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 L∞ 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 L∞ 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 ∇u . . . . . 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. For instance, the only sphere admitting a complex structure orthogonal to its round metric is S2. It is an open question whether any S6 admits a complex structure, but there are no complex structure which is close to being or- thogonal to the round metric [23]. Furthermore, no sphere other than S2 or S6 admit almost complex structures.
One particular class of complex structures that have been studied in depth are
K¨ahlercomplex structures. We say a complex structure is K¨ahlerif its K¨ahlerform
ω = g(J·, ·) is closed. We are primarily interested in the non-K¨ahlercase, as many of our results are already known or vacuous in the K¨ahler setting.
1.3 The Levi-Civita and Chern connections
Given a smooth manifold, there is canonical way to differentiate functions, but there is no way to differentiate vector fields without introducing a connection. A general affine connection on a manifold is a bilinear map ∇ satisfying the following conditions:
∇ : C∞(M, TM) × C∞(M, TM) → C∞(M, TM)
(X,Y ) 7→ ∇X Y
such that for all smooth functions f ∈ C∞M and all vector fields X,Y on M:
∞ • ∇fX Y = f∇X Y (∇ is C (M)-linear in the first variable);
• ∇X fY = df(X)Y + f∇X Y (∇ satisfies the product rule).
3 1.3.1 Curvature and torsion
Given an arbitrary connection ∇, it is possible to define the torsion and curvature.
If we let X and Y be smooth vector fields on our manifold, the torsion tensor τ is defined by
∇ τ (X,Y ) = ∇X Y − ∇Y X − [X,Y ].
Note that τ takes in two vector fields and produces a third. It is tensorial in X and Y , and measures how much the derivatives ∇X ∇Y fail to commute when acting on functions. The torsion can be interpreted as how much a frame must twist as it moves along auto-parallel curves.
We can also define the curvature of the connection. This is a tensor that takes three vector fields and produces a fourth. This measures how non-flat the connection is. If we let X,Y and Z be smooth vector fields, the curvature tensor R is defined in the following way:
∇ R (X,Y,Z) = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z
1.3.2 Metric Connections and Complex Connections
There is no a priori relation between a Riemannian metric and an affine connection.
In order to be able to theorems about the metric using connections, we really want to consider metric connections. A connection ∇ is said to be metric if ∇g = 0. This gives a natural compatibility condition between connections and Riemannian metrics.
There are reasons to study non-metric connections, but all of the connections that we consider here will be metric.
4 We can also define what it means for a complex structure and a connection to
be compatible. We say that a connection is complex if it satisfies ∇J = 0. Given a
Hermitian manifold, there are several canonical connections. We will primarily focus
on two of these, the Levi-Civita connection and the Chern connection.
The Levi-Civita connection is the unique metric connection which is torsion-free.
We denote this connection as ∇L.C.. It is the main connection used in Riemannian geometry.
The Chern connection is the unique connection which is metric, complex, and whose torsion is of type (2, 0) with respect to J. ∇h. This connection can be defined
h L.C. 1 using the following formula: g(∇X Y,Z) = g(∇X Y,Z) − 2 dω(JX,Y,Z) [2]. These two connections are the same if and only if the complex structure is K¨ahler,
in which case we have ∇L.C.J = 0. Since the Chern connection is metric, it admits
the same autoparallel curves as the Levi-Civita connection and will differ from the
∇h l Levi-Civita only by a contorsion tensor. Here, we denote τijk ≡ gklτij . We also Pn j denote the torsion one form ηi as j=1 Tij. It is worth noting that the Lie form, which is usually defined as θ = Jδω, is directly related to η. By computing in a
unitary frame, we have that θ = −2ηi − 2ηi.
It is straightforward to see that the torsion of the Chern connection controls
∇L.C.J. This observation will be crucial later on, and so we denote it as a lemma.
Lemma 1. There is a C1 estimate on the complex structure in terms of the torsion.
|∇L.C.J| ≤ 3|T ∇h |
To see this, note that we have the following formula:
1 1 − dω(JX,Y,Z) = (τ − τ + τ ) 2 2 JXYZ YZJX ZJXY
5 This characterization gives us an equivalent way of defining the K¨ahlercondition.
A complex structure is K¨ahlerif and only if ∇L.C.J ≡ 0.
For the rest of this thesis, we will use T to denote T ∇h to avoid overly stacking
indices.
1.4 A crucial lemma
Much of the work for this thesis owes its roots to Bo Yang and Fangyang Zheng’s
work [27]. By working in a unitary frame on a Hermitian manifold, they derived four
formulas relating the Chern curvature, the torsion of the Chern connection, and the
Riemannian curvature. For clarity, we recall the results of this lemma here. This
lemma was originally proved in [27]
n Theorem. (Lemma 7) Let (M , g) be a Hermitian manifold and let p ∈ M. Let {ei} be a unitary frame near p such that θ|p = 0. Furthermore, let R···· be components of
h the Riemannian curvature (in terms of the unitary frame), R···· be components of the
· Chern curvature and T·· be the components of the Chern torsion tensor. Then, at the point p we have:
k h h 2Tij,¯l = Rj¯lik¯ − Ri¯ljk¯
l l r l r Tij,k = Rijk¯l − TriTjk + TrjTik
l k r r k j l i l j k i 2Rijkl = Tij,k¯ − Tij,¯l + 2TijTkl + TriTrl + TrjTrk − TriTrk − TrjTrl
h j i r r j i l k Rk¯li¯j = Rk¯li¯j − Tik,¯l − Tjl,k¯ + TikTjl − TrkTrl − TriTrj
where r is summed through and h,i = ei(h) and h,¯i =e ¯i(h).
6 This lemma was invaluable to the rest of the work in this thesis, as it provided
ways of computing basic estimates. In particular, we are able to calculate various
different bounds on the torsion in terms of the curvature tensors.
1.5 Torsion bounds
In this section, we discuss ways to control the torsion. These are the initial
estimates that follow directly from the preceding lemma. In particular, we show that
we can bound the C1 norm of the torsion by two norms, which both measure the
“difference” between the Hermitian and Riemannian curvature. This section was originally published in [16].
To begin, we define the two norms that measure how much the metric fails to be
h 2 K¨ahler. This first norm, which we denote as kR − Rk?, is defined in the following way:
h 2 X h 2 X 2 kR − Rk? = |Ri¯jk¯l − Ri¯jk¯l| + 2 |Rijk¯¯l| i,j,k,l i,j,k,l h Recalling that RXY Z¯W¯ = 0 by Gray’s theorem [11], this can be thought of as a norm of the differences of Riemannian and Hermitian curvatures.
h 2 The second norm is denoted kR − Rk??, and is defined as follows:
h 2 X 2 kR − Rk?? = |Rijk¯l| i,j,k,l h Since RXYZW¯ = 0 [27], this can also be thought of as a norm. Using these two norms, we can measure how much a Hermitian metric fails to be K¨ahlerian.
Theorem. The following inequalities hold point-wise:
2 h 2 h ||T || ≤ |R − R|? and ||η|| ≤ |R − R|?
7 Proof. From equation 41 of [27],
1 h h X k 2 Y X (R + R ) − R ¯ ¯ = (|T | + 2Re(T T )) 2 XXY¯ Y¯ Y YX¯ X¯ XYY X XY kY kX k
We also have that RXY X¯Y¯ = RXXY¯ Y¯ − RXYY¯ X¯ . Therefore, we can rewrite the left- hand side of the above equation:
X k 2 Y X (|TXY | + 2Re(TkY TkX )) k
1 h h = (R + R ) − R ¯ ¯ − R ¯ ¯ 2 XXY¯ Y¯ Y YX¯ X¯ XXY Y XY XY 1 h 1 h = (R − R ¯ ¯ ) + (R − R ¯ ¯ ) − R ¯ ¯ 2 XXY¯ Y¯ XXY Y 2 Y YX¯ X¯ Y YXX XY XY
Pn X Y We want to gain a better understanding of k=1 2Re(TkX TkY ). We choose a
P i unitary frame and denote i Tki = ηk. Using this, we have the following equation
" !# X X i j X i X j Re(TkiTkj) = Re Tki Tkj i j i j 2 = |ηk|
That is to say,
X X i j X i 2 2 2Re(TkiTkj) = 2| Tki| = 2|ηk| i j i Thus, we have the following:
X X 1 h 1 h (R − R ¯ ¯) + (R − R ¯ ¯) − R ¯¯ 2 i¯ij¯j iijj 2 j¯ji¯i jjii ijij i j X X X k 2 i j = (|Tij| + 2Re(TkiTkj)) k i j = ||T ||2 + 2||η||2
8 2 h This then immediately proves that ||T || ≤ kR − Rk?.
h This theorem shows that |R −R|? measures how much a metric fails to be K¨ahler
h by directly bounding the torsion. There are many terms in kR − Rk? that are not
h needed to control the torsion, but we define |R − R|? as such in view of the next
lemma.
Lemma 2. Let ∇c00 be the derivative with respect to the Chern connection of a (0, 1)
c00 h vector. Then the following inequality holds: k∇ T k ≤ kR − Rk?
Proof. We use the first equation of Lemma 7 and the following Bianchi identity:
Ri¯jk¯l − Rk¯ji¯l = Ri¯jk¯l + R¯jki¯l
= −Rki¯j¯l = Rik¯j¯l
Combining these we find that:
j h h e¯l(Tik) = (Ri¯jk¯l − Ri¯jk¯l) − (Rk¯ji¯l − Rk¯ji¯l) − Rik¯j¯l
We can also bound the derivative with respect to a (1, 0) vector.
Lemma 3. The following inequality holds where ∇c0 is the derivative with respect to
the Chern connection of a (1, 0) vector:
c0 h h |∇ T | ≤ C(n)kR − Rk? + kR − Rk??
