Geometry Research

Some of our current and recent projects

• The Geometry of generalized flat ribbons. (M. Raffaelli and S. Markvorsen)
• Design of ruled surfaces and elastica-foliated surfaces for architecture. (A. Fisker, D. Brander and J. Gravesen). Part of an Innovation Fund supported project "Digital Factory". March 2015 - December 2018.
• Optimization on manifolds - with applications to shape optimization. (J. Møller-Andersen, J. Gravesen, A. Evgrafov and P. Nørtoft). 2013-2016.
• Finsler geometry and modelling of wildfires. (Steen Markvorsen).
• Geometry rationalization for hot-wire and hot-blade production of architectural formwork. (J. Gravesen, D. Brander, S. Markvorsen, T. Nørbjerg, K. Hornbak Steenstrup, and external partners). Part of the Innovation Fund project "BladeRunner". March 2013- May 2016.
• Integrable surfaces. (D. Brander). Surfaces that have a loop group representation, usually related to harmonic maps into symmetric spaces. These include minimal surfaces, constant mean curvature surfaces, constant Gaussian curvature surfaces and Willmore surfaces.
• Symmetry techniques in differential geometry. (A. Swann, D. Brander and M. Svensson). FNU project 2010-2013.
• Iso-geometric analysis and shape optimization in electromagnetism. (D.M. Nguyen and J. Gravesen). 2009-2012.
• Minimal metric skeletons. (S. Markvorsen).
• Iso-geometric analysis and shape optimization in fluid mechanics. (P. Nørtoft and J. Gravesen). 2009-2012.
• Geometric analysis : classification theory of Riemannian submanifolds. (S. Markvorsen and V. Palmer.)

Selected recent publications

• D. Brander and J. Gravesen: Monge surfaces and planar geodesic foliations. J. Geom. (2018) 109: 4. https://doi.org/10.1007/s00022-018-0413-7.
• D. Brander and Peng Wang: On the Björling problem for Willmore surfaces. J. Differential Geom. 108 (2018) 411-457.
• D. Brander, A. Bærentzen, A. Fisker, and J. Gravesen: Designing interactively with elastic splines. Computer Aided Geometric Design, 2018, doi.org/10.1016/j.cagd.2018.03.007.
• D. Brander, A. Bærentzen, A. Fisker, and J. Gravesen: Bézier curves that are close to elastica. Under submission. Related software.
• D. Brander, A. Bærentzen, K. Clausen, A. Fisker, J. Gravesen, M. Lund, T. Nørbjerg, K. Steenstrup and A. Søndergaard: Designing for hot-blade cutting: geometric approaches for high-speed manufacturing of doubly-curved architectural surfaces. Part of: Advances in Architectural Geometry 2016, pages: 306-327, 2016.
• J. Bohr and S. Markvorsen: Autorotation. Physica Scripta, vol: 91,issue: 2, 2016.
• D. Brander: Pseudospherical surfaces with singularities. Ann. Mat. Pura Appl., 2016, DOI: 10.1007/s10231-016-0601-8
• M. Raffaelli, J. Bohr and S. Markvorsen: Sculpturing Surfaces with Cartan Ribbons. Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture, pages: 457-460, 2016.
• D. Brander: Spherical Surfaces. Experimental Mathematics, 25:3, 2016, pp: 257-272.
  
