Desingularization of an immersed self-shrinker.

Transcription

1 Desingularization of an immersed self-shrinker. Xuan Hien Nguyen (joint work with S. Kleene and G. Drugan) Iowa State University May 29, / 28

2 Examples of gluing constructions Summary of a classical construction Minimal surfaces One of the difficulties An immersed configuration The problem The fix 2 / 28

3 Examples of Gluing Constructions 3 / 28

4 Desingularizations To desingularize: to transform an immersed surface into an embedded one 4 / 28

5 Building blocks: Scherk minimal surfaces Scherk surfaces are minimal surfaces: There is a one-parameter family of them: They enjoy symmetries. The wings tend to half-planes exponentially fast. 5 / 28

6 Minimal surfaces, H = 0 [Kapouleas, 1997] [Traizet, 1996] Self-translating surfaces: H + ez ν = 0 [Nguyen, 2012] [Dávila-Del Pino-N., almost done] 6 / 28

7 Self-shrinking surfaces: H 1 2 X ν = 0 [Kapouleas Kleene Møller], [N.] The non-compact surface is asymptotic to a cone at infinity. 7 / 28

8 Common features of the examples the equation is of the form the stuff above scales properly: H + stuff = 0. (1) X = 1 τ X, H = τ H Equation (1) becomes H + τ stuff = 0 it is a desingularization of two or more intersecting surfaces. 8 / 28

9 Immersed self-shrinking surfaces, [Drugan-Kleene, preprint] Desingularizing immersed surfaces? Same story? Yes and no. 9 / 28

10 Summary of a classical construction: minimal surfaces 10 / 28

11 Sketch of the proof: minimal surfaces 1. Construct an approximate solution: wrap a Scherk surface around a large circle scale by τ, a small constant (or rather, scale everything else by 1 τ ). 2. Look at graphs of functions over this approximate surface: position X X + f ν mean curvature H H+ f + A 2 f +Quadratic solve L(f ) := f + A 2 f = E on each of the 5 pieces finish with a fixed point theorem 11 / 28

12 Linear operator on the Scherk surfaces L(f ) = f + A 2 f is associated to normal perturbations of the mean curvature. The mean curvature is invariant under translations. Problem: e x ν, e y ν, and e z ν are in the kernel of L. Partial solution: We can impose one symmetry: e z ν. But, what do we do with e x ν and e y ν? Define z 1 = e x ν, z 2 = e y ν 12 / 28

13 Linear operator on the Scherk surfaces: a priori estimates We want exponential decay for f. We need a priori estimates for the linear problem on Σ. Lemma (A priori estimates) Let 0 < γ < 1 and E : Σ R with e γs E C η 0 : Σ R a cut-off function f be a bounded solution to f + A 2 f = E in Σ such that Σ f η 0z i = 0, i = 1, 2 then f C e γs E, f Ce γs ( e γs E ), 13 / 28

14 Linear operator on the Scherk surfaces z 1 = e x ν, z 2 = e y ν R > R 1 with R 1 large 0 < γ < 1 η 0 : Σ R a cut-off function Lemma (Existence) Given E : Σ R with e γs E C, there are constants c 1 and c 2, there is a unique f such that Σ R f η 0 z i = 0, i = 1, 2 f + A 2 f = E + c 1 η 0 z 1 + c 2 η 0 z 2 in Σ R f = 0 on Σ R Moreover, c 1, c 2, f C e γs E Σ is the entire Scherk surface; Σ R is Σ truncated at s = R. 14 / 28

15 Achieving exponential decay Solve on Σ R, then let R. We get a solution f on Σ. f e γs and f C On each wing, lim s f =constant L i f +c 1 z 1 + c 2 z 2 has zero limit on two adjacent wings. f + c 1 z 1 + c 2 z 2 + c 1 ζ 1 z 1 + c 2 ζ 2 z 1 has zero limit on all wings. ζ 1 and ζ 2 are smooth cut-off functions on separate wings. ζ i = 1 for s large, i = 1, 2 15 / 28

