Pedit, Franz
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Professor, Department of Mathematics and Statistics
Last Name
Pedit
First Name
Franz
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Geometry and Topology
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Differential geometry
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Publication Open Access Isothermic submanifolds of symmetric $R$-spaces(2010-01) Burstall, F; Donaldson, N; Pedit, F; Pinkall, UWe extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric $R$-spaces with essentially no loss of integrable structure.Publication Open Access Willmore tori in the 4-sphere with nontrivial normal bundle(2005-01) Leschke, K; Pedit, F; Pinkall, UPublication Metadata only DISCRETIZING CONSTANT CURVATURE SURFACES VIA LOOP GROUP FACTORIZATIONS - THE DISCRETE SINE-GORDON AND SINH-GORDON EQUATIONS(1995) Pedit, F; WU, HYThe sine- and sinh-Gordon equations are the harmonic map equations for maps of the (Lorentz) plane into the 2-sphere. Geometrically they correspond to the integrability equations for surfaces of constant Gauss and constant mean curvature. There is a well-known dressing action of a loop group on the space of harmonic maps. By discretizing the vacuum solutions we obtain via the dressing action completely integrable discretizations (in both variables) of the sine- and sinh-Gordon equations. For the sine-Gordon equation we get Hirota's discretization. Since we work in a geometric context we also obtain discrete models for harmonic maps into the 2-sphere and discrete models of constant Gauss and mean curvature surfaces.Publication Open Access Quaternionic holomorphic geometry: Plucker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori(2001-01) Ferus, D; Leschke, K; Pedit, F; Pinkall, UPublication Open Access Dressing orbits of harmonic maps(1995) Burstall, FE; Pedit, FWe study the harmonic map equations for maps of a Riemann surface into a Riemannian symmetric space of compact type from the point of view of soliton theory. There is a well-known dressing action of a loop group on the space of harmonic maps and we discuss the orbits of this action through particularly simple harmonic maps called {\em vacuum solutions}. We show that all harmonic maps of semisimple finite type (and so most harmonic $2$-tori) lie in such an orbit. Moreover, on each such orbit, we define an infinite-dimensional hierarchy of commuting flows and characterise the harmonic maps of finite type as precisely those for which the orbit under these flows is finite-dimensional.Publication Metadata only Weierstrass type representation of harmonic maps into symmetric spaces(1998) Dorfmeister, J; Pedit, F; Wu, HPublication Open Access Curved flats and isothermic surfaces(1997) Burstall, F; HertrichJeromin, U; Pedit, F; Pinkall, UPublication Open Access Conformal maps from a 2-torus to the 4-sphere(2007-01) Bohle, C; Leschke, K; Pedit, F; Pinkall, UWe study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus the kernel bundle can be extended to the compactification of the spectral curve and we obtain a linear 2-torus worth of algebraic curves in projective 3-space. The original conformal immersion of the 2-torus is recovered as the orbit under this family of the point at infinity on the spectral curve projected to the 4-sphere via the twistor fibration.Publication Open Access Discrete holomorphic geometry I. Darboux transformations and spectral curves(2009-01) Bohle, C; Pedit, F; Pinkall, UFinding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over “discrete surfaces”, by which we mean the vertex sets of triangulated surfaces with bi-colored set of faces. The resulting theory of discrete conformality is simultaneously Möbius invariant and based on linear equations. In the special case of maps into the 2-sphere we obtain a reinterpretation of the theory of complex holomorphic functions on discrete surfaces introduced by Dynnikov and Novikov. As an application of our theory we introduce a Darboux transformation for discrete surfaces in the conformal 4-sphere. This Darboux transformation can be interpreted as the space- and time-discrete Davey–Stewartson flow of Konopelchenko and Schief. For a generic map of a discrete torus with regular combinatorics, the space of all Darboux transforms has the structure of a compact Riemann surface, the spectral curve. This makes contact to the theory of algebraically completely integrable systems and is the starting point of a soliton theory for triangulated tori in 3- and 4-space devoid of special assumptions on the geometry of the surface.Publication Open Access Envelopes and osculates of Willmore surfaces(2007-01) Leschke, K; Pedit, FWe view conformal surfaces in the 4-sphere as quaternionic holomorphic curves in quaternionic projective space. By constructing enveloping and osculating curves, we obtain new holomorphic curves in quaternionic projective space and thus new conformal surfaces. Applying these constructions to Willmore surfaces, we show that the osculating and enveloping curves of Willmore spheres remain Willmore.