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On curvature, volume growth and uniqueness of steady Ricci solitons Cui, Xin 2016

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Xin Cui

A Dissertation Presented to the Graduate Committee of Lehigh University in Candidacy for the Degree of Doctor of Philosophy in Mathematics

Lehigh University May 2016 Copyright Xin Cui

ii Approved and recommended for acceptance as a dissertation in partial fulﬁll- ment of the requirements for the degree of Doctor of Philosophy.

Xin Cui On curvature, volume growth and uniqueness of steady Ricci solitons.

Date

Prof. Huai-Dong Cao, Dissertation Director, Chair

Accepted Date

Committee Members

Xiaofeng Sun

Terrence Napier

Justin Corvino

iii Acknowledgments

I would like to express my sincere gratitude to my advisor, Professor Huai- Dong Cao. This work could not have been done without his signiﬁcant inspiration, constant guidance and support. Many thanks to Professor Xiaofeng Sun, Professor Terrence Napier, Professor Justin Corvino, Professor Don Davis, Professor David Johnson, Professor Joseph Yukich and Professor Wei-Min Huang for their generous helping during my study at Lehigh University. I also would like to thank my friends Meng Zhu, Qiang Chen,Yingying Zhang, Yashan Zhang, Jianyu Ou, Chih-Wei Chen and Zhuhong Zhang. Also thank my college classmates, Qiang Guang, Xing Wang, Zhilin Ye, Hang Xu, Shi Wang for years of discussion on mathematics. There are many people whom I wish to thank. In particular, I want to thank Mary Ann Dent and Mary Ann Haller. They are always there whenever I meet a problem. Their help reminds me the feeling of a family. I would like to thank Mr. Dale S. Strohl ’58. His generous gift to Lehigh University granted me fellowship scholarship two times. I also would like to show my respect to Mr. Robert E. Zoellner ’54. I would like to especially mention that the door is always open at the Zoellner art center, I was able to practice piano at the center although I am not an music major student. Finally and most importantly, I also would like to thank my parents, Yuanshan Cui and Qiuhua Piao. My father had a dream to be a mathematician and he tried even harder than I do. My mother created chances for me to touch various form of art and never force me to study. Eventually I picked piano by myself. I deeply thank them for their open-mindedness and courage.

iv Contents

Abstract 1

1 Preliminary 2 1.1 Deﬁnition of Ricci solitons ...... 2 1.2 Curvature equations and inequalities ...... 3 1.3 Basic properties of solitons ...... 5 1.4 Some examples of steady solitons ...... 5

2 Curvature estimates for four- dimensional steady solitons 6 2.1 Background ...... 6 2.2 Main Results ...... 8 2.3 Preliminaries ...... 9 2.4 Case 1: steady soliton with Ric > 0...... 10

2.5 Case 2: steady soliton with limx→∞ R(x) = 0 ...... 14

3 Curvature and volume growth of steady K¨ahlerRicci solitons 23 3.1 Background ...... 23 3.2 Volume growth ...... 24 3.3 Curvature estimates ...... 27

4 Uniqueness under constraints of the asymptotic geometry. 32 4.1 Background ...... 32 4.2 Main Theorem ...... 33 4.3 Preliminary ...... 33

v 4.4 Calculations ...... 34 4.4.1 Killing vectors of the model metric ...... 34 4.4.2 Shifting preserves the assumption 1 ...... 37 4.4.3 Rigidity of the soliton vector ...... 39 4.4.4 Decay rate of Ricci of the model metric ...... 41 4.4.5 Main Argument ...... 43

Bibliography 45

Vita 52

vi Abstract

This thesis contains my work during my Ph.D. studies at Lehigh University under the guidance of my advisor Huai-Dong Cao. The work is related to objects called Ricci solitons which serve as singularity models of Ricci ﬂow. We are going to study Ricci solitons in this thesis from the following aspects:

1. Curvature properties.

2. Volume growth properties.

3. Uniqueness under constraints of the asymptotic geometry.

We ﬁrst explore the curvature estimate for four dimensional steady Ricci solitons. The main result is about control of the full curvature tensor Rm by scalar curvature R. We are then going to study curvature and volume growth properties of complete steady Kahler Ricci solitons with positive Ricci curvature. The main result is that 1 volume growth is at least half dimensional and scalar curvature behaves like r in average where r is the geodesic distance to some point. In the third part, we are going to study the uniqueness of the steady Kahler Ricci soliton constructed by Huai-Dong Cao under constraints of the asymptotic geometry. The main result says that it is unique if we ask that the metric tensor be C1 close in some sense to the model.

1 Chapter 1

Preliminary

1.1 Deﬁnition of Ricci solitons

The Ricci ﬂow is a geometric PDE introduced by R. Hamilton in 1982 [37]. It is a nonlinear weakly parabolic system which evolves the metric tensor by its Ricci tensor,

∂ g = −2R . ∂t ij ij It is a powerful tool in the study of the geometry of the underlying manifold where this PDE system evolves. For example, it is the primary tool used in G. Perelman’s solution of the Poincaréconjecture[51]. It has also been applied by R. Schoen and S. Brendle[5] in the proof of the differentiable sphere theorem. Singularity analysis is one of the main parts of studying the Ricci flow. Self similar solutions, called Ricci solitons arise during singularity analysis.

Deﬁnition 1.1.1. A complete Riemannian manifold (M, g) is called a Ricci soliton, if there exists a complete vector ﬁeld V , such that

1 R + L g = λg ij 2 V ij ij for some constant λ ∈ R.

2 Based on the sign of λ, they divide into three types, namely shrinking (λ > 0), steady (λ = 0) and expanding (λ < 0).

Moreover, if V is a gradient vector ﬁeld, i.e., V = ∇f, then we say it is a gradient Ricci soliton with potential function f. In the these we are going to focus on gradient steady solitons (V = ∇f, λ = 0)

Rij + ∇i∇jf = 0.

