Annals of Global Analysis and Geometry

, Volume 48, Issue 4, pp 397–422 | Cite as

On minimal immersions in Finsler space

  • Patrick Overath
  • Heiko von der MoselEmail author


We explore a connection between the Finslerian area functional and well-investigated Cartan functionals to prove new Bernstein theorems, uniqueness and removability results for Finsler-minimal graphs, as well as enclosure theorems and isoperimetric inequalities for minimal immersions in Finsler spaces. In addition, we establish the existence of smooth Finsler-minimal immersions spanning given extreme or graph-like boundary contours.


Finsler-minimal immersions Cartan functionals Spherical Radon transform 

Mathematics Subject Classification

44A12 49Q05 49Q10 53A35 53B40 53C60 



Substantial parts of this work are contained in the first author’s thesis which was partially supported by DFG grant no. Mo 966/3-1,2, and moreover by the Excellence Initiative of the German Federal and State Governments. Moreover, the second author would like to express his gratitude to Professor Seiki Nishikawa for inviting him to Tohoku University at Sendai, Japan, to give a series of talks on this line of research.


  1. 1.
    Álvarez Paiva, J.C., Berck, G.: What is wrong with the Hausdorff measure in Finsler spaces. Adv. Math. 204(2), 647–663 (2006). doi: 10.1016/j.aim.2005.06.007 zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bailey, T.N., Eastwood, M.G., Gover, A.R., Mason, L.J.: Complex analysis and the Funk transform. J. Korean Math. Soc. 40(4), 577–593 (2003). [Sixth International Conference on Several Complex Variables (Gyeongju, 2002)]Google Scholar
  3. 3.
    Bao, D., Chern, S.S., Shen, Z.: An introduction to Riemann–Finsler geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000)Google Scholar
  4. 4.
    Bergner, M., Fröhlich, S.: Existence, uniqueness and graph representation of weighted minimal hypersurfaces. Ann. Global Anal. Geom. 36(4), 363–373 (2009). doi: 10.1007/s10455-009-9164-x zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Busemann, H.: Intrinsic area. Ann. Math. 2(48), 234–267 (1947)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Busemann, H.: A theorem on convex bodies of the Brunn–Minkowski type. Proc. Nat. Acad. Sci. U. S. A. 35, 27–31 (1949)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chern, S.S., Shen, Z.: Riemann–Finsler geometry. In: Nankai Tracts in Mathematics, vol. 6. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005)Google Scholar
  8. 8.
    Clarenz, U.: Sätze über extremalen zu parametrischen funktionalen. Ph.D. thesis, Univ. of Bonn (1999)Google Scholar
  9. 9.
    Clarenz, U.: Enclosure theorems for extremals of elliptic parametric functionals. Calc. Var. Partial Differ. Equ. 15(3), 313–324 (2002). doi: 10.1007/s005260100128 zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Clarenz, U., von der Mosel, H.: Compactness theorems and an isoperimetric inequality for critical points of elliptic parametric functionals. Calc. Var. Partial Differ. Equ. 12(1), 85–107 (2001). doi: 10.1007/s005260000050 zbMATHCrossRefGoogle Scholar
  11. 11.
    Clarenz, U., von der Mosel, H.: Isoperimetric inequalities for parametric variational problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(5), 617–629 (2002). doi: 10.1016/S0294-1449(02)00096-3
  12. 12.
    Cui, N., Shen, Y.B.: Bernstein type theorems for minimal surfaces in \((\alpha,\beta )\)-space. Publ. Math. Debr. 74(3–4), 383–400 (2009)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Cui, N., Shen, Y.B.: Minimal rotational hypersurfaces in Minkowski \((\alpha,\beta )\)-space. Geom. Dedicata 151, 27–39 (2011). doi: 10.1007/s10711-010-9517-4 zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal surfaces. I. In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 295. Springer, Berlin (1992). (Boundary value problems)Google Scholar
  15. 15.
    Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
  16. 16.
    Funk, P.: Über eine geometrische Anwendung der Abelschen Integralgleichung. Math. Ann. 77(1), 129–135 (1915). doi: 10.1007/BF01456824 zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Gardner, R.J.: Geometric tomography. In: Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, Cambridge (1995)Google Scholar