This is an immediate consequence of the equation from Theorem 4 and the fol-
lowing equation from [27]:
l l r l r Tij,k = Rijk¯l + TrjTik − TriTjk
9 Using a straightforward computation, we can relate the derivatives of the torsion
tensor with respect to the Levi-Civita connection to the derivatives of the torsion
tensor with respect to the Chern connection and a quadratic expression in torsion.
Therefore, we also have the following result:
Theorem. Let T be the torsion tensor and ∇T the derivative of the torsion tensor with respect to the Levi-Civita connection. Then there exists a constant C0(n) that grows linearly in n such that the following inequality holds:
0 h h ||∇T || ≤ C (n)kR − Rk? + kR − Rk??
1.6 k-Gauduchon complex structures
A K¨ahlerform ω = g(J·, ·) is said to be k-Gauduchon if i∂∂¯(ωk) ∧ ωn−k−1 = 0.
This condition was introduced by Fu, Wang, and Wu in [8]. When k = n − 1, this is the condition ∂∂ω¯ n−1 which is more commonly known as Gauduchon, or standard
Gauduchon. A result due to Gauduchon shows that for any complex structure J, and compatible metric g, there is another metricg ˜ conformal to g for whichg ˜(J·, ·)
is Gauduchon. However, we will initially focus on k-Gauduchon for other values of k.
A result due to Gauduchon [9] shows that for any integrable complex structure,
the Lie form θ satisfies the following identity:
2n − 2 |θ|2 + 2δθ = S − 2hW (ω), ωi 2n − 1
In this equation, S is the scalar curvature, and W is the Weyl curvature, viewed
as a map of two forms. In light of this, we define the Gauduchon curvature G as
follows: 2n − 2 G = S − hW (ω), ωi 2n − 1 10 We now substitute the Gauduchon curvature into the k-Gauduchon condition. We
consider the following:
¯ k n−k−1 αk = i∂∂(ω ) ∧ ω
Using a slight generalization of the calculation in [20], we can solve for the right hand side to obtain the following.
n 2kω 2 2 α = −(n − 2)η ,¯ +(2k − 1)|η| + (n − k − 1)|T | k n(n − 1)(n − 2) i i
Substituting in the Gauduchon curvature, we obtain:
2kωn α = −(n − 2)G + (2k − n)|η|2 + (n − k − 1)|T |2 k n(n − 1)(n − 2)
¯ k n−k−1 n If the metric is k-Gauduchon, then 0 = i∂∂(ω ) ∧ ω . For n − 1 > k > 2 , this shows the following:
(n − 2)G = (2k − n)|η|2 + (n − k − 1)|T |2
Since (2k − n), (n − k − 1) > 0, this gives us point-wise bounds on torsion in terms
of G alone.
1.6.1 Weak k-Gauduchon structures
We can extend the notion of k-Gauduchon to C1 complex structures, using the
previous identity. We say that a complex structure is weakly k-Gauduchon if (n −
2)G = (2k − n)|η|2 + (n − k − 1)|T |2. This definition is equivalent to the standard
one for twice-differentiable complex structure, but has the advantage of extending the
definition to once-differentiable complex structures.
11 For weakly k-Gauduchon complex structures, the same estimate on the torsion applies. As such, they are uniformly C1 by the earlier lemma.
1.7 The W 1,2 estimate on conformal changes
Under a conformal changeg ˜ = eug of the metric, the Lie form θ satisfies the
following equation.
2 −2u 2 2 X |θe| = e {|θ| + (2n − 1)|∇u| + 2(2n − 1) (θiui) i We calculate the new scalar curvature under the conformal transformation.
S˜ = e−2u S + 2(2n − 1)4u − (2n − 2)(2n − 1)k∇u|2
Since we know that there is a unique unit volume metric in any conformal class that makes the complex structure Gauduchon (ie. δθ = 0), we can try to estimate this conformal change, which provides a natural W 1,2 estimate on the conformal factor.
−2u 2 2 X e {|θ| + (2n − 1)|∇u| + 2(2n − 1) (θiui)} i = 2n − 2 e−2u S + 2(2n − 1)4u − (2n − 2)(2n − 1)k∇u|2 − 2e−2uhW ω, ωi 2n − 1 We rewrite this in the following way.
(2n − 1) {|θ|2 + (2n − 1)|∇u|2 − 4(2n − 1)kθk2 − k∇uk} 2 ≤ 2n − 2 S + 2(2n − 1)4u − (2n − 2)(2n − 1)k∇u|2 + 2kW k 2n − 1
12 Simplifying this, we obtain W 1,2 bounds on u in terms of R, n, and W , which yields a bound on the total scalar curvature of the Gauduchon metric.
1.8 Monotonicity results
As a separate application of Lemma 7, we explore various monotonicity formu- las that compare the curvature of the Chern connection and the curvature of the underlying Riemannian manifold.
In some sense, the Hermitian curvature is greater than the Riemannian curvature on the same manifold, regardless of the choice of complex structure. It is well known that the Chern scalar curvature dominates the Riemannian scalar curvature [27], but there are other monotonicity formulae as well. We have not yet been able to use these formulas to control the geometry in any meaningful way, so at this point we have not found good applications for this work.
One example of such a monotonicity formula is the following. Rh We consider Bh(X,Y ) = XXY¯ Y¯ , which is also known as the holomorphic bisec- |X|2|Y |2 tional curvature.
We also consider its Riemannian counterpart:
aR ¯ ¯ + (1 − a)R ¯ ¯ B (X,Y ) = XXY Y XYY X a |X|2|Y |2
This curvature was discussed in [27] as a possible notion of Riemannian bisectional curvature. We generally denote B(X,Y ) = B0(X,Y ) for conciseness.
13 Using the results from Lemma 7, we have the following sequence of equalities.
X X X 1 Bh(e , e ) − B(e , e ) = (Rh + Rh ) − (R − 2R ) i j i j 2 eie¯iej e¯j ej e¯j eie¯i eie¯iej e¯j eiej e¯ie¯j i j6=i i,j i6=j n X X X k 2 i j = |Tij| + 2Re(TkiTkj) i i6=j k=1 n n X X X k 2 X X X i j = |Tij| + 2Re(TkiTkj) i i6=j k=1 k=1 i i6=j n n X X X k 2 X X i 2 X i 2 = |Tij| + (| Tki| − |Tki| ) i i6=j k=1 k=1 i i n n n X X X k 2 X 2 X X i 2 = |Tij| + |ηk| − |Tki| i j k=1 k=1 k=1 i n n X X X k 2 X X i 2 2 = |Tij| − |Tki| + |η| i j k=1 k=1 i n n X X X k 2 X X k 2 2 = |Tij| − |Tik| + |η| i j k=1 i=1 k X X X k 2 2 = |Tij| + |η| i j k6=j
Here, the fifth line follows from skew symmetry of the torsion and the second equality follows from Lemma 7.
Theorem 4. In any unitary frame, the following equation holds,
X X h X X X k 2 X X i 2 X X i 2 B (ei, ej) − B0(ei, ej) = |Tij| + | Tki| + |Tki| i j6=i i j6=i k6=i,j k i k i
Note that we also have the following.
n X Bh(e , e ) − B (e , e ) = |T k |2 + 2Re(T ei T ei ) i i 0 i i eiei kei kei k=1
14 We can simplify that as follows.
n X Bh(e , e ) − B(e , e ) = 2|T ei |2 i i i i kei k=1
Currently, we have not found a use for these estimates in our geometric analysis.
If it is possible to prove strong monotonicity formulas, it will be possible to compare the Hermitian and Riemannian geometries. However, scalar curvature and similar scalar quantities do not seem strong enough to effectively control the geometry, so stronger monotonicity results are needed.
15 c Chapter 2: The eigenvalues of
Given a metric connection ∇T on a Riemannian manifold, there is an associated
drift Laplace-type operator ∆T := ∆ − g(T, ∇·) where T is the torsion one-form.
Equivalently, ∆T is the trace of the covariant derivative of d. On a general Hermitian
manifold, this is intricately related with the complex geometry.
c If we let {ek} be a basis of an orthonormal frame and ∇k be the covariant Chern
derivative in the ek direction, this operator can be defined explicitly as
c X c c ∆ = − ∇k∇k k .
Equivalently, we have that ∆c = ∆ +ξ(∇·) where ξ is the Lie form of the complex
structure and ∆ is the Laplace-Beltrami operator. As such, for a Hermitian manifold,
∆c is exactly ∇ξ. When the complex structure is not balanced (dωn−1 6= 0), ξ is non-
zero so ∆c acts on functions as a drift-Laplacian. Furthermore, whenever the complex structure is not conformally balanced, the operator is not self-adjoint.
2.1 The drift Laplacian
In this section, we are interested in trying to estimate the principle eigenvalue of
∆c. In order to do this for an arbitrary J orthogonal to a given metric, we must assume minimal regularity on the drift.
16 The drift Laplacian is a natural operator that appears in physical applications. It
has been studied in various settings, including on a Riemannian manifold [7] and for
smooth domains in Euclidean space [12]
The associated heat equation has been studied and the drift term acts as convec-
tion (i.e. stirring). Often, though not always, stirring speeds up the diffusion process.
Therefore, we might expect to be able to derive lower bounds on the real part of the
spectrum for the drift Laplacian. Such an estimate would provide a lower bound on
the exponential decay of the heat equation.
The following theorem provides an example of such bounds. If the drift-Laplacian
is non-self-adjoint, the statement applies only to the real eigenvalues.Given a non-
selfadjoint operator ∆+v(∇·) on Riemannian manifold, the spectrum of this operator is not necessarily real, but are isolated points in a conic neighborhood of the real axis.
We expect that there are positive lower bounds on the real part of any eigenvalue, but we are currently only able to derive lower bounds for real eigenvalues.
2.1.1 Previous work
Conjecture 5. Given a compact Hermitian manifold (M n, h), there exists a constant
C depending only on the Riemannian geometry such that if u = λu then λ ≥ C.