Ricci solitons are natural generalization of Einstein manifolds (V = 0). The self similar solution generated by a Ricci soliton often appears as a singularity model, i.e., the parabolic dilation limit of Ricci ﬂow near a singularity. Therefore, the structure of Ricci soliton helps us know more about the Ricci ﬂow near itsp singularity.

1.2 Curvature equations and inequalities

n Lemma 1.2.1. (Hamilton [39]) Let (M , gij, f) be a complete gradient steady soliton satisfying Eq. (1.1). Then

R = −∆f, (1.1)

∇iR = 2Rij∇jf, (1.2)

2 R + |∇f| = C0 (1.3)

∇lRijkl = Rijkl∇lf (1.4)

We also collect several equations and inequalities of R, Ric and Rm (cf. [39],[54]).

n Lemma 1.2.2. Let (M , gij, f) be a complete gradient steady soliton satisfying Eq.

3 (1.1). Then, we have

2 ∆f R = −2|Ric| ,

∆f Ric = −2RijklRjl,

∆f Rm = Rm ∗ Rm,

where ∗ means linear combinations of contractions between tensors, ∆f is the f- Laplacian operatorp ∆ − ∇f · ∇.

n Lemma 1.2.3. Let (M , gij, f) be a complete gradient steady soliton satisfying Eq. (1.1). Then

2 2 2 ∆f |Ric| ≥ 2|∇Ric| − 4|Rm||Ric| , 2 ∆f |Rm| ≥ −c|Rm| , 2 2 3 ∆f |Rm| ≥ 2|∇Rm| − C|Rm| .

Here c > 0 is some universal constant depending only on the dimension n.

Remark 1.2.1. To derive the second diﬀerential inequality, one needs to use the Kato inequality |∇|Rm|| ≤ |∇Rm| as shown in [45]. To get nonnegativity of scalar curvature, we will need the following useful result by B.-L. Chen [22].

Proposition 1.2.1. (B.-L Chen [22]) Let gij(t) be a complete ancient solution to n the Ricci ﬂow on a noncompact manifold M . Then the scalar curvature R of gij(t) is nonnegative for all t.

Since gradient steady solitons generate self silimar solutions which are not just ancient but eternal, we have,

n Lemma 1.2.4. Let (M , gij, f) be a complete gradient steady soliton. Then it has nonnegative scalar curvature R ≥ 0.

n Remark 1.2.2. In fact, by Proposition 3.2 in [54], either R > 0 or (M , gij) is Ricci ﬂat.

4 1.3 Basic properties of solitons

Let us compare some previous results for complete gradient shrinking and steady Ricci solitons.

Properties Shrinking solitons Steady solitons (1) If Rc > 0, R attains√ maximum, Potential function f ∼ 1 r2, [15] c r − c ≤ −f ≤ R r + c , [16] 4 1 2 max√ 3 growth (2)infy∈∂Br(x) f(y) ∼ − Λr, [63] (1)V ol(B ) ≤ crn, [15] r √ a r Volume growth (2)V ol(Br) ≥ cr, [46] (1)c · r ≤ V ol(Bp(r)) ≤ c · e , [44] (3)If Rc ≥ 0, then (2) If f satisﬁes a uniform condition, V ol(Br) n limr→∞ rn = 0, [48] V ol(Br) ≤ r . [61]

1.4 Some examples of steady solitons

Steady solitons arise as certain Type II singularity models of the Ricci ﬂow. Recall a gradient steady Ricci soliton satisﬁes

Rij + fij = 0.

Therefore Ricci ﬂat spaces are steady solitons if we pick our potential function f to be 0. Indeed, by an argument of Hamilton [39], a compact steady soliton has to be trivial (Ricci ﬂat). Therefore a nontrivial steady soliton is noncompact. Many people have constructed nontrivial steady solitons and we list some of them.

Space R2 Rn(n ≥ 3) Cn(n ≥ 2) Metric Ansatz SO(2) or U(1) SO(n) U(n) Potential function f = − ln cosh(r) f ∼ −cr f ∼ −cr n+1 n Volume Growth V ol(Br) ∼ r V ol(Br) ∼ r 2 V ol(Br) ∼ r Curvature R(g) > 0, sec(g) > 0, sec(g) > 0, −2r 1 1 R = O(e ) R = O( r ) R = O( r ) Found by R. Hamilton, [38] R. Bryant, [8] H.-D. Cao, [11]

5 Chapter 2

Curvature estimates for four- dimensional steady solitons

2.1 Background

n A complete Riemannian metric gij on a smooth manifold M is called a gradient steady Ricci soliton if there exists a smooth function f on M n such that the Ricci tensor Rij of the metric gij satisﬁes the equation

Rij + ∇i∇jf = 0. (2.1)

The function f is called a potential function of the gradient steady soliton. Clearly, n when f is a constant the gradient steady Ricci soliton (M , gij, f) is simply a Ricci- flat manifold. Gradient steady solitons play an important role in Hamilton’s Ricci flow, as they correspond to translating solutions, and often arise as Type II singularity models. Thus one is interested in possibly classifying them or understanding their geometry. It turns out that compact steady solitons must be Ricci-flat. In dimension n = 2, Hamilton [36] discovered the first example of a complete noncompact gradient steady

