arbeit auch etliche neue Ergebnisse. Darunter unter anderem Existenz- und Regularit¨atss¨atzef¨ur Minimierer, Hebbarkeit von Singularit¨aten, Einschlusss¨atze,isoperimetrische Ungleichungen und Kr¨ummungsabsch¨atzungenf¨urFinslerminimalfl¨achen. v Abstract The present thesis generalizes results established in [ST03], [SST04] and [CS09] on Finsler- minimal hypersurfaces in Finsler manifolds equipped with the Busemann-Hausdorff volume wherefrom the Finsler area derives. Therefore, a Finsler structure F on Rm+1 is assumed and some conditions involving the m-symmetrization F(m) of F are formulated to guarantee the el- lipticity of the Finsler area. The present thesis generalizes especially Bernstein-type theorems of [SST04] and [CS09] beyond the class of (α; β)-metrics. There are even many new results some of which with no direct counterpart in the aforementioned references. These results in- clude existence and regularity of minimizers, removability of singularities, enclosure theorems, isoperimetric inequalities and curvature estimates for Finsler-minimal hypersurfaces. vi Acknowledgments First and foremost, it is my pleasure to express my deep gratitude to my advisor Professor Heiko von der Mosel for his constant guidance and patient support. I would like to thank AOR PD Dr. Alfred Wagner for accepting to be the second referee. Special thanks go to Stephanie Feddern for proofreading parts of the present thesis. I want to thank my friends and colleagues at the Institut f¨urMathematik of RWTH Aachen University, especially Dr. Frank Roeser, Dr. Armin Schikorra, Dr. Niki Winter, Martin Meurer and Sebastian Scholtes for active listening and taking part in many helpful discussions as well as creating a comfortable lively atmosphere which I really enjoyed. Last but not least, I am grateful to my family and friends for their support and motivation. vii viii Contents Introduction 1 1 Preliminaries of linear Algebra, Analysis and Finsler manifolds7 1.1 Linear Algebra . .7 1.1.1 Matrices . .7 1.1.2 Finite dimensional vector spaces and related topics . 12 1.2 Real and Functional Analysis . 20 1.2.1 Real Analysis and Measure theory . 20 1.2.2 Fr´echet spaces and some examples . 24 1.3 Manifolds . 28 1.4 Finsler metrics and Finsler manifolds . 35 1.5 Immersions into Euclidean space . 41 1.6 Cartan functionals . 46 1.6.1 Review of variational results of Cartan functionals . 46 1.6.2 Notions of mean curvature . 49 1.6.3 Existence of minimizers and regularity . 56 1.6.4 Enclosure results . 62 1.6.5 Isoperimetric inequalities . 63 1.6.6 Curvature estimates and Bernstein results . 64 2 Finsler Area 67 2.1 Finsler Area as a Cartan functional . 67 2.2 The spherical Radon transform . 76 2.3 Finsler area in terms of the spherical Radon transform . 100 2.4 Finsler area and ellipticity of the symmetrization . 108 2.5 Finsler mean curvature and Cartan integrand mean curvatures . 137 3 Application of Cartan functional theory on Finsler area 145 3.1 Introduction . 145 3.2 Existence and regularity for Finsler surfaces . 146 3.3 Enclosure theorems . 150 3.4 Isoperimetric inequalities . 151 3.5 Curvature estimates and Bernstein-type theorems for Finsler-minimal immersions158 3.5.1 Curvature estimates for Finsler-minimal immersions . 158 3.5.2 Bernstein-type theorems for Finsler-minimal immersions . 164 Nomenclature 167 ix Contents Bibliography 173 Index 179 Lebenslauf 183 Erkl¨arung 185 x Introduction The present thesis covers an investigation of Finsler-minimal immersions into Finsler man- ifolds and their properties. Finsler-minimal immersions generalize minimal immersions into Euclidean space, i.e., Finsler-minimal immersions are critical immersions to the Finsler area functional. In other words, the first variation of such an immersion w.r.t. Finsler area van- ishes. Finsler area in this context means the Busemann-Hausdorff definition of area in a Finsler manifold (see [Bus47]). There were only very few results known regarding Finsler- minimal immersions. Especially, Souza, Spruck and Tenenblat [SST04] established results for Finsler-minimal graphs in a Randers-Minkowski space, i.e. a Minkowski space