This would mirror the case for the Laplace-Beltrami operator, where the spectrum can be estimated using the dimension, diameter, and Ricci curvature.
In [16], we proved a similar result on the spectrum using a C1 bound on the drift.
Theorem 6. Let (M n, g) be a compact Riemannian manifold without boundary, sat-
isfying Ric M ≥ −(n − 1)k and v be a one-form on M. Suppose u ∈ C3(M) is a
solution to the equation ∆u + v(∇u) = λu with λ real. Let kvk be the C0 norm of
17 2 v and k∇vk be the C0 norm of ∇v (as a two tensor). Let D = 2nd where d is the
1 2 2 2 diameter of M and E = 2n ((16n − 32n + 5)kvk + 2(n − 1) k + 2(n − 1)k∇vk) Then we have: √ 1 (1 + 1 + 4DE)2 − DE λ ≥ √ D exp(1 + 1 + 4DE)
We then translated this estimate on the eigenvalue on the drift Laplacian into an estimate on the Laplacian on a Hermitian manifold.
Theorem 7. Suppose that (M n, h) is a compact, Hermitian manifold. Then there exists a uniform C > 0 such that any real eigenvalue λ of the Chern connection
Laplacian satisfies:
2 2 h h 1 d2 + 3Cn (k + kR − R k? + kR − R k??) λ ≥ 4n p 2 2 h h exp 1 + 1 + 4Cn d (k + kR − R k? + kR − R k??)
Using this estimate and a Harnack estimate, we we then estimate the principle eigenvalue solely in terms of the Riemannian geometry in the special case where the geometry was conformally Riemannian flat.
Theorem 8. Let (M n, g) be a compact globally conformally Riemannian-flat Hermi- tian manifold. Let K = infx∈M Ric M, k = supx∈M Ric M, R be the scalar curvature of M, d be the diameter of M and i be the injectivity radius of M. If λ1 is a real eigenvalue of the complex Laplacian , then we have the following estimate:
λ ≥ C(d, K, k, n, |∇R|,R2, i)
From the perspective of differential geometry, globally conformally flat metrics are well understood. However, such metrics admit a rich moduli of orthogonal complex structures [17], many of which are non-K¨ahler.As such, this is an interesting class of metrics to consider for this problem.
18 The first theorem can be proved using the Li-Yau estimate with an extra drift
term. The second theorem uses the results from Chapter 1 to show that bounds on
the Riemannian curvature and the two norms bounds the drift in C1 norm and then
applies the first theorem. Finally, the third theorem reduces to proving a Harnack
estimate for the conformal factor in terms of the curvature, which can be used to
control the torsion one-form in C1 sense. From there, we can apply the first theorem to obtain the desired estimate. The proofs of all three theorems are contained in [16].
2.2 Drift Laplacians with L∞ drift
In this section, we consider the case where we only assume L∞ bounds on the
drift. Under some fairly general assumptions on v, we can show that λ > 0 using a
basic maximum principle argument [16], but effective lower bounds are quite a bit
more challenging. The main result here here is to obtain semi-explicit lower bounds.
Theorem 9. Let (M n, g) be a compact Riemannian manifold. Suppose that there
exists u ∈ C2(M) ∆u + v(∇u) = λu with λ real. Here, v is some one form satisfying
kvk∞ < C. Then there exists some constant δ > 0 depending only kRk, k∇Rk,
C, diam(M) and n so that λ > δ.
The reason that we established the previous estimate was to understand the spec-
trum of k-Gauduchon Hermitian manifolds. The previous theorem immediately es-
c tablishes lower bounds on the spectrum of in terms of the Riemannian geometry when the complex structure is k-Gauduchon.
Theorem 10. Let (M n, g) be a compact Riemannian manifold. Suppose it admits a
n complex structure J so that ω = g(J·, ·) is k-Gauduchon for 2 < k ≤ n − 1. Consider
19 1 the complex Laplacian = 2 ∂z∂z¯ and suppose that it has a real eigenvalue λ. Then there exists some constant δ > 0 depending only kRk, k∇Rk, diam(M) and n so that
λ > δ. In particular, δ is independent of J.
This theorem follows from the preceding one because when (M n, g, J) is k-Gauduchon for an appropriate k, we have a point-wise bound on the torsion one-form [16] in terms of the curvature and so the complex Laplacian is a drift Laplacian with L∞ drift.
2.3 A general outline of the proof
The general strategy is to use the Li-Yau estimate, but we have to use some technical arguments and an observation of Hamel et. al. [12] to obtain the necessary smoothness. As an outline, we will do the following.
1. Step 1: We start by taking a sequence of drifts that minimize λ. We pick
some subsequence for which the associate drifts and functions converges in some
weak sense. We find that when the drift minimizes λ, the minimizing function
satisfies ∆u±C|∇u|+λu = 0 on an open set where the plus-minus corresponds
to whether u is positive or not. Essentially, this gives us C1 control over the
drift away from the zero locus of u and ∇u.
2. Step 2. We establish some regularity on u using general elliptic theory.
a) We use standard elliptic theory (Theorem 9.11 of [10]) to get a W 2,p estimate
on u. This gives C1,α control on u using the Sobolev embedding theorem. We
define the c-radius rc as the in the following way: rc = infx(d(x, p)|u(x) =
c, u(p) = 1), we can use standard elliptic theory to establish uniform lower
bounds on r3/4, r1/2 and r1/4 and show that r1/2 − r3/4 has a uniform lower
20 bound. It is worth noting that when we do this, we assume λ < 1 or else we
are done automatically. We refer to these as the global C1,α bounds.
b) We then use Theorem 6.2 of [10] and our W 2,p estimate on u. Away from the
zero locus of ∇u and u, this allows us to bootstrap the regularity of u to C3,α.
We refer to these as the Schauder bounds. Due to the nature of the argument,
we cannot use incorporate these bounds into our estimate of λ, but we need
them in order to use the Bochner technique.
3. Step 3: We consider the point x0 ∈ M which maximizes
|∇(u)|2 F (x) = (r2 − ρ2) (β − (u))2 1/2
and use the Li-Yau estimate to obtain an upper bound for F (x).
4. Step 4: We integrate out the previous gradient inequality. Using the previous
estimates, we choose β to obtain a lower bound on λ.
With this rough overview out of the way, we dive into the details.
2.4 Step 1: Finding the drift that minimizes the principle eigenvalue
Let (M n, g) be a compact manifold. We consider a drift (one-form) v with real eigenvector λi that solve the problem ∆u + v(∇u) = λu. We impose the condition
|v| < C point-wise and suppose that λ < 1, or else we are done trivially. Then we
2,p have that u ∈ Wloc for all finite p via the Calder´on-Zygmund estimates. This is Theorem 9.11 of [10], adapted to the manifold setting.
Now, consider the open manifold M + = {M|u > 0}. Note that we can show
2,p that this domain contains a uniform ball, by the Wloc estimate on u. We can also
21 show that its complement contains an open ball. However, we do not have any a
+ + priori regularity of the boundary of M . Therefore, we instead consider M so that
+ + + M ⊂ M and the boundary of M is smooth. We now consider the L∞ drift v0, which minimizes the principle eigenvalue λ(∆u+
+ + + 0 + v, M ) among all drifts v. Since M is larger than M , λ(∆u + v ,M ) is smaller
+ 0 + than λ(∆ + v(∇·),M ). Therefore, it suffices to estimate λ(∆u + v ,M ). We now prove that the minimal principle eigenvalue λ and it’s associated eigen- function u satisfy the Dirichlet problem for the following semi-linear equation on
+ M : ∆u + C|∇u| + λu = 0
0 ∇u + To do this, we assume that v 6= C |∇u| on some subset of M with non-zero measure. Then we know that u is a sub-solution to the following:
∇u ∆u − C · ∇u + λ(∆u + v0,M +) ≤ ∆u − v0(∇u)λu + λ(∆u + v0,M +) = 0 |∇u|
0 ∞ + 2,p Now, since v is L and M is smooth, we have a local W estimate on u, and
∇u ∞ 2,p hence ∇u is well defined. As such, C |∇u| is L and there exists a locally W solution to the Dirichlet problem.
∇u 0 = u0 − C · ∇u0 + λ0u0 |∇u|
Since we assumed that v0 minimizes λ, we know that λ ≤ λ0 which implies that u0
is a super-solution to the following problem:
∇u 0 ≤ u0 − C · ∇u0 + λu0 |∇u|
22 + ∞ Since M is smooth and the drift is L , the Hopf lemma holds and shows that ∇u 6= 0 on the boundary. From this, if we consider u − κu0, and choose κ so that it is the maximum such κ for which u−κu0 > 0, we can use a standard touching argument and either the maximum principle or the Hopf lemma to show that u ≡ κu0. In fact, this is exactly Lemma 2.1 of [12], applied to an open domain on a manifold. As such, we have proven the ansatz.
This ansatz is quite miraculous. It shows that in the worst case scenario, where all the drift is working to make the principle eigenvalue as small as possible, we obtain much stronger regularity than we initially assumed. This gives us very strong control of the drift away from the zero locus of u and ∇u. In essence, all the drift is working together and cannot be too irregular. This phenomena was first observed in [12], where they considered the drift-Laplacian with Dirichlet boundary conditions on C2,α open domains in Rn and proved a version of the Faber-Krahn inequality.
2.5 Step 2: The a priori Cα estimate and uniform radii esti- mates
To go further, we need to ensure that the function u does not vanish too quickly because we do not have C1 control of the drift on the zero locus of u. This section uses a Cα estimate instead of a C1,α estimate, to minimize the regularity assumptions.