6 soliton on R2, called the cigar soliton, where the metric is given by

dx2 + dy2 ds2 = . 1 + x2 + y2

The cigar soliton has potential function f = − log(1 + x2 + y2), positive curvature R = 4ef , and is asymptotic to a cylinder at infinity. Furthermore, Hamilton [36] showed that the only complete steady soliton on a two-dimensional manifold with bounded (scalar) curvature R which assumes its maximum at an origin is, up to scaling, the cigar soliton. For n ≥ 3, Bryant [10] proved that there exists, up to scaling, a unique complete rotationally symmetric gradient Ricci soliton on Rn; see, e.g., Chow et al. [28] for details. The Bryant soliton has positive sectional curvature, linear curvature decay and volume growth of geodesic balls B(0, r) on the order of r(n+1)/2. In the Kähler case, Cao [11] constructed a complete U(m)-invariant gradient steady Kähler-Riccisoliton on Cm, for m ≥ 2, with positive sectional curvature. It has volume growth on the order of rm and also linear curvature decay. Note that in each of these three examples, the maximum of the scalar curvature is attained at the origin. One can find additional examples of steady solitons, e.g., in [41, 43, 30, 31, 3] etc; see also [14] and the references therein. In dimension n = 3, Perelman [52] claimed that the Bryant soliton is the only complete noncompact, κ-noncollapsed, gradient steady soliton with positive sectional curvature. Recently, Brendle has affirmed this conjecture of Perelman (see [6]; and also [7] for an extension to the higher dimensional case). On the other hand, for n ≥ 4, Cao-Chen [16] and Catino-Mantegazza [20] proved independently, and using different methods, that any n-dimensional complete noncompact locally conformally n flat gradient steady Ricci soliton (M , gij, f) is either flat or isometric to the Bryant soliton (the method of Cao-Chen [16] also applies to the case of dimension n = 3). In addition, Bach-flat gradient steady solitons (with positive Ricci curvature) for all n ≥ 3 [19] and half-conformally flat ones for n = 4 [25] have been classified respectively. Inspired by the very recent work of Munteanu-Wang [45], in [17] we studied curvature estimates of four-dimensional complete noncompact gradient steady solitons.

7 In [45], Munteanu and Wang made an important observation that the curvature 4 tensor of a four-dimensional gradient Ricci soliton (M , gij, f) can be estimated in terms of the potential function f, the Ricci tensor and its ﬁrst derivates. In addition, the (optimal) asymptotic quadratic growth property of the potential function f proved in [15], as well as a key scalar curvature lower bound R ≥ c/f shown in [29] are crucial in their work. Even though gradient steady Ricci solitons in general don’t share these two special features (cf. [63, 44, 61] and [29, 34]), some of the arguments in [45] can still be adapted to prove certain curvature estimates for two classes of gradient steady solitons.

2.2 Main Results

4 Theorem 2.2.1. Let (M , gij, f) be a complete noncompact 4-dimensional gradient steady Ricci soliton with positive Ricci curvature Ric > 0 such that the scalar cur- 4 4 vature R attains its maximum at some point x0 ∈ M . Then, (M , gij) has bounded Riemann curvature tensor, i.e.

sup |Rm| ≤ C x∈M for some constant C > 0. If in addition R has at most linear decay, then

|Rm| sup ≤ C. x∈M R

4 Theorem 2.2.2. Let (M , gij, f), which is not Ricci-ﬂat, be a complete noncompact

4-dimensional gradient steady Ricci soliton. If limx→∞ R(x) = 0, then, for each 0 < a < 1, there exists a constant C > 0 such that

|Ric|2 ≤ CRa and sup |Rm| ≤ C. x∈M

Suppose in addition R has at most polynomial decay. Then, for each 0 < a < 1,

8 there exists a constant C > 0 such that

|Rm|2 ≤ CRa.

2.3 Preliminaries

It follows from Remark 1.2.2 that the constant C0 in (1.3) is positive whenever f is a non-constant function (i.e., the steady soliton is non-trivial). By a suitable scaling of the metric gij, we can normalize C0 = 1 so that

R + |∇f|2 = 1. (2.2)

In the rest of this chapter, we shall always assume this normalization (2.4). Combining (2.1) and (2.4), we obtain −∆f + |∇f|2 = 1. Thus, setting F = −f,

∆f F = 1. (2.3)

For gradient steady solitons with positive Ricci curvature Ric > 0,

n Proposition 2.3.1. (Cao-Chen [16]) Let (M , gij, f) be a complete noncompact gradient steady soliton with positive Ricci curvature Ric > 0 such that the scalar n curvature R attains its maximum Rmax = 1 at some point x0 ∈ M . Then, there exist some constants 0 < c1 ≤ 1 and c2 > 0 such that F = −f satisﬁes the estimates

c1r(x) − c2 ≤ F (x) ≤ r(x) + |F (x0)|, (2.4)

where r(x) = d(x0, x) is the distance function from x0.

Remark 2.3.1. In (2.4), only the lower bound on F requires the assumptions on Ric and R. Note that, under the assumption in Proposition 2.3.1, F (x) is proportional to the distance function r(x) = d(x0, x) from above and below. Throughout the

9 chapter, we denote

D(t) = {x ∈ M : F (x) ≤ t},

B(t) = B(x0, t) = {x ∈ M : d(x0, x) ≤ t}.

2.4 Case 1: steady soliton with Ric > 0

First of all, we need the following key fact, valid for 4-dimensional gradient steady Ricci solitons in general, due to Munteanu and Wang [45].

4 Lemma 2.4.1. (Munteanu-Wang [45]) Let (M , gij, f) be a complete noncompact gradient steady soliton satisfying (1.1). Then there exists some universal constant c > 0 such that |∇Ric| |Ric|2 |Rm| ≤ c( + + |Ric|). |∇f| |∇f|2 Proof. This follows from the same arguments as in the proof of Proposition 1.1 of [45], but without replacing |∇f|2 by f in their argument.

