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Fachbereich Mathematik

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Benutzerspezifische Werkzeuge

Prof. Dr. Gerhard Huisken

List of Publications

Forschungsprojekte

Geometrische Evolutionsgleichungen

Die Deformation geometrischer Strukturen durch Systeme nichtlinearer partieller Differentialgleichungen hat in den letzten Jahrzehnten in der Differentialgeometire und in der Mathematischen Physik zunehmend an Bedeutung gewonnen. Fortschritte im analytischen Verständnis nichtlinearer parabolischer Gleichungen und Systeme hat es ermöglicht, die Lösungen solcher geometrischen Evolutionsgleichungen auch auf langen Zeitintervallen zu kontollieren und mögliche Singularitäten zu verstehen. Mit Hilfe dieser Lösungen konnten neue geometrische Ungleichungen und Uniformisierungssätze bewiesen werden.

Übersichtsartikel:

  • G. Huisken, Heat diffusion in Geometry, in "Geometric Analysis", IAS/Park City Mathematical Series, H.-L. Bray, G. Galloway, R. Mazzeo, N. Sesum edts., (American Math. Soc. and Inst. of Adv. Studies), Vol. 22: pp1-13, (2016).
  • G. Huisken, Evolution Equations in Geometry. In B. Engquist, & W. Schmid (Eds.), Mathematics Unlimited - 2001 and Beyond (pp. 593-604). Berlin: Springer (2002).
  • G. Huisken, Evolution of hypersurfaces by their curvature in Riemannian manifolds, Doc. Math.J.DMV, Extra Volume ICM II (1998), 361-370.

Beipiele geometrischer Evolutionsgleichungen sind:

Fluss entlang der mittleren Krümmung

Die Deformation von Kurven, Hyperflächen und allgemein Untermannigfaltigkeiten einer umgebenden Riemannschen Mannigfaltigkeit in Richtung ihrer mittleren Krümmung führt zu einem quasilinearen System parabolischer Differentialgleichungen, das die typischen Glättungseigenschaften parabolischer Gleichungen aufweist, aber auch Lösungen mit Singularitäten aufweist.

  • S. Brendle and G. Huisken, Mean curvature flow with surgery of mean convex surfaces in three-manifolds, 21pp., (2015), submitted.
  • S. Brendle and G. Huisken, A fully nonlinear flow for 2-convex hypersurfaces, 41pp., (2015), arXiv:1507.04651.
  • S. Brendle and G. Huisken, Mean curvature flow with surgeries of mean convex surfaces in R3, Invent. Math. 203: 615-654, 2016.
  • G. Huisken and C. Sinestrari, Convex ancient solutions of the mean curvature flow, Journal Differential Geometry, 101 (2015), no. 2, 267–287.
  • J. Arnlind and G. Huisken, Pseudo-Riemannian geometry in terms of multi-linear brackets, Letters in Mathematical Physics (2014-10-17), Bd. 104, H. 12, S. (1507-1521).
  • J. Arnlind, J. Hoppe und G. Huisken, Multilinear formulation of differential geometry and matrix regularizations, Journal Differential Geometry , 91: 1-39, 2012.
  • G. Huisken and C. Sinestrari. Mean curvature flow with surgeries of two–convex hypersurfaces. Inventiones mathematicae, 175:137–221, 2009.
  • G. Huisken and C. Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Mathematica, 183(1):45–70, 1999.
  • G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. 8 (1999), 1-14.
  • G. Huisken, A Distance comparison principle for evolving curves, Asian J. of Math.2 (1998), 127-133.
  • G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Proc.Symp.in Pure Math. 54, part I (1993), 175-191.
  • K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547-569.
  • G. Huisken. Asymptotic behaviour for singularities of the mean curvature flow. Journal of Differential Geometry, 31:285–299, 1990.
  • K. Ecker and G. Huisken. Mean curvature evolution of entire graphs. Annals of Mathematics, 130:453–471, 1989.
  • G. Huisken, Non - parametric mean curvature evolution with boundary conditions, J.Differential Equations, 77(2) (1989) 369-378.
  • G. Huisken, The volume preserving mean curvature flow, Journal für die Reine und Angewandte Mathematik 382, (1987) 35-48.
  • G. Huisken, Deforming hypersurfaces of the sphere by their mean curvature, Mathematische Zeitschrift 195, (1987) 205 - 219.
  • G. Huisken, Mean curvature contraction of convex hypersurfaces, Proc.Symp.in Pure Math. 44, (1986) 275 - 280.
  • G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Inventiones Mathematicae 84, (1986) 463 - 480.
  • G. Huisken. Flow by mean curvature of convex surfaces into spheres. Journal of Differential Geometry, 20:237–266, 1984.