∇u Suppose u satisfies ∆u ± C k∇uk ∇u + λu = 0. Around each point p, we consider a ball of radius R where, in normal coordinates, the elliptic regularity of ∆ is between
1 2 and 2 and so that inj(M) < R. Such a ball can be chosen uniformly as the
1 k l 3 deformation away from gij = δij − 3 Rikjlx x + O(|x| ) Then by Corollary 9.24 of [10], we have the following estimate,
23 α 0 r osc u ≤ C (osc u + λ||u|| n R) Br(p) R BR(p) L
0 Here, C and α depend on R, n and C since by construction, in BR(p) the elliptic
1 constant is bounded between 2 and 2. To derive an explicit lower bound on λ, we would have to calculate C‘ and α explicitly, which we do not do here.
In any case, we know that oscBR(p)u ≤ 2 and we are trying to attain a lower bound on λ, so we can assume λ < 1, or else we have nothing to prove. If λ < 1, then we have the following estimate.
r α osc u ≤ C00 (2 + Rn+1) Br(p) R
This establishes Holder-α regularity, which we use in the next step.
2.5.1 Step 2a: Variation estimates
Suppose u(p) = 1. We define the c-radius rc as infx(d(x, p)|u(x) = c, u(p) = 1/α c 1). We define dc = R C00(2+Rn+1) . By the Holder continuity, we have rc > 1/α d = R 1−c , as the Holder continuity implies inf u(x) ≥ 1 − c. 1−c C00(2+Rn+1) x∈Bdc (p)
Essentially, dc is a lower bound for how far one must travel to get an oscillation of c.
This estimate scales badly as c goes to zero, but since we are working on a fixed scale
(3/4, 1/2 and 1/4), that is inconsequential. Trivially, we have that r3/4 < r1/2 < r1/4, but using this argument, we also have that r1/2 ≥ r3/4 + d1/4 and r1/4 ≥ r1/2 + d1/4.
This estimate only depends on the geometry of the manifold. Therefore, we can use the constant dc throughout the estimate.
24 2.5.2 Step 2b: Regularity away from the zero locus of ∇u
2,p Using the Calder´on-Zygmund estimates, we have a Wloc estimate on u for p large. Using a partition of unity and the Sobolev embedding theorem, this provides a C1,α- estimate on u. From this, there is in fact a Cα estimate on |∇u|. Thus, when |∇u| is
∇u non-zero, we have that u satisfies ∆u + C |∇u| · ∇u − λu = 0. The coefficients are now Cα, so we gain C2,α0 control on u away from where |∇u| = 0 by Schauder estimates.
Therefore, |∇u| ∈ C1,α in this neighborhood and hence using the Schauder interior estimates, we have that u ∈ C3,α in a possible smaller neighborhood. The size of this
neighborhood will decay as β converges to 1, but that’s okay because we essentially
only need to be able to take three derivatives at a single point.
This estimate does not depend on the geometry, but on how close to the zero locus
of u and ∇u we are. As such, we cannot use this estimate to establish bounds on λ.
2.6 Step 3: The Li-Yau Estimate
Suppose we have a function u ∈ W 2,p which satisfies the following:
∆u + C|∇u| + λu = 0. (2.1)
Suppose further that we have rescaled u so that sup u = 1. Given β > 1, to be
determined later, we consider the function G(x) defined by:
|∇u|2 F (x) = (r2 − ρ2) (2.2) (β − u)2 1/2
Here, β > 1 is a constant to be specified later. We observe that there is a point
1 x ∈ Br3/4 with |∇(u)| > 4d where d is the diameter of M. At such a point,
|∇(u)|2 1 (r2 − ρ2) > (cr ) (β − (u))2 1/2 16d2(β − 3/4)2 3/4
25 We consider the point x0 ∈ M which maximizes F (x). Our previous estimate
shows that the following:
(β − 1)2 1 |∇(u)|2 > (cr ) d2 16d2(β − 3/4)2 3/4
This estimate is not uniform in β, but we will later provide a value for β that only depends on the geometry of M, and not on the earlier parameters introduced into the problem, so this is not an issue. In particular, we do not need to estimate the size of the neighborhood in which |∇(u)|= 6 0, only show that we are in such a neighborhood.
Arranging everything to work so that we could specify a particular value for β at the end without circular reasoning was the most difficult technical element of this proof.
It’s the reason that we used a global C1,α-estimate and then bootstrapped to a local
C3,α-estimate.
1,α Using the C estimate from the previous argument, in a small ball around x0, we have that |∇u| 6= 0. As noted before, we can use Schauder theory to bootstrap our regularity twice. After doing so, we see that u ∈ C3,α” in a possibly very small neighborhood around the x0. The size of this neighborhood will certainly decay as β gets closer to 1, though. However, for a given β, this is enough regularity to apply the maximum principle. At this point, we have established global C1,α bounds on u and enough regularity to use the maximum principle at x0.
We consider an orthonormal frame around a point x0 ∈ M. Here, we assume that
Ric(M) > −(n − 1)K for some K.
26 2 X 2 X ∆(|∇u| ) = 2 uij + 2 ui(∆u)i + 2Ric(∇u, ∇u) i,j i X 2 X = 2 uij + 2 ui(−C|∇u| − λu)i + 2Ric(∇u, ∇u) i,j i X 2 X = 2 uij + 2 ui(−C|∇u| − λu)i + 2Ric(∇u, ∇u) i,j i X 2 X 2 ≥ 2 uij + 2 ui(−C|∇u| − λu)i − (n − 1)K|∇u| i,j i X 2 X 2 = 2 uij + 2 ui(−C|∇u|)i − ((n − 1)K + λ)|∇u| i,j i
We may choose normal coordinates at x so that u1(x0) = |∇u|, ui = 0 for i 6= 1.
2 P 2 This choice ensures that ∇j|∇u| = u1j and hence |∇(|∇u|)| = j u1j. Note that ∆(|∇u|2) = 2|∇u|∆(|∇u|) + 2|∇(|∇u|)|2.
We can substitute this equation into the preceding inequality.
X 2 X 2 X 2 |∇u|∆(|∇u|) ≥ uij − u1j − 2 ui(C|∇u|)i − ((n − 1)K + λ)|∇u| i,j j i
27 We now estimate the first two terms.
X 2 X 2 X 2 X 2 uij − u1j ≥ ui1 + uii i,j j i>1 i>1 X 1 X ≥ u2 + ( u )2 i1 n − 1 ii i>1 i>1 X 1 ≥ u2 + (∆u − u )2 i1 n − 1 11 i>1 X 1 ≥ u2 + (−C|∇u| − λu − u )2 i1 n − 1 11 i>1 X 1 u2 ≥ u2 + 11 − 2(C|∇u|)2 − 2(λu)2 i1 n − 1 2 i>1 1 X 2 ≥ u2 − (C|∇u|2 + (λu)2 2(n − 1) i1 n − 1 i 1 2 = |∇|∇u||2 − (C|∇u|)2 + (λu)2 2(n − 1) n − 1
This implies the following.
1 X ∆(|∇u|2) ≥ 2 + |∇|∇u||2 − 2 u (C|∇u|) (n − 1) i i i 2 −((n − 1)K + λ)|∇u|2 − (C|∇u|)2 + (λu)2 n − 1
2.6.1 An estimate using the maximum principle
We are now ready to estimate F (x) using a modified Li-Yau estimate with an extra cut-off function. Recall that we defined F (x) in the following way.
|∇u|2 F (x) = (r2 − ρ2) (β − u)2 1/2
28 Since F∂B (p) ≡ 0, we can find x0 inside this ball where F is maximized. We r1/2
can assume that x0 is not a cut point or else we can slightly alter our cut-off function as is done in Lectures in Differential Geometry [24]. Therefore, we assume that the cut-off function is smooth at this point.
At x0, we pick an orthonormal frame so that u1 = |∇u| and ui = 0 otherwise.
Then, since x0 maximizes F (and F is twice differentiable at x0 by the Schauder estimate), we have that ∇F (x0) = 0,
2u f u2u |∇u|2 (r2 − ρ2) 1i 1 − 2(r2 − ρ2) 1 i − 2ρρ = 0 1/2 (β − u)2 1/2 (β − u)3 i (β − u)2
We can simplify this equation to obtain two identities.
2 2 u1 |∇u| u1 ρ|∇u| u11 = + ρρ1 2 2 ≥ − 2 2 (β − u) (r1/2 − ρ ) (β − u) (r1/2 − ρ )
and
ρρi ui1 = u1i = 2 2 (r1/2 − ρ )
Now we have the following formula for the Laplacian of F .
2 2 ! 2 ! (β − u) (β − u) (β − u) 2 (∆F ) 2 2 + (∇F )∇ 2 2 + F ∆ 2 2 = ∆(|∇u| ) (r1/2 − ρ ) (r1/2 − ρ ) (r1/2 − ρ )
Therefore, at x0, where ∆F ≤ 0 and ∇F = 0, we have the following string of
inequalities. The goal of this is to obtain an estimate on F using geometric quantities
of M.