4 Proposition 2.4.1. Let (M , gij, f) be a complete noncompact gradient steady soliton with positive Ricci curvature such that R attains a maximum. Then, there exists some constant C > 0, depending on the constant c1 in (2.7), such that outside a compact set, |Rm| ≤ C(|∇Ric| + |Ric|2 + |Ric|).

Proof. This easily follows from Lemma 2.4.1 and the following fact shown by Cao- Chen [16]: 2 |∇f| ≥ c1 > 0. (2.5)

Remark 2.4.1. Note that, combining (2.5) with (2.2) and (2.3), we have

2 2 0 < c1 ≤ |∇F | = |∇f| ≤ 1. (2.6)

10 Now we are ready to prove our ﬁrst main result.

4 Theorem 2.4.1. Let (M , gij, f) be a complete noncompact gradient steady soliton with positive Ricci curvature Ric > 0 such that R attains its maximum at some 4 point x0 ∈ M . Then, there exists some constant C > 0, depending on c1 in (2.5), such that sup |Rm| ≤ C. x∈M Proof. First of all, from (2.4), we have R ≤ 1. Hence, since Ric > 0, it follows that

0 < |Ric| ≤ R ≤ 1. (2.7)

Thus, by Proposition 2.4.1 and (2.7), we see that

1 1 |∇Ric|2 ≥ |Rm|2 − (|Ric|2 + |Ric|)2 ≥ |Rm|2 − 4. (2.8) 2C2 2C2

Using the ﬁrst two inequalities in Lemma 1.2.3, we obtain

2 2 2 2 ∆f (|Rm| + λ|Ric| ) ≥ −C|Rm| + 2λ(|∇Ric| − 2|Rm||Ric| ). (2.9)

By (2.8), (2.9), and picking constant λ > 0 suﬃciently large (depending on the constant C in Proposition 2.4.1, hence on c1), it follows that

2 2 0 2 2 ∆f (|Rm| + λ|Ric| ) ≥ 2|Rm| − 4λ|Rm| − C ≥ (|Rm| + λ|Ric| ) − C. (2.10)

Next, let ϕ(t) be a smooth function on R+ so that 0 ≤ ϕ(t) ≤ 1, ϕ(t) = 1 for

0 ≤ t ≤ R0, ϕ(t) = 0 for t ≥ 2R0, and

t2 |ϕ0(t)|2 + |ϕ00(t)| ≤ c (2.11)

for some universal constant c and R0 > 0 arbitrary large. We now take ϕ = ϕ(F (x)) as a cut-oﬀ function with support in D(2R0). Note that

0 c 0 00 2 c |∇ϕ| = |ϕ ||∇F | ≤ and |∆f ϕ| ≤ |ϕ ∆f F | + |ϕ ||∇F | ≤ (2.12) R0 R 0

11 on D(2R0) \ D(R0) for some universal constant c. Setting u = |Rm| + λ|Ric|2 and G = ϕ2u, then direct computations, (2.10) and (2.12) yield

2 4 2 2 2 2 ϕ ∆f G = ϕ ∆f u + ϕ u∆f (ϕ ) + 2ϕ ∇u · ∇ϕ 4 2 2 2 2 2 ≥ ϕ u − C + ϕ u 2ϕ∆f ϕ + 2|∇ϕ| + 2∇G · ∇ϕ − 8|∇ϕ| G ≥ G2 + 2∇G · ∇ϕ2 − CG − C.

Now it follows from the maximum principle that G ≤ C on D(2R0) by some constant 2 C > 0 depending on c1 but independent of R0. Hence u = |Rm| + λ|Ric| ≤ C on

D(R0). Since R0 > 0 is arbitrary large, we see that

sup |Rm| ≤ sup |Rm| + λ|Ric|2 ≤ C. x∈M x∈M

This completes the proof of Theorem 2.4.1.

4 Proposition 2.4.2. Let (M , gij, f) be a complete noncompact gradient steady soli- 4 ton with positive Ricci curvature Ric > 0 and R attains its maximum at x0 ∈ M . |Rm|+λ|Ric|2 Then the function u = R , with λ > 0 sufficiently large, satisfies the differential inequality 2 ∆f u ≥ u R − CR − 2∇u · ∇(log R) for some constant C > 0 outside a compact set.

Proof. First of all, by an argument similar to that of (2.8)-(2.10) in the proof of Theorem 2.4.1, by choosing λ suﬃciently large we have

2 2 2 2 4 4 2 ∆f (|Rm| + λ|Ric| ) ≥ (|Rm| + λ|Ric| ) − 4λ |Ric| − λ(|Ric| + |Ric| ) ≥ (|Rm| + λ|Ric|2)2 − C|Ric|2 for some constant C > 0. Here we have also used the fact (2.8).

12 Thus, by a direct computation,

−1 2 −1 −1 ∆f u = R ∆f (|Rm| + λ|Ric| ) + (uR)∆f (R ) + 2∇(uR) · ∇(R ) (|Rm| + λ|Ric|2)2 − C|Ric|2 |Ric|2 |∇R|2 ≥ + (uR) 2 + 2 R R2 R3 2 − u|∇R|2 + R∇u · ∇R R2 ≥ Ru2 − CR − 2∇u · ∇ log R.