Fluss entlang der inversen mittleren Krümmung

Die Deformation von Flächen entlang der inversen mittleren Krümmung zeichnet sich dadurch aus, dass das Flächenelement an jeder Stelle exponentiell wächst. Schwache Lösungen mit nichtnegativer Krümmung erfüllen ein Variationsprinzip und erlauben Sprünge der Lösungsflächen. Der Fluss hat Anwendungen auf geometrische Ungleichungen und Energieungleichungen in der Allgemeinen Relativitätstheorie, siehe “Mathematische Relativitätstheorie”.

  • G. Huisken and T. Ilmanen. Higher regularity of the inverse mean curvature flow. J. Differential Geometry, 80:433–451, 2008.

Fluss entlang der Ricci-Krümmung

Die Deformation Riemannscher Metriken entlang der Ricci-Krümmung ist ein quasilineares parabolisches System, dass zur Uniformisierung Riemannscher Metriken genutzt werden kann und zum Beweis der Poincare-Vermutung durch Hamilton und Perelman geführt hat. In der Physik tritt der Fluss auf als 1-loop Renormalisierungsgruppenfluss in Sigma-Modellen der Stringtheorie.

  • S. Brendle, G. Huisken, and C. Sinestrari. Ancient solutions to the Ricci flow with pinched curvature. Duke Mathematical Journal, 158: 537–551, 2011.
  • G. Huisken, Geometric Flows and 3-Manifolds. Oberwolfach Preprints (Owp), 01, (2007) 1-9. http://www.mfo.de/scientific-programme/publications/owp/2007/OWP2007_01.pdf.
  • G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J.Differential Geometry 21, (1985) 47 - 62.

Mathematische Relativitätstheorie

In der Mathematischen Relativitätstheorie werden analytische und differentialgeometrische Methoden verwendet, um Lösungen der Einsteinschen Feldgleichung in der Allgemeinen Relativitätstheorie zu verstehen. Elliptische und parabolische Methoden erweisen sich dabei als wichtig bei der Untersuchung stationärer Lösungen, bei der Konstruktion guter Eichungen und natürlicher Untermannigfaltigkeiten in Zeit und Radialrichtung sowie beim Beweis von Energieungleichungen.

Dr. Carla Cederbaum

Christopher Nerz

 

  • G. Huisken and T. Ilmanen. Inverse mean curvature flow and the Riemannian Penrose inequality. Journal of Differential Geometry, 59:353–437, 2001.
  • G. Huisken and T. Ilmanen, The Riemannian Penrose Inequality, Int.Math.Research Notices, 20 (1997), 1045 -1058.
  • G. Huisken and S.-T. Yau. Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Inventiones mathematicae, 124:281–311, 1996.
  • K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Commun.Math.Phys. 135 (1991), 595-613.

Übersichtsartikel

  • G. Huisken, Mathematisierung der Gravitation: Die Schwarzschildlösung der Einsteingleichungen als Grundmodell vieler Phänomene der Gravitation, Schriftenreihe der Berlin-Brandenburger Akad., Debatte, Heft 4, Mathematisierung der Natur, (2006), 29-35.
  • G. Huisken and T. Ilmanen, Energy inequalities for isolated systems and hypersurfaces moving by their curvature, Proc. of the 16th Int. Conference on General Relativity and Gravitation in Durban, World Scientific, New Jersey (2002), 162–173.
  • G. Huisken, Geometric Concepts for the Mass in General Relativity , Trends in Mathematical Physics, Proc.Conf.Univ.Tennessee, Knoxville 1998, V.Alexiados and G.Siopsis (eds.), Intl.Press (1999), 299-306.

Multilineare Strukturen in der Differentialgeometrie

Viele Strukturen der Differentialgeometrie lassen sich in rein algebraischer Form fassen, insbesondere besteht eine enge Beziehung zwischen der Geometrie von Flächen und Poissonalgebren.

  • J. Arnlind, J. Hoppe und G. Huisken, Multilinear formulation of differential geometry and matrix regularizations, accepted J. Diff. Geometry, 91: 1-39, 2012.
  • J. Arnlind, G. Huisken, On the geometry of Kähler-Poisson structures, arXiv:1103.5862, (2011).
  • J. Arnlind, J. Hoppe, G. Huisken, Discrete curvature and the Gauss-Bonnet theorem, arXiv:1001.2223, (2010).
  • J. Arnlind, J. Hoppe, G. Huisken, On the classical geometry of embedded surfaces in terms of Poisson brackets, arXiv:1001.1604, (2010).
  • J. Arnlind, J. Hoppe, G. Huisken, On the classical geometry of embedded manifolds in terms of Nambu brackets, arXiv:1003.5981, (2010).