16 Linear operator on the Scherk surfaces Lemma Given E : Σ R with e γs E C, there are constants c 1, c 2, c 1, and c 2, there is a function f such that L f = E+ c 1 L(ζ 1 z 1 ) + c 2 L(ζ 2 z 1 ) + c 1 η 0 z 1 + c 2 η 0 z 2 in Σ lim f = 0 on each wing s f Ce γs e γs E 16 / 28

17 Dislocations of the Scherk surface We have to compensate for the four extra terms with dislocations of the Scherk surface. L f = E+ c 1 L(ζ 1 z 1 ) + c 2 L(ζ 2 z 1 ) + c 1 η 0 z 1 + c 2 η 0 z 2 17 / 28

18 Dislocations of the Scherk surface L f = E+ c 1 L(ζ 1 z 1 ) + c 2 L(ζ 2 z 1 ) + c 1 η 0 z 1 + c 2 η 0 z 2 +d 1 L(z 1 trans) + d 2 L(z 2 trans) 18 / 28

19 Dislocations of the Scherk surface L f = E+ c 1 L(ζ 1 z 1 ) + c 2 L(ζ 2 z 1 ) + c 1 η 0 z 1 + c 2 η 0 z 2 +d 1 L(z 1 trans) + d 2 L(z 2 trans) + θ 1 L(z 1 rot) + θ 2 L(z 2 rot) 19 / 28

20 If you remember one thing... The linear problem on the Scherk surfaces is well-studied [Kapouleas, 1997]. We can use it! For a construction to work, one just needs flexibility of the initial setting. solve the associated linear problem on all the non-scherk pieces. Setting up the fixed point theorem at the end is also well-documented. 20 / 28

21 An immersed configuration 21 / 28

22 The problem 22 / 28

23 The flexibility is still there First unwind the bowtie to a segment [s, s + ], s < 0 < s +. Lf d = 0, f d (0) = 1, ḟ d (0) = 0 Record the Dirichlet and Neumann data at the ends Lf n = 0, f n (0) = 0, ḟ n (0) = 1 Record the Dirichlet and Neumann data at the ends 23 / 28

24 The flexibility is still there We can impose Dirichlet and Neumann defects ( ) d2 d iff det 1 d 4 d 3 0 n 2 n 1 n 4 n 3 iff there are no nontrivial solution to Lf = 0 globally. 24 / 28

25 Remarks There is no guarantee that the Scherk angle, position, rotation are preserved. We are using up all the degrees of freedom. One can solve ( + A 2 )f = E on Σ R with f C 2,α e R E C 0,α We are working on a Dirichlet to Neumann map to finish the construction. 25 / 28

26 The bowtie exists! 26 / 28

27 Immediate work and questions Work in progress The Dirichlet to Neumann map is invertible for τ small. Proof of the existence of the bowtie. Questions for you Extensions of the work? Related problems? Applications? 27 / 28

28 Thank you for your attention! 28 / 28

29 Angenent, S. B. (1992). Shrinking doughnuts. In Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), volume 7 of Progr. Nonlinear Differential Equations Appl., pages Birkhäuser Boston, Boston, MA. Drugan, G. and Kleene, S. J. ( 13). Immersed self-shrinkers. preprint, arxiv: Kapouleas, N. (1997). Complete embedded minimal surfaces of finite total curvature. J. Differential Geom., 47(1): Kapouleas, N., Kleene, S., and Møller, N. M. ( 14). Mean curvature self-shrinkers of high genus: Non-compact examples. preprint, arxiv: Møller, N. M. ( 11). Closed self-shrinking surfaces in r 3 via the torus. preprint, arxiv: Nguyen, X. H. ( 12). 28 / 28

30 Doubly periodic self-translating surfaces for the mean curvature flow. preprint, arxiv: Nguyen, X. H. (2012). Complete embedded self-translating surfaces under mean curvature flow. Journal of Geometric Analysis. Nguyen, X. H. (to appear). Construction of complete embedded self-similar surfaces under mean curvature flow. Part III. Duke Math. J., arxiv: Traizet, M. (1996). Construction de surfaces minimales en recollant des surfaces de Scherk. Ann. Inst. Fourier (Grenoble), 46(5): Wang, X.-J. (2011). Convex solutions to the mean curvature flow. Ann. of Math. (2), 173(3): / 28