29 2 ! 2 (β − u) 0 ≥ ∆(|∇u| ) − F ∆ 2 2 (r1/2 − ρ ) 1 X ≥ 2 + |∇|∇u||2 − 2 u (C|∇u|) − ((n − 1)K + λ)|∇u|2 (n − 1) i i i 2 ! 2 2 2 (β − u) − (C|∇u|) + (λu) − F ∆ 2 2 n − 1 (r1/2 − ρ ) 1 X ≥ 2 + |u |2 − 2 u (C|∇u|) − ((n − 1)K + λ)|∇u|2 (n − 1) 11 i i i 2 ! 2 2 2 (β − u) − (C|∇u|) + (λu) − F ∆ 2 2 n − 1 (r1/2 − ρ ) 1 X X ≥ 2 + (|u |2 + |u |2) − 2 u (C|∇u|) (n − 1) 11 i1 i i i6=1 i 2 2 2 2 2 |∇u| −((n − 1)K + λ)|∇u| − (C|∇u|) + (λu) − 2F 2 2 n − 1 (r1/2 − ρ )
∆u ∇u · ∇ρ 2 2 2 −1 −2F (β − u) 2 2 + 8F (β − f)ρ 2 2 2 − F (β − u) ∆(r1/2 − ρ ) (r1/2 − ρ ) (r1/2 − ρ )
30 2 1 |u1| ρ1ρu1 X ≥ 2 + ( + 2 )2 + |u |2) (n − 1) β − u (r2 − ρ2) i1 1/2 i6=1 X 2 −2 ui(C|∇u|)i − ((n − 1)K + λ)|∇u| i 2 2 2 2 |∇u| ∆u − (C|∇u|) + (λu) − 2F 2 2 − 2F (β − u) 2 2 n − 1 (r1/2 − ρ ) (r1/2 − ρ )
∇u · ∇ρ 2 2 2 −1 +8F (β − u)ρ 2 2 2 − F (β − u) ∆(r1/2 − ρ ) (r1/2 − ρ ) 1 1 |u |4 ≥ 2 + 1 − 1 (n − 1) 4(n − 1) (β − u)2 1 ρρ1u1 2 −(4(n − 1) − 1) 2 + (2 2 2 ) (n − 1) (r1/2 − ρ ) 1 X X + 2 + |u |2 − 2 u (C|∇u|) − ((n − 1)K + λ)|∇u|2 (n − 1) i1 i i i6=1 i 2 2 2 2 |∇u| ∆u − (C|∇u|) + (λu) − 2F 2 2 − 2F (β − u) 2 2 n − 1 (r1/2 − ρ ) (r1/2 − ρ )
∇u · ∇ρ 2 2 2 −1 +8F (β − u)ρ 2 2 2 − F (β − u) ∆(r1/2 − ρ ) (r1/2 − ρ )
Canceling the leading terms, we find the following:
1 1 |u |4 0 ≥ − 1 2(n − 1) 4(n − 1)2 (β − u)2 !2 1 ρρ1u1 −(4(n − 1) − 1) 2 + 2 2 2 (n − 1) (r1/2 − ρ ) 2 2 2 1 X ρ ρ |u1| X + 2 + i − 2 u (C|∇u|) (n − 1) (r2 − ρ2)2 i i i6=1 1/2 i 2 −((n − 1)K + λ)|∇u|2 − (C|∇u|)2 + (λu)2 n − 1 ∆u ∇u · ∇ρ −2F (β − u) 2 2 + 8F (β − u)ρ 2 2 2 (r1/2 − ρ ) (r1/2 − ρ ) 2 2 2 −1 −F (β − u) ∆(r1/2 − ρ )
31 Substituting in ∆u+C|∇u|+λu = 0 into the third line and then simplifying, this yields:
1 1 |u |4 0 ≥ − 1 2(n − 1) 4(n − 1)2 (β − u)2 1 ρρ1u1 2 −(4(n − 1) − 1) 2 + (2 2 2 ) (n − 1) (r1/2 − ρ ) 2 2 2 1 X ρ ρ |u1| X + 2 + i − 2 u (C|∇u|) − ((n − 1)K + λ)|∇u|2 (n − 1) (r2 − ρ2)2 i i i6=1 1/2 i
2 2 2 −C|∇u| − λu − (C|∇u|) + (λu) − 2F (β − u) 2 2 n − 1 (r1/2 − ρ ) u1ρ1 2 2 2 −1 +8F (β − u)ρ 2 2 2 − F (β − u) ∆(r1/2 − ρ ) (r1/2 − ρ ) 1 1 |u |4 ≥ − 1 2(n − 1) 4(n − 1)2 (β − u)2 1 ρu1 2 −(4(n − 1) − 1) 2 + (2 2 2 ) (n − 1) (r1/2 − ρ ) X 2 −2 ui(C|∇u|)i − ((n − 1)K + λ)|∇u| i
2 2 2 −C|∇u| − λu − (C|∇u|) + (λu) − 2F (β − u) 2 2 n − 1 (r1/2 − ρ ) u1ρ1 2 2 2 −1 +8F (β − u)ρ 2 2 2 − F (β − u) ∆(r1/2 − ρ ) (r1/2 − ρ )
2 2 4 (r1/2−ρ ) Multiplying through by (β−u)2 , we have the following.
32 1 1 |u |4 (r2 − ρ2)4 0 ≥ − 1 1/2 2(n − 1) 4(n − 1)2 (β − u)2 (β − u)2 (r2 − ρ2)4 1 ρu1 2 1/2 −(4(n − 1) − 1) 2 + (2 2 2 ) 2 (n − 1) (r1/2 − ρ ) (β − u) 2 2 4 2 2 4 (r1/2 − ρ ) X (r1/2 − ρ ) −2 u (C|∇u|) − ((n − 1)K + λ)|∇u|2 (β − u)2 i i (β − u)2 i 2 (r2 − ρ2)4 − (C|∇u|)2 + (λu)2 1/2 n − 1 (β − u)2 2 2 4 −C|∇u| − λu (r1/2 − ρ ) −2F (β − u) 2 2 2 (r1/2 − ρ ) (β − u) 2 2 4 u1ρ1 (r1/2 − ρ ) +8F (β − u)ρ 2 2 2 2 (r1/2 − ρ ) (β − u) (r2 − ρ2)4 −F (β − u)2∆(r2 − ρ2)−1 1/2 1/2 (β − u)2 1 1 1 = − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) 2 2 4 (r1/2 − ρ ) X −2 u (C|∇u|) − ((n − 1)K + λ)F (r2 − ρ2)2 (β − u)2 i i 1/2 i 2 (r2 − ρ2)4 (r2 − ρ2)3 − (C|∇u|)2 + (λu)2 1/2 + F (C|∇u| + λu) 1/2 n − 1 (β − u)2 (β − u) u ρ +8F ρ 1 1 (r2 − ρ2)2 − F ∆(r2 − ρ2)−1(r2 − ρ2)4 (β − u) 1/2 1/2 1/2 1 1 1 ≥ − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) (r2 − ρ2)4 −2 1/2 u (C|∇u|) − ((n − 1)K + λ)F (r2 − ρ2)2 (β − u)2 1 1 1/2 2 (r2 − ρ2)4 (r2 − ρ2)3 − (C|∇u|)2 + (λu)2 1/2 + F (C|∇u| + λu) 1/2 n − 1 (β − u)2 (β − u) 3/2 2 2 3/2 2 2 −1 2 2 4 −8F ρ(r1/2 − ρ ) − F ∆(r1/2 − ρ ) (r1/2 − ρ )
33 1 1 1 0 ≤ − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) (r2 − ρ2)4 −2 1/2 u Cu − ((n − 1)K + λ)F (r2 − ρ2)2 (β − u)2 1 11 1/2 2 (r2 − ρ2)4 (r2 − ρ2)3 − (Cu )2 + (λu)2 1/2 + F (Cu + λu) 1/2 n − 1 1 (β − u)2 1 (β − u) 3/2 2 2 3/2 2 2 −1 2 2 4 −8F ρ(r1/2 − ρ ) − F ∆(r1/2 − ρ ) (r1/2 − ρ ) 1 1 1 ≥ − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) (r2 − ρ2)4 2 ! 1/2 u1 ρ|∇u| −2 2 u1C + 2 2 (β − u) (β − u) (r1/2 − ρ ) 2 2 2 −((n − 1)K + λ)F (r1/2 − ρ ) 2 (r2 − ρ2)4 2 − (λu)2 1/2 − C2F (r2 − ρ2)3 n − 1 (β − u)2 n − 1 1/2 (r2 − ρ2)3 +F (λu) 1/2 + F 3/2C(r2 − ρ2)5/2 (β − u) 1/2 3/2 2 2 3/2 2 2 −1 2 2 4 −8F ρ(r1/2 − ρ ) − F ∆(r1/2 − ρ ) (r1/2 − ρ ) 1 1 1 = − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) (r2 − ρ2)4 ρ(r2 − ρ2)3 −2 1/2 u3C − 2 1/2 u2C (β − u)3 1 (β − u)2 1 2 2 2 −((n − 1)K + λ)F (r1/2 − ρ ) 2 (r2 − ρ2)4 2 − (λu)2 1/2 − C2F (r2 − ρ2)3 n − 1 (β − u)2 n − 1 1/2 (r2 − ρ2)3 +F (λu) 1/2 + F 3/2C(r2 − ρ2)5/2 (β − u) 1/2 3/2 2 2 3/2 2 2 −1 2 2 4 −8F ρ(r1/2 − ρ ) − F ∆(r1/2 − ρ ) (r1/2 − ρ )
Therefore, we have the following inequality.
34 1 1 1 0 ≤ − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) 2 2 4 3/2 2 2 3 2 2 2 −2C(r1/2 − ρ ) F − 2Cρ(r1/2 − ρ ) F − ((n − 1)K + λ)F (r1/2 − ρ ) 2 (r2 − ρ2)4 2 − (λu)2 1/2 − C2F (r2 − ρ2)3 n − 1 (β − u)2 n − 1 1/2 (r2 − ρ2)3 +F (λu) 1/2 + F 3/2C(r2 − ρ2)5/2 (β − u) 1/2 3/2 2 2 3/2 2 2 −1 2 2 4 −8F ρ(r1/2 − ρ ) − F ∆(r1/2 − ρ ) (r1/2 − ρ )
Now, we estimate the final term. We use the Laplace comparison theorem [24] to show the following:
2 2 2 X 2ρiiρ 8ρ ρ 2ρ ∆(r2 − ρ2)−1 = + i + i 1/2 (r2 − ρ2)2 (r2 − ρ2)3 (r2 − ρ2)2 i 1/2 1/2 1/2 n − 1 2ρ 8ρ2 2 ≤ (1 + Kρ) 2 2 2 + 2 2 3 + 2 2 2 ρ (r1/2 − ρ ) (r1/2 − ρ ) (r1/2 − ρ )
Therefore, we can estimate the last term in the following way:
2 2 −1 2 2 4 2 2 2 2 2 2 F ∆(r1/2 − ρ ) (r1/2 − ρ ) ≤ 2(n − 1)(1 + Kρ)F (r1/2 − ρ ) + 8F ρ (r1/2 − ρ )
2 2 2 +2F (r1/2 − ρ )
We combine all of this into a single inequality.