where the Finsler metric is a Randers metric. A Randers metric is a Riemannian metric plus an additive linear perturbation term. In [SST04], some results regarding (local) uniqueness, a Bernstein- type theorem in dimension 3, and removability of singularities are given for such graphs. To the author's knowledge, most of the formerly established results including [SST04] were re- stricted to graphs and target Finsler manifolds of a very specific structure and there were essentially no results regarding existence, isoperimetric inequalities or curvature estimates for Finsler-minimal immersions. In the beginning, we name some of the basic notions necessary to speak of Finsler-minimal immersions. The Busemann-Hausdorff volume, an extension of the notion of volume on Rie- mannian manifolds, is one of two main choices of volume on Finsler manifolds. Here, the most influential work was that of Busemann [Bus47] from 1947. Therein, he not only defined this Finsler volume, but also showed that it coincides on reversible Finsler manifolds with the Hausdorff measure induced by the Finsler structure. This property gives it a basic geometric interpretation and makes it a natural choice of volume.1 Most of the founding definitions and basic results on that type of Finsler volume are due to Busemann. Busemann especially derived from his volume a Finsler area for immersions, which leads to Finsler-minimal im- mersions in Finsler manifolds (see [She98]). Shen gave in [She98] a Finsler version of mean curvature, stemming from the Euler-Lagrange equation and the first variation of the Finsler area. Then, Finsler-minimal immersions are immersions of vanishing Finsler mean curva- ture. So, Finsler-minimal immersions can be characterized by a differential equation and they are critical points of Finsler area. Further investigations on the topic have been carried out by [ST03], [SST04], where these characterizing differential equations are computed for Finsler- minimal graphs in a Randers-Minkowski space setting, i.e. in a Minkowski space with Finsler metric F = α + β with a Riemannian metric α and a linear 1-form β. These differential equations are identifiedp as of mean curvature type (cf. [GT01]) and being elliptic as such up to a threshold 1= 3 on the Riemannian norm of the linear perturbation term β. An example in [SST04] shows even more that this threshold is sharp. It is a cone, which is a solution to the characterizing differential equations of Finsler-minimal graphs for β of norm equal to 1Notice that the alternative Holmes-Thompson volume (see [APB06´ ]) leads to a different notion of Finsler- minimal immersions that we do not address here. 1 Introduction p 1= 3. For elliptic differential equations of mean curvature type Jenkins and Serrin [JS63] as well as Simon [Sim77a] proved Bernstein-type theorems, which where then applied by Souza et al. to the Finsler-minimal graphs as the solution of such an equation. Thereby, [SST04] obtained, among other results, Bernstein-type theorems for critical graphs to Finsler area in Randers-Minkowski space. Cui and Shen [CS09] applied this method to more general (α; β)- metrics (see Definition 1.4.13). Again, they classify the characterizing differential equation of critical graphs of Finsler area as being elliptic and of mean curvature type. Notice that a Randers metric is a special type of (α; β)-metric. The purpose of this thesis is to investigate minimizers and critical immersions of Busemann's Finsler area, generalize results established in [SST04] and [CS09] as well as present some entirely new results not restricted to graphs. In contrast to those sources, we use a variational approach as the problem naturally arises by a variation of Finsler area. The drawback of the approach in [SST04] and [CS09] is that the characterizing differential equations are highly non-linear in nature. This is the reason why their coefficients can only be computed in an involved manner. Therefore, classifying the differential equations as elliptic becomes also quite complicated, which restricts their approach to very specific Finsler spaces. So, the main idea of the present