Publications:

  1. S. Brendle and G. Huisken, Mean curvature flow with surgery of mean convex surfaces in three-manifolds, 21pp., (2015), submitted.
  2. S. Brendle and G. Huisken, A fully nonlinear flow for 2-convex hypersurfaces, 41pp., (2015), arXiv:1507.04651.
  3. S. Brendle and G. Huisken, Mean curvature flow with surgeries of mean convex surfaces in R3, Invent. Math. 203: 615-654, 2016.
  4. G. Huisken and C. Sinestrari, Convex ancient solutions of the mean curvature flow, Journal Differential Geometry, 101 (2015), no. 2, 267–287.
  5. J. Arnlind and G. Huisken, Pseudo-Riemannian geometry in terms of multi-linear brackets, Letters in Mathematical Physics (2014-10-17), Bd. 104, H. 12, S. (1507-1521).
  6. J. Arnlind, J. Hoppe und G. Huisken, Multilinear formulation of differential geometry and matrix regularizations, accepted J. Diff. Geometry, 91: 1-39, 2012.
  7. S. Brendle, G. Huisken, and C. Sinestrari. Ancient solutions to the Ricci flow with pinched curvature. Duke Mathematical Journal, 158: 537–551, 2011.
  8. G. Huisken and C. Sinestrari. Mean curvature flow with surgeries of two–convex hypersurfaces. Inventiones mathematicae, 175:137–221, 2009.
  9. G. Huisken and T. Ilmanen. Higher regularity of the inverse mean curvature flow. J. Differential Geometry, 80:433–451, 2008.
  10. G. Huisken, Geometric Flows and 3-Manifolds. Oberwolfach Preprints (Owp), 01, (2007) 1-9. http://www.mfo.de/publications/owp/2007/OWP2007_01.pdf.
  11. G. Huisken and T. Ilmanen. Inverse mean curvature flow and the Riemannian Penrose inequality. Journal of Differential Geometry, 59:353–437, 2001.
  12. G. Huisken and W. Klingenberg. Flow of real hypersurfaces by the trace of the Levi form. Math. Research Letters, (6):645–661, 1999.
  13. G. Huisken and A. Polden, Geometric evolution equations for hypersurfaces, Calculus of Variations and Geometric Evolution Problems, CIME Lectures at Cetraro of 1996, S.Hildebrandt and M.Struwe (eds.), Springer (1999).
  14. G. Huisken and C. Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Mathematica, 183(1):45–70, 1999.
  15. G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc.Var. 8 (1999), 1-14.
  16. G. Huisken, Evolution of hypersurfaces by their curvature in Riemannian manifolds, Doc. Math.J.DMV, Extra Volume ICM II (1998), 361-370.
  17. G. Huisken, A Distance comparison principle for evolving curves, Asian J. of Math.2 (1998), 127-133.
  18. G. Huisken and T. Ilmanen, The Riemannian Penrose Inequality, Int.Math.Research Notices, 20 (1997), 1045 -1058.
  19. U. Dierkes and G. Huisken, The n-dimensional analogue of the catenary: prescribed area, Geometric Analysis and the Calculus of Variations, edited by J. Jost, (1997), Intl. Press,Boston.
  20. G. Huisken and S.-T. Yau. Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Inventiones mathematicae, 124:281–311, 1996.
  21. G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature, Proc.Symp.in Pure Math. 54, part I (1993), 175-191.
  22. K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), 547-569.
  23. K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Commun.Math.Phys. 135 (1991), 595-613.
  24. K. Ecker and G. Huisken, A Bernstein result for minimal graphs of controlled growth, J.Differential Geometry 31, (1990) 397-400.
  25. G. Huisken. Asymptotic behaviour for singularities of the mean curvature flow. Journal of Differential Geometry, 31:285–299, 1990.
  26. U. Dierkes and G. Huisken, The n - dimensional analogue of the catenary: Existence and non - existence, Pacific J. Math. 141 (1990), 47-54.
  27. K. Ecker and G. Huisken. Mean curvature evolution of entire graphs. Annals of Mathematics, 130:453–471, 1989.
  28. K. Ecker and G. Huisken, Interior curvature estimates for hypersurfaces of prescribed mean curvature, Anal. nonlineaire, Ann. Inst. H. Poincare 6 (1989), 251-260.
  29. K. Ecker and G. Huisken, Immersed hypersurfaces with constant Weingarten curvature, Mathematische Annalen, 283 (2) (1989), 329-332.
  30. G. Huisken, Non - parametric mean curvature evolution with boundary conditions, J.Differential Equations, 77(2) (1989) 369-378.
  31. G. Huisken, The volume preserving mean curvature flow, Journal für die Reine und Angewandte Mathematik 382, (1987) 35-48.
  32. G. Huisken, Deforming hypersurfaces of the sphere by their mean curvature, Mathematische Zeitschrift 195, (1987) 205 - 219.
  33. G. Huisken, Mean curvature contraction of convex hypersurfaces, Proc.Symp.in Pure Math. 44, (1986) 275 - 280.
  34. G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Inventiones Mathematicae 84, (1986) 463 - 480.
  35. G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J.Differential Geometry 21, (1985) 47 - 62.
  36. G. Huisken, Capillary surfaces over obstacles, Pacific J.Math. 117, (1985) 121 - 141.
  37. G. Huisken. Flow by mean curvature of convex surfaces into spheres. Journal of Differential Geometry, 20:237–266, 1984.
  38. G. Huisken, Capillary surfaces in negative gravitational fields, Mathematische Zeitschrift 185, (1984) 449 - 464.
  39. G. Huisken, On pendent drops in a capillary tube, Bull. Austral.Math.Soc. 28, (1983) 343 - 354.