35 1 1 1 0 ≥ − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) 2 2 4 3/2 2 2 3 2 2 2 −2C(r1/2 − ρ ) F − 2Cρ(r1/2 − ρ ) F − ((n − 1)K + λ)F (r1/2 − ρ ) 2 (r2 − ρ2)4 2 − (λu)2 1/2 − C2F (r2 − ρ2)3 n − 1 (β − u)2 n − 1 1/2 (r2 − ρ2)3 +F (λu) 1/2 + F 3/2C(r2 − ρ2)5/2 (β − u) 1/2 3/2 2 2 3/2 −8F ρ(r1/2 − ρ )
2 2 2 2 2 2 2 2 2 −2(n − 1)(1 + Kρ)F (r1/2 − ρ ) + 8F ρ (r1/2 − ρ ) + 2F (r1/2 − ρ )
2 2 2 We know that r1/2, ρ < d so (r1/2 −ρ ) < d . We substitute this into our inequality.
1 1 1 0 ≥ − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) −2Cd8F 3/2 − 2Cd7F − ((n − 1)K + λ)F d4 2 d8 2 − (λu)2 − C2F d6 n − 1 (β − u)2 n − 1 (r2 − ρ2)3 +F (λu) 1/2 + F 3/2C(r2 − ρ2)5/2 (β − u) 1/2 −8F 3/2d4 − 2(n − 1)(1 + Kρ)F d4 + 8F d4 + 2F d4
The only positive term that needed is the first one. The others act on lower orders so sharpen the estimate but are not necessary.
36 1 1 1 0 ≥ − F 2 − (4(n − 1) − 1) 2 + 2ρF 2(n − 1) 4(n − 1)2 (n − 1) −2Cd8F 3/2 − 2Cd7F − ((n − 1)K + λ)F d4 2 d8 2 − (λu)2 − C2F d6 n − 1 (β − u)2 n − 1 d6 −F (λu) − F 3/2Cd5 (β − u) −8F 3/2d4 − 2(n − 1)(1 + Kρ)F d4 + 8F d4 + 2F d4
With this done, we are finally finished with the hardest part of the estimate.
2.6.2 Using the inequality
At this point, it seems reasonable to ask how this gives any hope of providing a
lower bound on λ.
We have uniform control of the cutoff function in Br3/4 from the results using |∇u| standard elliptic theory. If we integrate out the resulting inequality on along (β − u) 3 a geodesic from x with u(x) = 4 to x0, we get a bound
β − 3/4 λ1/2 log ≤ C + c d β − 1 (β − 1)1/2
However, if we choose β close to 1, both the left hand side and the right hand side must blow up, which can be utilized to get a lower bound on λ.
In order to make this precise we observe the following. As β goes to 1, our C3,α control on u weakens. Therefore, we do not take the limit but set β − 1 at some small but fixed scale depending on the geometry. This allows us to maintain enough control to use the Bochner identity and use the maximum principle. We needed the
37 C3 estimates to be able to do this but crucially, neither C, c or d depend on the bounds from the Schauder estimates. √ We say that f = F , for convenience. We can rewrite the previous inequality in terms of f
1 1 1 0 ≥ − f 4 − (4(n − 1) − 1) 2 + 2df 2 2(n − 1) 4(n − 1)2 (n − 1) −2Cd8f 3 − 2Cd7f 2 − ((n − 1)K + λ)d4f 2 2 d8 2 − (λu)2 − C2f 2d6 n − 1 (β − u)2 n − 1 d6 −f 2(λu) − f 3Cd5 (β − u) −8f 3d4 − 2(n − 1)(1 + Kρ)f 2d4 + 10f 2d4
1 u We write α = β−1 and observe that α > β−u . We define the following constants:
1 1 A = − , 2(n − 1) 4(n − 1)2
B = 2Cd8 + Cd5 + 8d4,
d6 D = (λ) , (β − 1)
1 D = (4(n − 1) − 1) 2 + 2d + 2Cd7 + ((n − 1)K + λ)d4 (n − 1) 2 + C2d6 + 2(n − 1)(1 + Kρ)d4 + 10d4 n − 1 2λ2 d8 E = n − 1 (β − 1)2
38 From the previous inequality, we have the following estimate:
0 ≥ Af 4 − Bf 3 − Df 2 − Df 2 − E
Note that the calligraphic terms are the only terms where the coefficients aren’t
uniform in β and these both contain a λ. This is important.
Now we use a lemma about the roots of quartics.
4 3 2 Lemma 11. Suppose A1,A2,A3 > 0 and x satisfies P (x) = x −A1x −A2x −A3 ≤ 0. p √ Then x ≤ A1 + A2 + A3.
2 2 Note that P (x) + A3 = x (x − A1x − A2) so it is sufficient to show that P (a) > 0, because increasing x will only increase both factors if the latter is already positive.
Then we have the following:
4 q p P (a) = A1 + A2 + A3 q q p 3 p 2 −A1(A1 + A2 + A3) − A2(A1 + A2 + A3) − A3 2 q p = A1 + A2 + A3 × q q p 2 p (A1 + A2 + A3) − A1(A1 + A2 + A3) − A2 − A3
2 q p = A1 + A2 + A3 × √ q q 2 p 2 p A1 + A2 + A3 + 2A1 A2 + A3 − A1 − A1 A2 + A3) − A2 − A3 q √ q p 2 p = (A1 + A2 + A3) A3 + A1 A2 + A3 − A3 > 0
Thus the lemma is proved. In order to make future calculations more feasible, we note that the following inequality holds: q p 1/2 1/4 A1 + A2 + A3 ≤ A1 + (2A2) + (4A3)
39 This then shows us that
1 √ √ √ f ≤ B + 2 D + D + 2E 1/4 A 1 √ 1 √ √ ≤ B + 2 D + 2 D + 2E 1/4 A A
Thus, after making some of these inequalities less sharp using the fact that n ≥ 2,
we find the following: √ 1 √ λ f ≤ B + 2 D + 8(n − 1) d3 + d2 √ A β − 1
Now, using step 2, we have r1/2 ≥ r3/4 + d1/4 and so in Br3/4 , the following inequality holds:
2 2 2 2 2 (r1/2 − ρ ) ≥ (r1/2 − r3/4) ≥ 3d1/4
Using the definition of f, this implies that in Br3/4 , the following estimate holds:
√ √ |∇u| 1 8(n − 1) 3 2 λ ≥ 2 B + 2 D + 2 d + d √ β − u 3Ad1/4 3d1/4 β − 1 √ 1 8(n−1) 3 2 Setting C = 2 B + 2 D and c = 2 (d + d ), this shows that 3Ad1/4 3d1/4
√ |∇u| λ ≤ C + c√ β − u β − 1
2.7 Step 4: A lower bound on λ
3 We pick x ∈ Br3/4 with u(x) = 4 and a minimal geodesic γ between x and p (recall that p is the point where u(p) = 1). This integral is well defined because u has enough continuity for ∇u to be defined pointwise.
40 We can estimate this integral in the following way. √ ! β − 3/4 Z |∇u| λ log ≤ ≤ C + c√ d β − 1 γ β − u β − 1
Solving for λ, we have the final inequality.
√ p(β − 1) 1 β − 3/4 λ ≥ log − C c d β − 1
To finish this off with a genuine lower bound, we find a particular value for β that
β−3/4 dC+d gives us a positive lower bound. We let x = β−1 and set x = e . This then shows the following:
1 λ ≥ (edC+d − 1)−1 4c2
Thus, we have proven the theorem.
2.8 Future work
Apart from the application to complex geometry, this theorem is of interest in its
own right. We expect the assumptions can be loosened, and that the estimate does
not depend on the C1 norm of the Riemannian curvature, only on the Ricci curvature.
We also expect to be able to remove all the constants that emerge from the embedding theorems. In particular, we put forward the following conjectural estimate.
Conjecture 12. Let (M n, g) be a compact Riemannian manifold. Suppose that
there exists u ∈ C2(M) ∆u + v(∇u) = λu with λ real. Here, v is some one
form satisfying kvk∞ < C. Then there exists some constant δ > 0 depending only
Ric(M), C, diam(M) and n so that λ > δ.
41 2.9 A concrete example
In order to understand how the size of the drift affects the minimal principle
eigenvalue, we calculated the minimal eigenvalue explicitly in a simple case. To do
this, we considered the circle R/4Z, and considered the problem
00 0 x + f · x + λx = 0 with kfkL∞ ≤ C
To simplify the calculation, we set C = 2b. Therefore, we wish to minimize λ under the constraint that kfk∞ ≤ 2b. Our previous work (and symmetry) shows that we should consider the eigenvalue of the Dirichlet problem on the domain [−1, 1]. Using symmetry and the drift ansatz, x will satisfying the following ordinary differential equation:
x00 + 2b · x0 + λx = 0
on the domain [0, 1] under the constraints:
1. x(0) = 1
2. x0(0) = 0
3. x(1) = 0
4. λ is the minimal eigenvalue so that the a solution exists.
We can solve this ordinary differential equation explicitly, and then and then solve for λ in terms of b so that the boundary conditions are satisfied. Doing so, we find the following.
42 For b large, the principle eigenvalue satisfies the following equation:
√ 2 1 + e−2 b −λ p √ 1 − λ/b2 = 1 1 − e−2 b2−λ
For b small, λ satisfies:
√ √ λ − b2 = b tan λ − b2
Note that the minimal principle eigenvalue decreases roughly exponentially as the drift grows.
