Variational formulas for immersions into 3-manifolds

Abstract

In this thesis we introduce the S-theory. We apply the S-theory to variational problems of immersed Riemann surfaces, Elasticity theory and Spin theory. From classical mechanics it is known that geometric problems can be substantially simplified by symmetry considerations. The Noether theorem describes in this case how to find conservation laws from the invariance property of the observed energy. At first, the motivation was to give the space of immersions M = {f : M → R}, for an oriented Riemannian manifold M of dimension less than or equal to three, an Euclidean invariant description. For geometric energies, i.e. energies which are invariant under Euclidean transformations, one would be able to derive conservation laws for the respective variational problems. It is a well known fact that immersed space curves are uniquely determined by their curvature functions (κ1, κ2, τ) up to Euclidean transformations. In the case of immersed surfaces and immersed 3-manifolds the situation becomes more complicated and leads to a non conformal deformation theory (S-Theorie). The S-Theory builds the foundations of this thesis. Starting from a Riemannian manifold M and a reference metric ⟨ , ⟩ one can modell any other Riemmanian metric g through a positive definite and self adjoint operator S via g = ⟨S, S⟩. The operator S is an isometry, i.e. g = S∗⟨ , ⟩. From this point of view we compute the LeviCivita connection with respect to g out of Levi-Civita connection of the reference metric and the operator S. Further we introduce Spin bundles over two and three dimensional oriented Riemannian manifolds. Thereby Spin bundles, in contrast to most of the literature, have additionally a quaternionic structure. Many formulas become clearer and more accessible. We show that any Spin bundle Σ over (M, ⟨ , ⟩) has a unique Spin connection and compute via the S-Theory the deformed Spin connection with respect to the metric g. Eventually we consider the induced Spin bundle of an immersion f : M → R and compute the corresponding Spin connection and find a new interpretation of the Gauss-Codazzi equation. We introduce the Dirac operator and compute its deformation with respect to the metric g. We are now able to give the space of immersed surfaces M = {f : M → R} the desired geometric description. We compute the normal space of M and formulate the Noether theorem. As an application we compute conservation laws for the Willmore functional. Finally we introduce the intrinsic version of the famous distance-squared energy on a n-dimensional Riemannian manifold. We introduce the corresponding stress tensor and show that it’s closeness is a characterization for critical points of the energy. Thereby the S-theory provides a fundamental concept for the understanding of the stress tensor. In the case of a Riemann surface, the S-theory provides a canonical harmonic rotation 1-form for critical points of the distance-squared energy.