• A. Søndergaard, J. Feringa, T. Nøbjerg, K. Steenstrup, D. Brander, J. Gravesen, S. Markvorsen, J.A. Bærentzen, K. Petkov, J. Hattel, K. Clausen, K. Jensen, L. Knudsen, J. Kortbek: Robotic hot-blade cutting. Robotic Fabrication in Architecture, Art and Design 2016, proceedings of RobArch 2016.
• D. Brander, J. Gravesen and T. Nørbjerg: Approximation by planar elastic curves. Adv. Comput. Math., 2016, DOI: 10.1007/s10444-016-9474-z
• A. Hurtado, S. Markvorsen and V. Palmer: Estimates of the first Dirichlet eigenvalue from exit time moment spectra. Math. Ann., (2016) 365:1603-1632 DOI 10.1007/s00208-015-1316-7.
• V. Gimeno and S. Markvorsen: Ends, fundamental tones and capacity of minimal submanifolds via extrinsic comparison theory. Potential Analysis, vol: 42, 749--774, 2015.
• J. Gravesen: The Metric of Colour Space. Graphical Models, (82), issue: November, pages 77-86, 2015.
• S. Markvorsen: A Finsler geodesic spray paradigm for wildfire spread modelling. Nonlinear Analysis: Real World Applications, vol: 28, 208--228, 2015.
• D. Brander and S. Markvorsen: Surfaces with Natural Ridges. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture.
• P. Nørtoft, J. Gravesen and M. Willatzen: Isogeometric analysis of sound propagation through laminar flow in 2-dimensional ducts. Computer Methods in Applied Mechanics and Engineering, vol: 284, 1098--1119, 2015.
• A. Hurtado, S. Markvorsen and V. Palmer: Estimates of the first Dirichlet eigenvalue from exit time moment spectra. Mathematische Annalen, vol: 365, issue: 3, pp 1603-1632, 2015.
• D. Brander, J.A. Bærentzen, A. Evgrafov, J. Gravesen, S. Markvorsen, T. Nørbjerg, P. Nørtoft and K. Steenstrup: Hot blade cuttings for the building industries. In: Math for the Digital Factory (2014), 2016, Springer.
• D. Brander and Josef F. Dorfmeister: Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XIV, 645-675, 2015.
• D.M. Nguyen, A. Evgrafov, J. Gravesen and D. Lahaye: Iso-geometric shape optimization of magnetic density separators. Compel, vol: 33, issue: 4, 1416-1433, 2014.
• J. Bohr and S. Markvorsen: Ribbon Crystals. PLoS One, vol: 8, issue: 10, 2013.
• D. Brander and M. Svensson: Timelike constant mean curvature surfaces with singularities. The Journal of Geometric Analysis vol: 24, issue 3 1641-1672, 2014.
• D. Brander and M. Svensson: The geometric Cauchy problem for surfaces with Lorentzian harmonic Gauss maps. Journal of Differential Geometry 93, 37-66, 2013.
• P. Nørtoft and J. Gravesen: Isogeometric shape optimization in fluid mechanics. Structural and Multidisciplinary Optimization, vol: 48, 909-925, 2013.
• A. Hurtado, S. Markvorsen and V. Palmer: Comparison of exit time moment spectra for extrinsic metric balls. Potential Analysis, vol: 36, issue: 1, pages: 137-153, 2012.
• M. Willatzen and J. Gravesen: Strong curvature effects in Neumann wave problems. Journal of Mathematical Physics, vol: 53, issue: 8, pages 083507, 2012.
• D.M. Nguyen, A. Evgrafov, A. Gersborg and J. Gravesen: Isogeometric shape optimization of vibrating membranes. Computer Methods in Applied Mechanics and Engineering, vol: 200,issue:13-16, pages 1343-1353, 2011.
• P. Nielsen, A. Gersborg, J. Gravesen and N. Pedersen: Discretizations in isogeometric analysis of Navier-Stokes flow. Computer Methods in Applied Mechanics and Engineering, vol: 200,issue:45-46, pages 3242-3253, 2011.
• D. Brander: Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space. Math. Proc. Cambridge Philos. Soc. 150, 527-556, 2011.
• J. Gravesen, A. Evgrafov and D.M. Nguyen: On the sensitivities of multiple eigenvalues. Structural and Multidisciplinary Optimization, vol: 44,issue:4, pages 583-587, 2011.
• D. Brander and Josef F. Dorfmeister: The Björling problem for non-minimal constant mean curvature surfaces. Communications in Analysis and Geometry, 18, 171-194, 2010.
• S. Markvorsen and V. Palmer: Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below. Journal of Geometric Analysis, vol. 20, issue 2, pp 388-421, 2010.
• D. Brander, W. Rossman and N. Schmitt: Holomorphic representation of constant mean curvature surfaces in Minkowski space: consequences of non-compactness in loop group methods. Advances in Mathematics, 223, 949-986, 2010.
• A. Hurtado, S. Markvorsen and V. Palmer: Torsional rigidity of submanifolds with controlled geometry. Mathematische Annalen, vol. 344, issue 3, pp 511-542, 2009.
• I. Holopainen, S. Markvorsen and V. Palmer: p-Capacity and p-Hyperbolicity of Submanifolds. Rev. Mat. Iberoamericana, vol. 25, issue 2, pp 709-738, 2009.
• D. Brander: Loop group decompositions in almost split real forms and applications to soliton theory and geometry. Journal of Geometry and Physics, vol. 58, issue 12 pp 1792-1800, 2008.
• J. Gravesen and B. Jüttler and Z. Sir,: On rationally supported surface. Computer Aided Geometric Design, vol. 25, pp 320-331, 2008.
• D. Brander: Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane. Electron. Res. Announc. Math. Sci., vol. 15, pp 8-16, 2008.
• Z. Sir, J. Gravesen and B. Jüttler: Curves and surfaces represented by polynomial support functionss. Theoretical Computer Science, vol. 392, pp 141-157, 2008.
• S. Markvorsen and V. Palmer: Torsional rigidity of minimal submanifolds. Proc. London Math. Soc. (3), vol. 93, pp 253-272, 2006.
• J. Gravesen and M. Willatzen: Quantum eigenstates of curved nanowire structures. Physica B: Condensed Matter, vol. 371, pp 112-119, 2006.
• J. Gravesen, M. Willatzen and L.C. Lew Yan Voon: Schrödinger problems for surfaces of revolution -- the finite cylinder as a test example. J. Math. Phys., vol. 46, pp 012107, 2005.
• S. Markvorsen and V. Palmer: How to obtain transience from bounded radial mean curvature. Trans. Amer. Math. Soc., vol. 357, issue 3, pp 3459-3479, 2005.
• J. Gravesen, S. Markvorsen, R. Sinclair and M. Tanaka: The cut locus of a torus of revolution. Asian J. Math., vol. 9, pp 103-120, 2005.
• J. Gravesen, M. Willatzen: Eigenstates of Möbius nanostructures including curvature effects. Phys. Rev. A, vol. 72, issue 3, pp 032108, 2005.
• J. Gravesen: Third order invariants on surfaces. Computational Methods for Algebraic Spline Surfaces (COMPASS)., pages: 193-211, Springer, 2004.
• S. Markvorsen and V. Palmer: Transience and capacity of minimal submanifolds. Geom. Funct. Anal., vol. 13, issue 4, pp 915-933, 2003.
• J. Gravesen and M. Ungstrup: Constructing invariant fairness measures for surfaces. Advances in Computational Mathematics., vol. 17, issue 67, pp 67-88, 2002.
• S. Markvorsen and V. Palmer: Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds. Journal für die reine und angewandte Mathematik., vol. 2002, issue 551, pp 101-121, 2002.
• J. Gravesen and C. Henriksen: The Geometry of the Scroll Compressor. SIAM Review., vol. 43, issue 1, pp 113-126, 2001.