43 Chapter 3: Moduli of Orthogonal complex structures
One of the central questions in complex geometry is to understand the moduli
space of complex structures on a given manifold. This Kodaira-Spencer theory shows
that the moduli space of complex structures on a given manifold is locally a finite
dimensional analytic manifold. However, this space may have singularities and its
global geometry is not well understood. We do not expect to have a full picture of
the moduli space, except in some special cases.
In order to understand the geometry of the moduli space, one must first try to
understand the singularities. A general strategy might be to try to understand the
complex structures as one approaches a singularity. In all the known examples, these
singularities occur when the Riemannian geometry also becomes singular. As such, we
believe that the Riemannian geometry will be helpful in understanding the geometry
of the moduli space. Given a Riemannian metric g, we consider Mg as the moduli space of complex structures orthogonal to a given metric. If we vary the metric and consider the orthogonal complex structures over each metric, this gives a way to understand the full moduli space.
We start with several test cases, to show what we know and what may be true more generally. Much of what is proposed here are moonshot conjectures that may be true only in special cases.
44 3.1 Two dimensional metrics and Teichmuller theory
We start with the simplest case, which is two dimensional surfaces. For Riemann surfaces, the study of complex structures on a surface is known as Teichmuller theory
[15]. Teichmuller theory is a rich field of study; the moduli space has interesting ge- ometry in its own right. As a more thorough discussion of this topic would lead us far astray from our original project, we will not spend much time discussing Teichmuller theory except to note the following facts.
First, given a Riemannian metric, there is a unique complex structure orthogonal to that metric. Therefore, if we vary through the space of metrics, we will obtain every complex structure. However, this is an unsatisfying answer because given two different
Riemannian metrics, the unique complex structures may be the same. Those familiar with Teichmuller theory will note that the moduli space of complex structures moduli the action of homeomorphisms is finite dimensional whereas the space of Riemannian metrics is infinite-dimensional, so clearly this description contains a lot of redundancy.
To use the Riemannian geometry effectively, we must be able to choose a canonical metric for each complex structure. On a surface, the natural choice of metrics are the ones of constant curvature. In the conformal class of any metric, there is a unique unit volume metric of constant curvature. As a result, for each complex structure, there is a unique metric of constant curvature orthogonal to it. Therefore, there is a bijection between the moduli space of complex structures on a surface and the moduli space of constant curvature metrics on that surface.
This geometry is well understood. Furthermore, the singularities of the moduli space correspond to singularities of the metric. For instance, on the torus one can
45 construct a sequence of flat metrics which collapse. As this happens, the complex
structure similarly becomes singular.
3.2 Higher dimensions
In higher dimensions, the situation is quite a bit more complicated, but one hopes
that the general phenomena holds true.
Conjecture 13. Singularities of the Kodaira-Spencer moduli space correspond to the singularities of the metric.
Unlike the two dimensional case, there are some added complications. Firstly, not all complex structures are K¨ahlerand non-K¨ahlerHermitian metrics are quite a bit more difficult to understand. Secondly, most metrics don’t admit a complex structure at all.
Observation 14. Generically, a Riemmanian metric does not admit an orthogonal
complex structure, and the obstruction is a point-wise generic condition.
This is a consequence of Gray’s identity, which shows that the Weyl tensor must
have small rank for a metric to admit an orthogonal complex structure[11].
However, there can also be a rich moduli of orthogonal complex structure. For
instance, if the metric is Riemannian flat, there is a moduli of SO(2n)/U(n) of K¨ahler
complex structures, and in six dimensions and higher, there are families of non-K¨ahler
structures as well.
3.2.1 Generating orthogonal metrics and canonical metrics
Given a particular complex structure J and an orthogonal metric g, there are two
ways to generate another metric compatible to that complex structure. The first is
46 to consider a conformal metric ef g. The second is to consider another metric of the form g + i∂∂f¯ . Here, f is a smooth function and in the latter case, sufficiently small
in C2 norm so that the new metric is positive-definite. In general, there is an entire
cone of metrics orthogonal to a given complex structure. One of the central projects
of non-K¨ahler complex geometry is trying to find good canonical metrics in this cone.
A large amount of work has been in the search for canonical metrics in complex
geometry. The most famous result in this direction is Yau’s proof of the Calabi
conjecture [28]. In some sense, the work on the non-K¨ahler Calabi Yau manifolds and
papers such as [25] generalize this to the non-K¨ahlercase. The search for canonical
metrics is somewhat outside of the scope of this work, but having canonical metrics
would be very helpful to gain an understanding of the geometry of the moduli space.
Going back to the main goal of this project, we describe the conjectural picture
near flat metrics in the six dimensional case as a road map for what we hope to be
true more generally.
3.3 Flat six dimensional metrics
Given a flat compact six dimensional metric, there is a SO(6)/U(3) family of
K¨ahlerianstructures on it. If we vary the metric by choosing a different lattice in C3 to quotient by, then each lattice admits such a family of K¨ahleriancomplex structures.
As it turns out, this is a full description of the moduli space of K¨ahlercomplex
structures due to the following theorem of Baus and Cortes [4].
Theorem 15. Let X be a compact complex manifold which is aspherical with virtually
solvable fundamental group and assume X supports a K¨ahlermetric. Then X is
biholomorphic to a quotient of Cn by a discrete group of complex isometries.
47 .
Therefore, we can view the moduli space of K¨ahlerstructures on the torus as a
fiber bundle over the moduli space of flat metrics. Over each flat metric, there is a smooth compact fiber of orthogonal complex structures and the singularities of the moduli space correspond to the flat metric collapsing. Even stronger, in the large of a complex tori, all of the complex structures are again complex tori [6]. Therefore,
fiber bundle parametrizes a full connected component of the Kodaira-Spencer moduli space and we cannot deform a complex torus to a non-K¨ahlercomplex manifold.
If the metric is not a product of an elliptic curve and a four-dimensional torus, then all orthogonal complex structures are K¨ahler [17]. However, if the metric is a product of an elliptic curve and a four dimensional torus, then there are non-Kahler structures as well [5]. The moduli of these structures is isomorphic to the space of doubly-periodic maps from the elliptic curve to the projective plane. From this we can see that there are infinitely many connected components (depending on the degree of the map) but that each connected component orthogonal to a given metric is a compact smooth space. Once again, if we vary the elliptic curve or the four dimensional torus, the complex structures are distinct and if we collapse either, the moduli space becomes singular.
The natural conjecture is that, as in the K¨ahleriancase, each of these connected components is actually a connected component of the moduli space of complex struc- tures on the torus. This would mean that there are no hidden complex structures nearby. We do not know of any other complex structures on the topological six-torus and conjecture that the known list of examples is exhaustive. To prove this, one would need to show that there are no “non-flat” complex structures. To do this, it
48 is necessary to understand complex structures on exotic tori, so this seems to be a difficult problem.
3.3.1 More general phenomena
The example of the non-K¨ahler tori, which we call “BSV-tori,” shows two phe- nomena that may hold more generally.
Generically a metric does not admit a complex structure so most Riemannian metrics near flat metrics do not admit an orthogonal complex structure. If a metric has an orthogonal complex structure, generically such a structure is unique [23]. In light of these examples though, it seems likely that the both the number and dimension of complex structures orthogonal to a given metric are upper-semicontinuous functions as one varies the metric. This would mirror the situation in general deformation theory, where the dimension of the space of holomorphic deformations of a fiber is upper semi-continuous in the Zariski topology.
More strongly, a somewhat overoptimistic conjecture is that each connected com- ponent of Mg is compact for any metric g. Going further, it may even be the case that Mg is smooth for all g and only becomes singular as the metric becomes sin- gular. Of course, one must be careful to define what it means for the moduli space to be singular. If the conjectured upper-semicontinuity holds, then there are ways to formulate this. Since not admitting a complex structure is a generic condition, doing so requires some care.
If one were able to find a canonical metric for each complex structure, then one could view the moduli space of complex structures as having a natural surjection to
49 a family of special metrics. Using this, one could gather a better understanding of
the geometry of the moduli space.
3.4 Obstructions
The natural way to start would be to obtain estimates on the torsion in terms of
the Riemannian geometry alone. However, the only estimate we have in full generality
is an L2 estimate on the torsion one-form due to Gauduchon.
Within each conformal class, there is a unique metric which makes the complex structure Gauduchon. With the stronger assumption that the complex structure is
Gauduchon, we have a pointwise estimate on the torsion one-form. However, this is not a strong enough estimate to prove any of the above conjectures. In fact, the example of the BSV-tori show that one can find a complex structure on a flat metric whose torsion tensor is arbitrarily large in L2-norm. If we let f be the doubly periodic
1 map from the elliptic curve to CP and v2 be the volume of the four-torus M2., then we have the following formula [17]:
Z c 2 |T | dv = 32πv2deg(f), (3.1) M Therefore, one can make the L2 norm arbitrarily large by increasing the degree
of the doubly-periodic map. As a result, there is no hope to estimate the torsion
solely from the Riemannian geometry alone. In each connected component of the
moduli space of BSV-tori, the L2-norm of the torsion tensor is constant. Therefore,
the slightly less naive approach would be to try to prove an estimate on the torsion
for complex structures in the same connected component of the moduli space. If,
as for the flat 6-dimensional metrics, each connected component corresponds to a
50 topologically distinct class of maps, then such estimates may be possible within each connected component of the moduli space.