4 Theorem 2.4.2. Let (M , gij, f) be a complete noncompact gradient steady Ricci soliton with Ric > 0 such that R attains its maximum. Suppose R has at most linear decay, i.e. for some c > 0, R(x) ≥ c/r(x), outside a compact set. Then

|Rm| sup ≤ C. x∈M R

|Rm|+λ|Ric|2 Proof. Fix λ suﬃcient large so that Proposition 2.4.2 holds and set u = R . + d−t Next, let ϕ(t) be a Lipschitz function on R so that ϕ(t) = d for 0 ≤ t ≤ d, ϕ(t) = 0 for t ≥ d. Let ϕ = ϕ(F ) and G = ϕ2u. Then on D˚(d) \ D(1) ϕ satisﬁes,

1 |∇ϕ| = |ϕ0∇F | ≤ , d 1 ∆ ϕ = ϕ0∆ F = − . f f d

Then outside D(1), we have,

2 4 2 2 2 2 ϕ 4f (G) = ϕ (4f u) + ϕ u(4f ϕ ) + 2ϕ ∇ϕ · ∇u ≥ ϕ4 Ru2 − cR − 2∇u · ∇ log R 2 2 2 +2 ϕ4f ϕ + |∇ϕ| G + 2ϕ ∇ϕ · ∇u ≥ RG2 − cR + 4ϕ (∇ϕ · ∇ log R) G 8 − G + 2∇ϕ2 − 2ϕ2∇ log R · ∇G. d

13 Ric(∇f) Now by Lemma 1.2.3 and Ric > 0, we have |∇ log R| = 2| R | ≤ 2. Also, when a R has at most linear decay outside some D(t0) and for d > t0, we have R ≥ d in D(d)\D(t0) for some constant a > 0 independent d. Therefore there exists c independent of d such that on D˚(d) \ D(1), following inequalities holds,

c ϕ24 (G) ≥ RG2 − cR − G + 2∇ϕ2 − 2ϕ2∇ log R · ∇G f d 1 ≥ RG2 − cR + 2∇ϕ2 − 2ϕ2∇ log R · ∇G. 2

Recall u > 0, therefore the maximum of Gd must attains in the interior of D(d). Then it follows from maximum principle argument that u ≤ C on M 4, hence |Rm| ≤ CR on M 4.

2.5 Case 2: steady soliton with limx→∞ R(x) = 0

In this section, we prove our second main result, Theorem 2.2.2. Throughout the sec- 4 tion we assume (M , gij, f) is a complete noncompact, non Ricci-ﬂat, 4-dimensional gradient steady Ricci soliton such that

lim R(x) = 0. (2.13) x→∞

4 Note that, by Remark 1.2.2, (M , gij, f) necessarily satisﬁes R > 0. First of all, we need the following useful Laplacian comparison type result for gradient Ricci solitons.

n Lemma 2.5.1. Let (M , gij, f) be any gradient steady Ricci soliton and let r(x) = n d(x0, x) denote the distance function on M from a ﬁxed base point x0. Suppose that

Ric ≤ (n − 1)K

on the geodesic ball B(x0, r0) for some constants r0 > 0 and K > 0. Then, for any

14 n x ∈ M \ B(x0, r0), we have

2 ∆ r(x) ≤ (n − 1) Kr + r−1 . f 3 0 0

Remark 2.5.1. Lemma 2.5.1 is a special case of a more general result valid for solutions to the Ricci ﬂow due to Perelman [51], see, e.g., Lemma 3.4.1 in [18]. Also see [33] and [62] for a diﬀerent version.

4 Theorem 2.5.1. Let (M , gij, f) be a complete noncompact gradient steady Ricci soliton which is not Ricci-ﬂat,. If limx→∞ R(x) = 0, then, for each 0 < a < 1, there exists a constant C > 0 such that

sup |Ric|2 ≤ CRa and sup |Rm| ≤ C. x∈M x∈M

Proof. The proof is similar to that of Munteanu-Wang [45] except we need to use the distance function to cut-oﬀ rather than the potential function since the potential function may not be proper.

Since limx→∞ R(x) = 0, it follows from (2.4) that

|∇f| ≥ c1 > 0

for some 0 < c1 < 1 outside a compact set. By Lemma 1.2.1 and Lemma 2.4.1, we have

2 2 2 ∆f |Ric| ≥ 2|∇Ric| − C|Rm||Ric| ≥ 2|∇Ric|2 − C |∇Ric| + |Ric|2 + |Ric| |Ric|2.

Also, since R > 0 on M 4, by using the ﬁrst identity in Lemma 2.3 we have

1 |Ric|2 |∇R|2 ∆ = 2a + a(a + 1) . f Ra Ra+1 Ra+2

15 Hence,

|Ric|2 ∆ |Ric|2 1 1 ∆ = f + |Ric|2∆ + 2∇|Ric|2 · ∇ f Ra Ra f Ra Ra 2|∇Ric|2 (|∇Ric| + |Ric|2 + |Ric|) |Ric|2 ≥ − C Ra Ra |Ric|2 |∇R|2 |Ric| |∇|Ric|| |∇R| +|Ric|2 2a + a(a + 1) − 4a Ra+1 Ra+2 Ra+1

Apply Cauchy’s inequality to the last term

|Ric| |∇|Ric|| |∇R| |Ric| |∇Ric| |∇R| −4a ≥ −4a Ra+1 Ra+1 |Ric|2|∇R|2 4a |∇Ric|2 ≥ −a(a + 1) − . Ra+2 a + 1 Ra

Thus, we have

2(1 − a) |∇Ric|2 |∇Ric||Ric|2 ∆ |Ric|2R−a ≥ − C f 1 + a Ra Ra |Ric|4 + |Ric|3 |Ric|4 −C + 2a Ra Ra+1 CR |Ric|4 |Ric|3 ≥ (2a − ) − C . 1 − a Ra+1 Ra

|Ric|2 Therefore, for u = Ra , we have derived the diﬀerential inequality

CR ∆ u ≥ (2a − )u2Ra−1 − Cu3/2Ra/2. (2.14) f 1 − a

Since R → 0, for any 0 < a < 1, we can choose a ﬁxed d0 > 0 depending on a and suﬃciently large so that

CR (2a − ) ≥ a (2.15) 1 − a outside the geodesic ball B(x0, d0).