Übersichtsartikel und Konferenzbeiträge

  1. G. Huisken, Heat diffusion in Geometry, in "Geometric Analysis", IAS/Park City Mathematical Series, H.-L. Bray, G. Galloway, R. Mazzeo, N. Sesum edts., (American Math. Soc. and Inst. of Adv. Studies), Vol. 22: pp1-13, (2016).
  2. G. Huisken, Mathematisierung der Gravitation: Die Schwarzschildlösung der Einsteingleichungen als Grundmodell vieler Phänomene der Gravitation, Schriftenreihe der Berlin-Brandenburger Akademie, Debatte, Heft 4, Mathematisierung der Natur, (2006), 29-35.
  3. G. Huisken, Finitely many surgeries for mean curvature flow and Ricci flow, Oberwolfach Report, Conf. Nonlinear evolution problems, (2005).
  4. G. Huisken and T. Ilmanen, Energy inequalities for isolated systems and hypersurfaces moving by their curvature, Proc. of the 16th Int. Conference on General Relativity and Gravitation in Durban, World Scientific, New Jersey (2002), 162–173.
  5. G. Huisken, Evolution Equations in Geometry. In B. Engquist, & W. Schmid (Eds.), Mathematics Unlimited - 2001 and Beyond (pp. 593-604). Berlin: Springer (2002).
  6. G. Huisken and C. Sinestrari, Formazione di singolarità nel moto per curvatura media. Bollettino della Unione Matematica Italiana, 4, (2001), 107-119.
  7. G. Huisken, Geometric Concepts for the Mass in General Relativity , Trends in Mathematical Physics, Proc.Conf.Univ.Tennessee, Knoxville 1998, V.Alexiados and G.Siopsis (eds.), Intl.Press (1999), 299-306.
  8. G. Huisken, Lectures on geometric evolution equations, Tsing-Hua Lectures on Geometry and Analysis (1992), edited by S.T. Yau, International Press, Boston.
  9. G.Dziuk, G.Huisken and J.Hutchinson (eds.), Theoretical and Numerical Aspects of Geometric Variational Problems, Proc. Centre Math. Anal., Austr. Nat. Univ. 26 (1991).
  10. G. Huisken, Singularity formation in geometric evolution equations, Proc. Centre for Math. Anal., Austral. Nat. Univ. 26 (1991) 128-139.
  11. G. Huisken, \(C^{1,1}\)-Regularity of solutions to variational inequalities, Proc. Centre for Math. Anal., Austral. Nat. Univ. 5 (1984) 85-90.

Manuskripte

  1. J. Arnlind, J. Hoppe, G. Huisken, Discrete curvature and the Gauss-Bonnet theorem, arXiv:1001.2223, (2010).
  2. J. Arnlind, J. Hoppe, G. Huisken, On the classical geometry of embedded surfaces in terms of Poisson brackets, arXiv:1001.1604, (2010).
  3. J. Arnlind, J. Hoppe, G. Huisken, On the classical geometry of embedded manifolds in terms of Nambu brackets, arXiv:1003.5981, (2010).
  4. J. Arnlind, G. Huisken, On the geometry of Kähler-Poisson structures, arXiv:1103.5862, (2011).