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On the Hermitian Geometry of K-Gauduchon Orthogonal Complex Structures

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On the Hermitian Geometry of K-Gauduchon Orthogonal Complex Structures On the Hermitian Geometry of k-Gauduchon Orthogonal Complex Structures Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Gabriel Khan, B.A., M.S. Graduate Program in Mathematics The Ohio State University 2018 Dissertation Committee: Fangyang Zheng, Advisor Bo Guan King-Yeung Lam John Lafont c Copyright by Gabriel Khan 2018 Abstract This work deals with various phenomena relating to complex geometry. We are particularly interested in non-K¨ahlerHermitian manifolds, and most of the work here was done to try to understand the geometry of these spaces by understanding the torsion. Chapter 1 introduces some background material as well as various equations and inequalities on Hermitian manifolds. We are focused primarily on the inequalities that are useful for the analysis that we do later in the thesis. In particular, we focus on k-Gauduchon complex structures, which were initially defined by Fu, Wang, and Wu [8]. Chapter 2 discusses the spectral geometry of Hermitian manifolds. In particular, we estimate the real eigenvalues of @@¯ from below. In doing so, we prove a theorem on non-self-adjoint drift Laplace operators with L1 drift. This result is of independent interest, apart from its application to complex geometry. The work in this section is largely based on the Li-Yau estimate [24] as well as an ansatz due to Hamel, Nadirashvili and Russ [12]. Chapter 3 considers orthogonal complex structures to a given Riemannian metric. Much of the work in this section is conjectural in nature, but we believe that this ii is a promising approach to studying Hermitian geometry. We do prove several con- crete results as well. In particular, we show how the moduli space of k-Gauduchon orthogonal complex structures is pre-compact. iii For Kori iv Acknowledgments I am very grateful to Fangyang Zheng for his guidance and for being an excellent advisor. I feel very fortunate to have found an advisor who was willing to help and invest in me. I'm also deeply appreciative that he is always willing to listen and allow me the room to also work on my own projects. Thanks to Adrian Lam, for his help with the partial differential equations in Chapter 2. His insights were essential for this project. Further thanks to Bo Guan and John Lafont for their support and for serving on my committee. On a separate note, I would like to thank Jun Zhang for he reaching out and providing me the opportunity to work on his project next year. For this I am truly grateful. I would also like to thank Glen Richard Hall, who was my advisor as an undergraduate. He was very helpful and a great source of problems to work on. The time spent writing this would have been significantly less enjoyable and more difficult without the help of others. I would like to thank all my friends from Columbus and elsewhere who have supported me through the past six years. I'd also like to thank the creators of the OSU Dissertation LATEX class. This thesis would have been significantly more difficult to write without it. I owe a lot to my parents, Mizan and Therese. They have been tremendously helpful throughout my life, both in mathematics and otherwise. I would also like to thank Nathan, Nicholas, and Isabel, for being wonderful siblings. Many thanks are v due to Karen and Micky, for all their help with keeping our lives in order the past few years. Finally, and most importantly, I would like to thank Kori for her love and strength. The last few months have been crazy, and I could not have done any of this without her. She is my editor, organizer, and life partner. vi Vita July 17, 1990 . Born -Washington D.C., USA 2012 . .B.S. in Mathematics Boston University 2014 . .M.S. Mathematics The Ohio State University 2012-present . .Ph.D. Student The Ohio State University Publications Research Publications G. Khan \Hall's Conjecture on extremal sets for random triangles". Preprint. (2018) G. Khan, B. Yang, and F. Zheng. \The Set of All Orthogonal Complex Structures on the Flat 6-Tori". Adv. Math, 319, 451-471, 2017 G. Khan \The Drift Laplacian and Hermitian Geometry". AGAG, 53(2), 233-249, 2017 T. Hart, G. Khan, and M. Khan, \Revisiting Toom's Proof of Bulgarian Solitaire". Annales des Sciences Math´ematiquesdu Qu´ebec 36(2). (December 2011) Fields of Study Major Field: Mathematics vii Studies in: Complex Geometry Geometric Analysis viii Table of Contents Page Abstract . ii Dedication . iv Acknowledgments . v Vita . vii 1. Introduction . 1 1.1 Almost complex structures . 