Next, for any D0 > 2d0, we choose a function ϕ(t) as follows: 0 ≤ ϕ(t) ≤ 1 is a

16 smooth function on R such that   1, 2d0 ≤ t ≤ D0, ϕ(t) =   0, t ≤ d0 or t ≥ 2D0.

Also,

2 00 0 c t |ϕ (t)| ≤ c and 0 ≥ ϕ (t) ≥ − , if 2d0 ≤ t ≤ 2D0. (2.16) D0

Now we use ϕ = ϕ(r(x)) as a cut-oﬀ function whose support is in B(x0, 2D0) \

B(x0, d0). Note that by lemma 2.5.1, we get

2 0 2 c 0 00 C |∇ϕ| = |ϕ | ≤ 2 and ∆f ϕ = ϕ ∆f r(x) + ϕ ≥ − . (2.17) D0 D0 on B(x0, 2D0) \ B(x0, 2d0) respectively. Setting G = ϕ2u, then by our choice of ϕ and (2.17), we see that

2 4 2 2 2 2 ϕ ∆f G = ϕ ∆f u + ϕ u∆f ϕ + 2ϕ (∇u · ∇ϕ ) 4 2 a−1 3/2 a/2 2 2 2 2 ≥ ϕ au R − Cu R + 2ϕ u(∆f ϕ ) − 8|∇ϕ| G + 2∇G · ∇ϕ ≥ aG2Ra−1 − CG3/2Ra/2 − CG + 2∇G · ∇ϕ2.

Assume G achieves its maximum at some point p ∈ B(x0, 2D0). If p ∈ B(x0, 2D0)\

B(x0, 2d0), then it follows from the maximum principle that

0 ≥ aG2(p)Ra−1(p) − CG3/2(p)Ra/2(p) − CG(p).

On the other hand, noticing that the fact 0 < a < 1 and R uniformly bounded from above, implies. G(p) ≤ C for some constant C depending on a but independent of D0.

17 Thus, max u ≤ max G ≤ max C, max u ≤ C0 B(x0,D0) B(x0,2D0) B(2d0)

0 2 a 4 for some C > 0 indepedent of D0. Therefore |Ric| ≤ CR on M . 4 It remains to show |Rm| ≤ C on M . However, once we know supx∈M Ric ≤ C, |Rm| ≤ C follows essentially from the same argument as in the proof of Theorem 2.4.1. We leave the details to the reader.

4 Lemma 2.5.2. Let (M , gij, f), which is not Ricci-ﬂat, be a complete noncompact gradient steady Ricci soliton with limx→∞ R(x) = 0. Then for each 0 < a < 1 and µ > 0, there exist constants λ > 0 and D > 0 so that function

|Rm|2 + λ|Ric|2 v = Ra satisﬁes the diﬀerential inequality

∆f v ≥ µv − D.

Proof. By Lemma 1.2.2 and Theorem 2.5.1,

∆ (|Rm|2 + λ|Ric|2) 1 ∆ v = f + vRa∆ ( ) + 2∇(vRa) · ∇(R−a) f Ra f Ra 2|∇Rm|2 + 2λ|∇Ric|2 |Rm|2 + λ|Ric|2 ≥ − c Ra Ra 4 R |∇R|2 +(|Rm|2 + λ|Ric|2) −a f + a(a + 1) Ra+1 Ra+2 |Rm||∇Rm||∇R| |Ric||∇Ric||∇R| −4a − 4aλ . Ra+1 Ra+1

By applying Cauchy’s inequality to terms with |∇R|,

18 2|∇Rm|2 + 2λ|∇Ric|2 |Rm|2 + λ|Ric|2 ∆ v ≥ − c f Ra Ra 4a |∇Rm|2 4aλ |∇Ric|2 − − a + 1 Ra a + 1 Ra 2λ(1 − a) |∇Ric|2 |Rm|2 + λ|Ric|2 ≥ − c . 1 + a Ra Ra

Now by Proposition 2.4.1, for some constant > 0, we have

2|Rm|2 ≤ |∇Ric| + |Ric|2 + |Rc|2 ≤ 2|∇Ric|2 + 2(|Ric|2 + |Ric|)2.

Thus,

2λ(1 − a) |Rm|2 1 − a |Ric|2 ∆ v ≥ − c − 2λ (|Ric| + 1)2 + cλ f 1 + a Ra 1 + a Ra |Ric|2 |Ric|2 ≥ [λ(1 − a) − c](v − λ ) − λ 2(1 − a)(|Ric| + 1)2 + c . Ra Ra

Therefore, by Theorem 2.5.1, for each 0 < a < 1 and µ > 0 one can choose λ ≥ C/(1 − a), with C > 0 depending on µ and suﬃciently large, so that

∆f v ≥ µv − D for some constant D > 0 depending on λ.

4 Theorem 2.5.2. Let (M , gij, f), which is not Ricci-ﬂat, be a complete noncompact gradient steady Ricci soliton with limr→ ∞ R = 0. Suppose R has at most polynomial k decay, i.e., R(x) ≥ C/r (x) outside B(r0) for some ﬁxed r0 > 1, some constant c > 0 and positive integer k. Then, for each 0 < a < 1, there exists a constant C such that |Rm| ≤ CRa/2.

19 k + Proof. Let p = 2 . Consider the following function on R :

 p d−t 0 ≤ t ≤ d  d ϕ(t) =  0 t ≥ d.

