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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Construction of complete embedded self-similar surfaces under mean curvature flow. Part I.
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by Xuan Hien Nguyen PDF
Trans. Amer. Math. Soc. 361 (2009), 1683-1701 Request permission

Abstract:

We carry out the first main step towards the construction of new examples of complete embedded self-similar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of self-similar surfaces and desingularizing the intersection circle using an appropriately modified singly periodic Scherk surface, called the core. Using an inverse function theorem, we show that for small boundary conditions on the core, there is an embedded surface close to the core that is a solution of the equation for self-similar surfaces. This provides us with an adequate central piece to substitute for the intersection.
References
  • Sigurd B. Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21–38. MR 1167827
  • S. Angenent, T. Ilmanen, and D. L. Chopp, A computed example of nonuniqueness of mean curvature flow in $\mathbf R^3$, Comm. Partial Differential Equations 20 (1995), no. 11-12, 1937–1958. MR 1361726, DOI 10.1080/03605309508821158
  • David L. Chopp, Computation of self-similar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 1–15. MR 1302814
  • Ulrich Dierkes, Stefan Hildebrandt, Albrecht Küster, and Ortwin Wohlrab, Minimal surfaces. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 295, Springer-Verlag, Berlin, 1992. Boundary value problems. MR 1215267
  • Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. MR 1030675
  • Nicolaos Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. MR 1043269, DOI 10.2307/1971494
  • Nikolaos Kapouleas, Complete embedded minimal surfaces of finite total curvature, J. Differential Geom. 47 (1997), no. 1, 95–169. MR 1601434
  • Sebastián Montiel and Antonio Ros, Schrödinger operators associated to a holomorphic map, Global differential geometry and global analysis (Berlin, 1990) Lecture Notes in Math., vol. 1481, Springer, Berlin, 1991, pp. 147–174. MR 1178529, DOI 10.1007/BFb0083639
  • X. H. Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. Part II, preprint.
  • X. H. Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. Part III, in preparation.
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Additional Information
  • Xuan Hien Nguyen
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
  • Address at time of publication: Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, Kansas 66506
  • MR Author ID: 857138
  • Received by editor(s): September 13, 2006
  • Published electronically: November 25, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 1683-1701
  • MSC (2000): Primary 53C44
  • DOI: https://doi.org/10.1090/S0002-9947-08-04748-X
  • MathSciNet review: 2465812