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Alexander I. Bobenko
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In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on Mobius invariant building blocks such as circles and spheres. Concrete problems considered in these lectures include the Willmore energy as well as conformal and curvature line parametrizations of surfaces. In particular we discuss geometric properties of a recently found discrete Willmore energy. The convergence to the smooth Willmore functional is shown for special refinements of triangulations originating from a curvature line parametrization of a surface. Further we treat special classes of discrete surfaces such as isothermic and minimal. The construction of these surfaces is based on the theory of circle patterns, in particular on their variational description.
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We find all analytic surfaces in space $mathbb{R}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of
We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is related to the differential geometry of planar sections of the surface parallel to and close to the tangent planes, and to the symmetry sets of isophote curves, that is level sets of intensity in a 2-dimensional image. We investigate also the relationship of the vertex curve with the parabolic and flecnodal curves, and the evolution of the vertex curve in a generic 1-parameter family of smooth surfaces.
$alpha$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $alpha$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $alpha >1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $alpha$-harmonic maps for $alpha >1$ and then letting $alpha to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $alpha$-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth. By $varepsilon$-regularity and suitable perturbations, we can then show that such a sequence of perturbed $alpha$-Dirac-harmonic maps converges to a smooth nontrivial $alpha$-Dirac-harmonic map.
We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $alpha$-(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial.
We consider circles on a translation surface $X$, consisting of points joined to a common center point by a geodesic of length $R$. We show that as $R to infty$ these circles distribute to a measure on $X$ which is equivalent to the area. In the last section we consider analogous results for closed geodesics.
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