4 ˚ Next, let ϕ = ϕ(r(x0)) on M . Then on B(d) \ (Cut(x0) ∪ B(r0)) we have,

p d − rp−1 p |∇ϕ| = |∇r| = ϕ, d d d − r p d − rp−1 p(p − 1) d − rp−2 4 ϕ = − 4 r + |∇r|2 f d d f d2 d p p(p − 1) = − 4 r + ϕ d − r f (d − r)2

D |Rm|2+λ|Ric|2 Consider w = v − µ with v = Ra , µ and D as in Lemma 2.5.2. Then, w satisﬁes

4f w ≥ µw.

2 ˚ Let G=ϕ w, then on B(d) \ B(r0), we have

2 2 2 4f G = (∆f ϕ )w + ϕ ∆f w + 2(∇ϕ ) · ∇w G ≥ 2ϕ∆ ϕ + 2|∇ϕ|2 w + µϕ2w + 4ϕ∇ϕ · ∇ f ϕ2 24 ϕ |∇ϕ|2 4 ≥ µ + f − 6 G + h∇G, ∇ϕi. (2.18) ϕ ϕ2 ϕ

Recall that G = 0 outside B(d). Now consider a maximum point q of G.

Case 1. G(q) ≤ 0. Then, max w ≤ 0. B(d)

20 1 Case 2. G(q) > 0 and q ∈ B(r0). Then, on Ω = B((1 − 21/p )d), we have

1 max w ≤ max · G(q) Ω Ω ϕ2 ≤ 4G(q) ≤ 4 max w. B(r0)

Case 3. G(q) > 0 and q∈ / B(r0), q∈ / Cut(x0). Then we could apply by (2.18) and Lemma 2.5.1, at q,

4 ϕ |∇ϕ|2 0 ≥ µ + 2 f − 6 ϕ ϕ2 1 1 ≥ µ − 2pK − (4p2 + 2p) 0 d − r (d − r)2

1 for some constant K0 > 0 depending on r0 and maxB(r0) |Ric|. Hence d−r(q) > C for some constant C depending on µ, p = k/2 and K0. Thus, we have

d − r(q) ≤ c (2.19) for some constant c > 0 independent of d. Therefore,

1 max w ≤ max · G(q) Ω Ω ϕ2 ≤ 4G(q) (d − r(q))2p (|Rm|2 + λ|Ric|2) ≤ 4 (q) d2p Ra rak(q) ≤ C d2p ≤ Cd(a−1)k ≤ C for some constant C > 0 independent of d.

Case 4. G(q) > 0, q∈ / B(r0), q ∈ Cut(x0). Then we could not apply (2.18) directly since d(x0, −) is not smooth at p. Now consider the support function G

21 constructed by the following procedure. Firstly, pick any minimal geodesic γ from x0 to p, then choose a point x1 ∈ γ very close to x0. Notice that x1 ∈/ Cut(p). Let

= d(x0, x1), consider r(x) = d(x1, x). Then we have,

• r(q) + = r(q)

• r(x) + ≥ r(q)

• r(x) is smooth near q

2 Now consider G(x)=ϕ(r(x) + ) w where ϕ was deﬁned in the beginning of the section. Then G(x) ≤ G(x) ≤ G(q) = G(q). Then we could apply maximum principle at q since G is smooth at q.

2 4f ϕ |∇ϕ| 0 ≥ µ + 2 − 6 2 ϕ ϕ 1 2 1 ≥ µ − 2pK0(x1) − (4p + 2p) 2 d − r − (d − r − )

Hence 1 > C for some constant C depending on µ, p = k/2 and K (x ). d−r(q)− 0 1 In order to get rid of the dependence of x1, we let → 0, then we have

d − r(q) ≤ c (2.20) for some constant c > 0 independent of d. Now follow the exact same argument of Case 3, we get an uniform estimate of 1 max w on Ω = B((1 − 1/p )d) which is independent of d. Ω 2 2 a 4 Therefore supM w ≤ C, and hence |Rm| ≤ CR on M for each 0 < a < 1.

22 Chapter 3

Curvature and volume growth of steady K¨ahlerRicci solitons

3.1 Background

We call a Riemannian metric gij K¨ahlerif there exists a (1,1) tensor J such that

2 • J = −IdTM

• g(JX,JY ) = g(X,Y ) for any X,Y ∈ TM

• ∇J = 0

If a steady Ricci soliton is Kähler,we call it a steady KählerRicci soliton. For properties of Kählermanifold readers may consult [2], [32]. We are going to list some notations we will use in this chapter. Firstly consider the complexified tangent bundle TM C = TM ⊗ C. Then extend J by complex linearity. Denote

• T 1,0M = {X − iJX|X ∈ TM},

• T 0,1M = {X + iJX|X ∈ TM}.

C Now extend g using complex linearily to TM = TM ⊗ C. Then gC satisﬁes

23 • gC(X, Y ) = gC(X,Y )

C • gC(X, X) > 0 for X ∈ TM − 0

1,0 • gC(X,Y ) = 0 for X,Y ∈ T M

1,0 gC becomes a Hermitian metric on T M, for simplicity we use gij for this Her- mitian metric in this chapter. Since ∇J = 0, Ric also shares the similar symmetry.

We denote Ricij for the non vanishing part of the complexified tensor. In this part we are going to analyze the asymptotic behaviour of steady Kähler Ricci solitons with positive Ricci curvature. We are going to focus on two parts; the first one is volume growth and the second one is curvature decay. For the volume growth, when the manifold has nonnegative Ricci curvature the classical Bishop comparison theorem implies the volume growth is at most Euclidean. And under the same condition, the volume growth is at least linear by a result of Yau and Calabi [64]. If furthermore the manifold has positive holomorphic bisectional curvature, Bing-Long Chen and Xi-Ping Zhu showed [23] that the volume growth is at least half Euclidean growth and curvature has to decay in the average sense. Applying their method, we showed that if the manifold is a steady KählerRicci soliton metric, then similar results hold when the metric has positive Ricci curvature.

