# Jacob Bernstein - bernstein/PDF/Thesis.pdfآ  Jacob Bernstein Submitted to the Department of Mathematics

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• Conformal and Asymptotic Properties of

Embedded Genus-g Minimal Surfaces with One

End

by

Jacob Bernstein

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2009

The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document

in whole or in part.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Department of Mathematics May 18, 2009

Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tobias H. Colding Professor of Mathematics

Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Jerison

Chairman, Department Committee on Graduate Students

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• Conformal and Asymptotic Properties of Embedded Genus-g Minimal Surfaces with One End

by Jacob Bernstein

Submitted to the Department of Mathematics on May 18, 2009, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

Using the tools developed by Colding and Minicozzi in their study of the structure of embedded minimal surfaces in R3 [12, 19–22], we investigate the conformal and asymptotic properties of complete, embedded minimal surfaces of finite genus and one end. We first present a more geometric proof of the uniqueness of the helicoid than the original, due to Meeks and Rosenberg [45]. That is, the only properly embedded and complete minimal disks in R3 are the plane and the helicoid. We then extend these techniques to show that any complete, embedded minimal surface with one end and finite topology is conformal to a once-punctured compact Riemann surface. This completes the classification of the conformal type of embedded finite topology minimal surfaces in R3. Moreover, we show that such s surface has Weierstrass data asymptotic to that of the helicoid, and as a consequence is asymptotic to a helicoid (in a Hausdorff sense). As such, we call such surfaces genus-g helicoids. In addition, we sharpen results of Colding and Minicozzi on the shapes of embedded minimal disks in R3, giving a more precise scale on which minimal disks with large curvature are “helicoidal”. Finally, we begin to study the finer properties of the structure of genus-g helicoids, in particular showing that the space of genus-one helicoids is compact (after a suitably normalization).

Thesis Supervisor: Tobias H. Colding Title: Professor of Mathematics

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• 4

• Acknowledgments

I would like to thank my advisor, Tobias Colding, for giving me the problem that was the genesis of this thesis and for giving me support and help in navigating the vagaries of mathematical research. I would also like to thank Bill Minicozzi for his constant enthusiasm and for his always extremely helpful suggestions.

There are many other people at MIT I would like to acknowledge (in no particular order): Richard Melrose for teaching me analysis and for humoring some of my more ill-posed queries; Hans Christianson and Brett Kotschwar for handling most of the rest; Linda Okun and the rest of the math department staff for helping me understand some of MIT’s more arcane policies; Chris Kottke being a great office-mate and for our many procrastination driven conversations; Gigliola Staffilani for helping me in my job search; Michael Eichmair for energizing geometric analysis at MIT and for being a great friend; Tom Mrowka and William Lopes for discussing some elementary Morse theory with me, the results of which are in Chapter 6; thanks also to Vedran, YankI, Lu, James and Max for many great mathematical discussions.

Away from MIT, my collaborator Christine Breiner must be mentioned, both as a great conference co-attendee and friend. Special thanks to my fianceé Jessie Howell for putting up with me, even at my most slovenly and unshaven. I couldn’t have done it with out you. I would also like to thank my parents. I am especially grateful to my father exposing me to deep mathematics at young age, and thank him for his copy of Kelley’s General Topology, my first encounter with abstract mathematics.

Finally, I would like to dedicate this thesis to the memory of Juha Heinonen without whom I very likely would have left mathematics. His contribution to my mathematical education cannot be overstated.

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• 6

• Contents

Abstract 3

Acknowledgments 5

1 Introduction 11

2 Background 15

2.1 Minimal Surface Theory in R3 . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Classical and modern constructions . . . . . . . . . . . . . . . 18

2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Colding-Minicozzi Theory 23

3.1 Structure of Embedded Minimal Disks . . . . . . . . . . . . . . . . . 24 3.1.1 Points of large curvature . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Extending the sheets . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.3 Finding large curvature . . . . . . . . . . . . . . . . . . . . . . 26 3.1.4 One-sided curvature estimate . . . . . . . . . . . . . . . . . . 27

3.2 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Lamination theory . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 The Calabi-Yau conjecture . . . . . . . . . . . . . . . . . . . . 30 3.2.3 Generalizations to non-trivial topology . . . . . . . . . . . . . 31

4 Uniqueness of the Helicoid 33

4.1 Meeks and Rosenberg’s Approach . . . . . . . . . . . . . . . . . . . . 33 4.2 Outline of the Argument . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Geometric Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.1 Initial sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3.2 Blow-up pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.3 Asymptotic helicoids . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.4 Decomposition of Σ . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 Conformal Structure of a Complete Embedded Minimal Disk . . . . . 40 4.4.1 Conformal structure . . . . . . . . . . . . . . . . . . . . . . . 40 4.4.2 Conformal mapping properties of the Gauss map . . . . . . . 40 4.4.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7

• 4.5 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.5.1 Blow-up sheets . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.5.2 Geometry near a blow-up pair . . . . . . . . . . . . . . . . . . 43

5 Structure Near a Blow-up Pair 45

5.1 Lipschitz Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Scale of the Approximation . . . . . . . . . . . . . . . . . . . . . . . 48

6 Genus-g Helicoids 51 6.1 Outline of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Geometric Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2.1 Structural results . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2.2 Blow-up sheets . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.2.3 Blow-Up pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.2.4 Decomposing Σ . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3 Conformal Structure of the end . . . . . . . . . . . . . . . . . . . . . 58 6.3.1 Winding number of the Gauss map . . . . . . . . . . . . . . . 58 6.3.2 Conformal structure of the end . . . . . . . . . . . . . . . . . 60 6.3.3 The proofs of Theorem 6.0.6 and Corollary 6.0.8 . . . . . . . . 60

6.4 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4.1 Topological structure of Σ . . . . . . . . . . . . . . . . . . . . 61 6.4.2 Proofs of Proposition 6.2.1 and 6.2.2 . . . . . . . . . . . . . . 62 6.4.3 One-sided curvature in Σ . . . . . . . . . . . . . . . . . . . . . 63 6.4.4 Geometric Bounds near blow-up pairs . . . . . . . . . . . . . . 64

7 The Space of Genus-g Helicoids 65 7.1 Outline of Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.2 Weak Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2.1 Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.2.2 Proof of Theorem 7.1.1 . . . . . . . . . . . . . . . . . . . . . . 71

7.3 The Intrinsic and Extrinsic Normalization . . . . . . . . . . . . . . . 74 7.3.1 Two-sided bounds on the genus . . . . . . . . . . . . . . . . . 75 7.3.2 Intrinsic normalization . . . . . . . . . . . . . . . . . . . . . . 78 7.3.3 Extrinsic normalization . . . . . . . . . . . . . . . . . . . . . . 79

7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.4.1 Compactness of E(1, 1) . . . . . . . . . . . . . . . . . . . . . . 80 7.4.2 Geometric Structure of E(1, 1, R) . . . . . . . . . . . . . . . . 84

8 Conclusion 87

8

• List of Figures

2-1 The catenoid (Courtesy of Matthias Weber) . . . . . . . . . . . . . . 19 2-2 The helicoid (Courtesy of Matthias Weber) . . . . . . . . . . . . . . . 19 2-3 A genus one helicoid (Courtesy of Matthias Weber)

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