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Construction of complete embedded self-similar surfaces under mean curvature flow. I. (English) Zbl 1166.53046
The author describes an original way to construct new examples of complete embedded surfaces which evolve self-similarly under the mean curvature flow. The strategy is following: The author takes some pair of known self-similar solutions (planes, spheres, cylinders) and replaces a neighborhood of their intersection with an appropriately bent scaled singly periodic Scherk surface (the core). The resulting non-smooth surface represents an approximate solution. The task is to find a function whose graph over the constructed surface satisfies the equation for self-similar solutions. The author proposes to consider Dirichlet problems for graphs of functions on different pieces of the constructed surface. The first step is carried out: it is shown 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. Dirichlet problems on other pieces as well as the smooth gluing of solutions will be studied in forthcoming papers.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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[1] 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. · Zbl 0762.53028
[2] S. Angenent, T. Ilmanen, and D. L. Chopp, A computed example of nonuniqueness of mean curvature flow in \?³, Comm. Partial Differential Equations 20 (1995), no. 11-12, 1937 – 1958.
[3] David L. Chopp, Computation of self-similar solutions for mean curvature flow, Experiment. Math. 3 (1994), no. 1, 1 – 15. · Zbl 0811.53011
[4] 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. Ulrich Dierkes, Stefan Hildebrandt, Albrecht Küster, and Ortwin Wohlrab, Minimal surfaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 296, Springer-Verlag, Berlin, 1992. Boundary regularity. · Zbl 0777.53012
[5] Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285 – 299. · Zbl 0694.53005
[6] Nicolaos Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239 – 330. , https://doi.org/10.2307/1971494 Nicolaos Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33 (1991), no. 3, 683 – 715. · Zbl 0699.53007
[7] Nikolaos Kapouleas, Complete embedded minimal surfaces of finite total curvature, J. Differential Geom. 47 (1997), no. 1, 95 – 169. · Zbl 0936.53006
[8] 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. · Zbl 0744.58007
[9] X. H. NGUYEN, Construction of complete embedded self-similar surfaces under mean curvature flow. Part II, preprint. · Zbl 1200.53061
[10] height 2pt depth -1.6pt width 23pt, Construction of complete embedded self-similar surfaces under mean curvature flow. Part III, in preparation.
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