1 1.2 Complex Structures and the Newlander Nirenberg theorem . 2 1.3 The Levi-Civita and Chern connections . 3 1.3.1 Curvature and torsion . 4 1.3.2 Metric Connections and Complex Connections . 4 1.4 A crucial lemma . 6 1.5 Torsion bounds . 7 1.6 k-Gauduchon complex structures . 10 1.6.1 Weak k-Gauduchon structures . 11 1.7 The W 1;2 estimate on conformal changes . 12 1.8 Monotonicity results . 13 c 2. The eigenvalues of ............................. 16 2.1 The drift Laplacian . 16 2.1.1 Previous work . 17 2.2 Drift Laplacians with L1 drift . 19 2.3 A general outline of the proof . 20 2.4 Step 1: Finding the drift that minimizes the principle eigenvalue . 21 2.5 Step 2: The a priori Cα estimate and uniform radii estimates . 23 2.5.1 Step 2a: Variation estimates . 24 2.5.2 Step 2b: Regularity away from the zero locus of ru . 25 ix 2.6 Step 3: The Li-Yau Estimate . 25 2.6.1 An estimate using the maximum principle . 28 2.6.2 Using the inequality . 37 2.7 Step 4: A lower bound on λ ...................... 40 2.8 Future work . 41 2.9 A concrete example . 42 3. Moduli of Orthogonal complex structures . 44 3.1 Two dimensional metrics and Teichmuller theory . 45 3.2 Higher dimensions . 46 3.2.1 Generating orthogonal metrics and canonical metrics . 46 3.3 Flat six dimensional metrics . 47 3.3.1 More general phenomena . 49 3.4 Obstructions . 50 3.5 K¨ahlerand k-Gauduchon complex structures . 51 3.6 Stability under deformations . 53 Bibliography . 55 x Chapter 1: Introduction This section is dedicated to the basics of complex geometry, with a view towards the theorems that we will prove later. In particular, we are studying how the various curvature tensors and torsion tensors on a complex manifold are related. From this viewpoint, we can use theorems from analysis and Riemannian geometry to under- stand complex geometry. For this thesis, we will consider a compact Riemannian manifold (M n; g) which admits an integrable complex structure J. We are largely interested in results that can be proven about the Hermitian manifold (M n; g; J) but are independent of the par- ticular choice of J. The results that hold in this direction prove that the Riemannian geometry control various quantities about the Hermitian geometry. Alternatively, we can think of these as being results about the moduli space of orthogonal complex structures. However, before introducing these notions, we will first introduce some of the background. 1.1 Almost complex structures Complex numbers can be defined by adjoining the real numbers by an element i, which satisfies i2 = −1. Complex geometry generalizes this idea with the notion of an almost complex structure. Given a Riemannian manifold M, an almost complex 1 structure is a smoothly varying endomorphism of TM (denoted J) which satisfies J 2 = −Id. Intuitively, J corresponds to multiplication by i. We say that an almost complex structure is compatible with, or orthogonal to a metric, if it satisfies g(X; Y ) = g(JX; JY ) for all tangent vectors X and Y . Intuitively, this corresponds to the fact that multiplication by i does not change the norm of a number. In this case, the triple (M; g; J) is called an almost-Hermitian manifold. 1.2 Complex Structures and the Newlander Nirenberg the- orem In order for a manifold to be genuinely complex, J must satisfy a further condition. There needs to be be a local biholomorphism between our manifold and Cn. In other words, we want there to be local holomorphic coordinates. Given an arbitrary almost complex structure, it may not be possible to do this. A straightforward computation shows that if J is induced from complex coordinates, then Nijinhuis tensor must 2 vanish. This is a tensor defined by NJ (X; Y ) = −J [X; Y ] + J[X; JY ] + J[JX; Y ] − [JX; JY ], and it vanishes for any complex structure. Due to the Newlander-Nirenberg theorem, an almost complex structure is inte- grable if and only if its Nijinhuis tensor vanishes [21]. This theorem was proven for analytic almost complex structures in 1957. In this work, we are interested in complex structures with less regularity, so extensions of the Newlander-Nirenberg theorem are important for our work. The theorem was extended to the C1,α complex structures in [19]. However, it is possible to prove the Newlander-Nirenberg theorem in even weaker regularity. The result that we make use of here was originally proved in [13], which shows the Nijinhuis tensor is well defined and that the Newlander-Nirenberg theorem holds for complex structures that are Cα for α > 1=2. 2 We focus on complex manifolds, so we will only consider integrable complex struc- tures. This is a non-trivial restriction, many familiar manifolds do not admit orthog- onal complex structures.
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