3.5 K¨ahlerand k-Gauduchon complex structures
If we assume that the complex structure is K¨ahler,then it is possible to show that the moduli space of K¨ahlerorthogonal complex structures is compact. It is a straightforward application of the Arzela-Ascoli theorem to show that any sequence of K¨ahlercomplex structures compatible with a given metric must have a convergent sequence, and that the limit is again K¨ahler(since all of the complex structures
L.C. Ji satisfy ∇ Ji ≡ 0). Without too much difficulty, this can be extended to a sequence of Riemannian metrics, so long as the metrics are bounded in C2-sense and are non-collapsing. Since any small deformation of a K¨ahlercomplex structure is again K¨ahler[18], this shows that the singularities of the moduli space occur only when the compatible metrics also become singular. The study of deformation theory is rich and largely outside the scope of this work. However, with the model of K¨ahler structures as a guide, we wish to recreate this analysis in the non-K¨ahlercase.
These results are in stark contrast to the work of Hironaka [14], which showed that the K¨ahlerproperty of compact complex manifolds of complex dimension ≥ 3 is not closed under holomorphic deformations. What is happening here is that the
Riemannian geometry is able to detect the singularities, so even though we may be able to deform to a non-K¨ahler complex structure, we cannot do some without affecting the Riemannian geometry.
51 In the non-K¨ahlercase, the analysis is more difficult and it is not readily appar-
ent what remains true. However, with the additional assumption that the complex
structure is k-Gauduchon, then we can repeat some of this analysis.
From the earlier lemma, if J is k-Gauduchon for some k with n/2 < k < n−1, then
we have a bound on the entire torsion tensor of J, and as such we have bounds on ∇J.
From this, given a sequence of metrics gi whose metric tensors converge to a smooth
metric tensor in C2 norm and whose injectivity radius is uniformly bounded from
below and a sequence of weakly k-Gauduchon complex structures Ji compatible with
1 gi, then this sequence is of Ji is equibounded in C -norm. Therefore, the sequence of
r 1 Ji is compact in C for 2 < r < 1 . By the Arzela-Ascoli theorem, Ji converges along
r a subsequence to some C J∞ compatible to g∞. The formal Nijinhuis tensor must
vanish for J∞, using the results from Denson and Taylor. Appealing to the main
result of their work [13], we know that J∞ is integrable, and so is genuinely complex.
Therefore, the limit of a k-Gauduchon complex structure is a complex structure.
We can further refine our subsequence so that J∞ is Lipschitz, but the complex
structures might not converge in Lipschitz sense. As a result of this, the space of k-
Gauduchon complex structures is precompact inside the space of orthogonal complex
structures, but we are not able to show that it is compact at this time. We wish to
show that k-Gauduchon complex structures are compact, and that we can extract
a sequence that converges to an orthogonal weakly k-Gauduchon complex structure.
However, to do this we would need to extend the definition of k-Gauduchon to Cr
complex structures, and at present there is not a natural way to do so.
52 3.6 Stability under deformations
One of the key results of Kodaira-Spencer theory is that a small deformation of
a K¨ahler complex structure is again K¨ahler[18]. However, for non-K¨ahlercomplex
structures, determining which conditions are stable is often a difficult problem.
It is known that the balanced condition (dωn−1) is not necessarily stable under
deformations [1]. However, if a weaker version of the ∂∂¯ lemma holds, the balanced
condition is stable [3]. Related results are known for strongly Gauduchon complex
structures (∂ωn−1 is ∂¯ exact). Theorem 3.1 of [22] shows that the strongly Gauduchon property of compact complex manifolds is open under holomorphic deformations. It remains an open question as to whether the k-Gauduchon condition is open under deformation. The standard Gauduchon (k = n − 1) condition is open as complex structures are Gauduchon for some compatible metric. However, this is unknown for other k.
It is worth noting that since we are consider complex structures orthogonal to a given metric, we should also address whether orthogonality to a metric is a open or closed condition. It is closed, and not open. For a simple example that shows that it is not open, the deformation of an elliptic curve is not generally compatible with the original metric. Closure can be seen since DJ (X,Y ) = g(X,Y ) − g(JX,JY ) is
continuous in J for any choice of smooth vector fields X and Y .
At this point, we have a preliminary picture. Given a Riemannian manifold
(M n, g), we let { OCS } be the set of orthogonal complex structures. From the
previous results, we have the following inclusions and properties.
53 { K¨ahlerOCS} ⊂ {k-Gauduchon OCS|k = 1, . . . , n − 1}
⊂ { OCS} ⊂ { Orthogonal Almost Complex Structures}
For a compact manifold the orthogonal almost complex structures may form an
infinite dimensional manifold1. The orthogonal complex structures will form a locally
finite dimensional subset of the larger space. The k-Gauduchon complex structures
will form yet another subset of the orthogonal complex structures. There is not
natural inclusion of k-Gauduchon complex structures within each other, and for k =
n − 1 it is possible that the standard Gauduchon orthogonal complex structures are
exactly the same as the orthogonal complex structures, and that both have infinitely
many connected components of arbitrary high dimension. However, for n/2 < k <
n−1, {k-Gauduchon OCS} is precompact in OCS, and we wish to show it is compact
and smooth. Finally, {K¨ahlerOCS} is a subset of all of the {k-Gauduchon OCS}, and is compact.
1For a given Riemannian manifold, the set of orthogonal almost complex structures can also be empty.
54 Bibliography
[1] Lucia Alessandrini and Giovanni Bassanelli. Small deformations of a class of com- pact non-k¨ahlermanifolds. Proceedings of the American Mathematical Society, 109(4):1059–1062, 1990.
[2] Adrian Andrada, Maria Laura Barberis, and Isabel Dotti. Complex connections with trivial holonomy. Lie Groups: Structure, Actions, and Representations, pages 25–39, 2013.
[3] Daniele Angella and Luis Ugarte. On small deformations of balanced manifolds. Differential Geometry and its Applications, 54:464–474, 2017.
[4] Oliver Baues and Vicente Cort´es. Aspherical k¨ahlermanifolds with solvable fundamental group. Geometriae Dedicata, 122(1), 2006.
[5] Lev Borison, Simon Salamon, and Jeff Viaklovsky. Twistor geometry and warped product orthogonal complex structures. Duke Mathematical Journal, 156(1):125– 166, 2011.
[6] Fabrizio Catanese. Deformation types of real and complex manifolds. Contem- porary trends in algebraic geometry and algebraic topology, pages 195–238, 2002.
[7] Ana Cristina Ferreira and Isabel Salavessa. Dirichlet principal eigenvalue com- parison theorems in geometry with torsion. Journal of Mathematical Analysis and Applications, 453(2):700–723, 2017.
[8] Jixiang Fu, Zhizhang Wang, and Damin Wu. Semilinear equations, the gammak function, and generalized gauduchon metrics. Journal of the European Mathe- matical Society, 15(2):659–680, 2013.
[9] Paul Gauduchon. Structures complexes sur une variet´econforme de type nega- tive. Complex Analysis and Geometry, Lect. Notes Pure Appl. Math, 173, 1995.
[10] David Gilbarg and Neil S Trudinger. Elliptic partial differential equations of second order. Springer, 1983.
55 [11] Alfred Gray. Curvature identities for hermitian and almost hermitian manifolds. Tohoku Mathematical Journal, Second Series, 28(4):601–612, 1976.
[12] Fran¸coisHamel, Nikolai Nadirashvili, and Emmanuel Russ. An isoperimetric inequality for the principal eigenvalue of the laplacian with drift. Comptes Rendus Mathematique, 340(5):347–352, 2005.
[13] C Denson Hill and Michael Taylor. Integrability of rough almost complex struc- tures. The Journal of Geometric Analysis, 13(1):163–172, 2003.
[14] Heisuke Hironaka. An example of a non-kahlerian complex-analytic deformation of kahlerian complex structures. Annals of Mathematics, pages 190–208, 1962.
[15] John H. Hubbard. Teichm¨uller theory and applications to geometry, topology, and dynamics. Matrix Editions, 2016.
[16] Gabriel Khan. Eigenvalues of the complex laplacian on compact non-k¨ahler manifolds. Annals of Global Analysis and Geometry, pages 1–17, 2016.
[17] Gabriel Khan, Bo Yang, and Fangyang Zheng. The set of all orthogonal complex structures on the flat 6-tori. Advances in Mathematics, 319:451–471, 2017.
[18] Kunihiko Kodaira and Donald Clayton Spencer. On deformations of complex analytic structures, iii. stability theorems for complex structures. Annals of Mathematics, pages 43–76, 1960.
[19] Bernard Malgrange. Sur l’int´egrabilit´e des structures presque-complexes. S´eminaire Jean Leray, pages 1–8, 1969.
[20] Koji Matsuo and Takao Takahashi. On compact astheno-k¨ahlermanifolds. Col- loquium Mathematicum, 89(213–221), 2001.
[21] August Newlander and Louis Nirenberg. Complex analytic coordinates in almost complex manifolds. Annals of Mathematics, pages 391–404, 1957.
[22] Dan Popovici. Stability of strongly gauduchon manifolds under modifications. Journal of Geometric Analysis, 23(2):653–659, 2013.
[23] Simon Salamon. Orthogonal complex structures. Proceedings of the 6th Inter- national Conference on Differential Geometry, Brno, pages 103–117, 1995.
[24] Richard Schoen and Shing-tung Yau. Lectures on differential geometry, vol. 1 of conference proceedings and lecture notes in geometry and topology. 1994.
[25] G´abor Sz´ekelyhidi, Valentino Tosatti, and Ben Weinkove. Gauduchon metrics with prescribed volume form. Acta Mathematica, 219(1):181–211, 2017.
56 [26] Yu Wang and Xiangwen Zhang. An alexandroff–bakelman–pucci estimate on riemannian manifolds. Advances in Mathematics, 232(1):499–512, 2013.
[27] Bo Yang and Fangyang Zheng. On curvature tensors of hermitian manifolds. arxiv preprint, 2016.
[28] Shing-Tung Yau. On the ricci curvature of a compact k¨ahlermanifold and the complex monge-amp´ereequation, i. Comm. Pure Appl. Math., 31(3):339–411, 1978.
57