  18. 18.
    Giaquinta, M., Hildebrandt, S.: Calculus of variations. II. In: Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 311. Springer, Berlin (1996). (The Hamiltonian formalism)Google Scholar
  19. 19.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001). (Reprint of the 1998 edition)Google Scholar
  20. 20.
    Groemer, H.: Geometric applications of Fourier series and spherical harmonics. In: Encyclopedia of Mathematics and its Applications, vol. 61. Cambridge University Press, Cambridge (1996). doi: 10.1017/CBO9780511530005
  21. 21.
    Helgason, S.: The Radon transform. In: Progress in Mathematics, vol. 5. Birkhäuser Boston, Mass (1980)Google Scholar
  22. 22.
    Helgason, S.: Groups and geometric analysis. In: Pure and Applied Mathematics, vol. 113. Academic Press Inc., Orlando (1984). (Integral geometry, invariant differential operators, and spherical functions)Google Scholar
  23. 23.
    Hildebrandt, S., von der Mosel, H.: Dominance functions for parametric Lagrangians. In: Geometric Analysis and Nonlinear Partial Differential Equations, pp. 297–326. Springer, Berlin (2003)Google Scholar
  24. 24.
    Hildebrandt, S., von der Mosel, H.: Conformal representation of surfaces, and Plateau’s problem for Cartan functionals. Riv. Mat. Univ. Parma (7) 4*, 1–43 (2005)Google Scholar
  25. 25.
    Hildebrandt, S., Sauvigny, F.: An energy estimate for the difference of solutions for the \(n\)-dimensional equation with prescribed mean curvature and removable singularities. Analysis (Munich) 29(2), 141–154 (2009). doi: 10.1524/anly.2009.1042 zbMATHMathSciNetGoogle Scholar
  26. 26.
    Jenkins, H.B.: On two-dimensional variational problems in parametric form. Arch. Ration. Mech. Anal. 8, 181–206 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Overath, P.: Minimal immersions in finsler spaces. Ph.D. thesis, RWTH Aachen University (2014)Google Scholar
  28. 28.
    Overath, P., von der Mosel, H.: Plateau’s problem in Finsler 3-space. Manuscripta Math. 143(3–4), 273–316 (2014). doi: 10.1007/s00229-013-0626-x zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. In: 75 years of Radon transform (Vienna, 1992), Conf. Proc. Lecture Notes Math. Phys., IV, pp. 324–339. Int. Press, Cambridge (1994)Google Scholar
  30. 30.
    Räwer, K.: Stabile extremalen parametrischer doppelintegrale in \({\mathbb{R}}^3\). Ph.D. thesis, Univ. of Bonn (1993)Google Scholar
  31. 31.
    Rudin, W.: Functional Analysis. McGraw-Hill Book Co., New York (1973). (McGraw-Hill Series in Higher Mathematics)Google Scholar
  32. 32.
    Shen, Z.: On Finsler geometry of submanifolds. Math. Ann. 311(3), 549–576 (1998). doi: 10.1007/s002080050200 zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Simon, L.: On a theorem of de Giorgi and Stampacchia. Math. Z. 155(2), 199–204 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Simon, L.: On some extensions of Bernstein’s theorem. Math. Z. 154(3), 265–273 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Souza, M., Spruck, J., Tenenblat, K.: A Bernstein type theorem on a Randers space. Math. Ann. 329(2), 291–305 (2004). doi: 10.1007/s00208-003-0500-3 zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Souza, M., Tenenblat, K.: Minimal surfaces of rotation in Finsler space with a Randers metric. Math. Ann. 325(4), 625–642 (2003). doi: 10.1007/s00208-002-0392-7 zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    White, B.: Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on \(3\)-manifolds. J. Differential Geom. 33(2), 413–443 (1991).
  38. 38.
    Winklmann, S.: Isoperimetric inequalites involving generalized mean curvature. Analysis (Munich) 22(4), 393–403 (2002)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Winklmann, S.: Existence and uniqueness of \(F\)-minimal surfaces. Ann. Global Anal. Geom. 24(3), 269–277 (2003). doi: 10.1023/A:1024765105599 zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Winklmann, S.: A Bernstein result for entire \(F\)-minimal graphs. Analysis (Munich) 27(4), 375–386 (2007). doi: 10.1524/anly.2007.27.4.375

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institut für MathematikRWTH Aachen UniversityAachenGermany

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