3.2 Volume growth

2n Theorem 3.2.1. For any K¨ahlerRicci soliton (M , gij, f) with positive Ricci curvature and scalar curvature attaining its maximum, volume growth is at least half Euclidean, i.e., n V ol(Br) ≥ cr here r is the geodesic distance to some point x0.

− F Proof. Fix r > 1, and consider positive function ϕr = e r where F = −f. Here we

24 consider the Ricci form Ω = Ric(JX,Y ) and K¨ahlerform ω = g(JX,Y ).

Z √ n n n (ϕr − δ) ( −1) (∂∂F ) {ϕr>δ} Z √ n−1 n n−1 = − n(ϕr − δ) (∂ϕr ∧ ∂F ) ∧ ( −1) (∂∂F ) {ϕr>δ} Z √ n−1 ϕr n n−1 = n(ϕr − δ) ( ∂F ∧ ∂F ) · ( −1) (∂∂F ) {ϕr>δ} r Z √ n−1 ϕr 2 n−1 n−1 ≤ n(ϕr − δ) |∇F |g ∧ ( −1) (∂∂F ) ∧ ω {ϕr>δ} r Z n−1 √ (ϕr − δ) ϕr n−1 n−1 ≤ C1(n, R0) ( −1) (∂∂F ) ∧ ω {ϕr>δ} r Z n−2 2 √ (ϕr − δ) ϕr n−2 n−2 2 ≤ C2(n, R0) 2 ( −1) (∂∂F ) ∧ ω {ϕr>δ} r ··· Z n ϕr n ≤ Cn(n, R0) n ω {ϕr>δ} r

2 Here R0 is the maximum value of scalar curvature, recall we have |∇F | ≤ R0. Let δ → 0 we get Z Z n n C(n, R0) n n ϕr Ω ≤ n ϕr ω M r M Left hand side has a strictly lower bound since Ricci form is a positive (1,1) form. Therefore we have,

Z C(n, R0) n n c ≤ n ϕr ω r M √ Recall from (2.4) the potential function estimate c1d(x) − c2 ≤ F ≤ R0d(x) +

|F (x0)| which gives the following estimate,

c1d(x)−c2 d − −c1 ϕr ≤ e r ≤ Ce r

25 outside B(x0, 1). Therefore we have,

Z Z d n n 0 −c1n r n ϕr ω ≤ C e ω M M ∞ Z Z 0 X −c n d n 0 −c n d n ≤ C e 1 r ω + C e 1 r ω

i=0 B2i+1r−B2ir Br ∞ Z i Z 0 X −c n 2 r n 0 −c n r n ≤ C e 1 r ω + C e 1 r ω

i=0 B2i+1r Br ∞ i Z 0 X −2 c1n n 0 −c1n = C e ω + C e V ol(Br) i=0 B2i+1r ∞ i X −2 c1n 0 −c1n ≤ e V ol(B2i+1r) + C e V ol(Br) i=0 ∞ i X −2 c1n (i+1)2n 0 −c1n ≤ e 2 V ol(Br) + C e V ol(Br) i=0 00 ≤ C (c1, c2, n) · V ol(Br)

Here c1, c2 come from the estimate for ϕr from the previous page. Therefore Z C(n, R0) n n V olBr c ≤ n ϕr ω ≤ C(n, R0, c1, c2) n r M r

26 3.3 Curvature estimates

2n Theorem 3.3.1. For a steady K¨ahlerRicci soliton (M , gij, fij) with positive Ricci curvature such that the scalar curvature attains its maximum, for any x0 there exists C such that 1 Z C R(x) ≤ , V ol(B(r)) B(r) 1 + r here B(r) is a geodesic ball of radius r to any point.

Proof. We are going to use the following theorem by H¨ormander. The version we are going to use is in Chapter VIII, Theorem 6.5. [32] With the natural function F our manifold is Stein. Furthermore we have the following inequality,

√ −1∂∂(F ) + c1(KM ) + Ric = Ric > 0

Now fix a base point x0 and some cut off function θ near x0. Let be some √ small number such that −1∂∂(F + 2θ log |z − x0|) is still positive. Then for m 2 m large enough such that [m] − n > 0 we find a nontrivial L section S of KM with prescribed value at x0, say S(x0). Furthermore

Z 2 −mF |S|he dVg < ∞ M

√ 2 −1∂∂ log |S|h = [S = 0] + mRic ≥ mRic

Since the Ricci curvature is nonnegative, the mean value inequality for subhar- monic functions gives,

c(n) Z |S|2(x) ≤ |S|2(x) V ol(B(x, 3d(x0, x))) B(x,3d(x0,x)) c(n) Z = |S|2(x)e−mF emF V ol(B(x, 3d(x0, x))) B(x,3d(x0,x)) √ ≤ C0em R0d(x,x0)

27 Now consider Mf = M × C2, with the product metric. This space has nonnegative Ricci curvature, therefore parabolicity translates to volume growth.([42] Theorem 5.2). The volume growth of Mf is at least n + 4, therefore there exists a positive 0 0 Green function Ge(x, y) on Mf. Now consider Ge(x) = Ge(x0, x) where x0 = (x0, 0) where x0 is the point that we can prescribe S. Recall that we have,

4 log(|S|2) ≥ mR,

and log(|S|2) has singularity along S = 0. Therefore we consider log(|S|2 + δ).

|S|2 4 log(|S|2 + δ) ≥ mR |S|2 + δ

Now pull back functions |S|2, R on M through map Mf → M, we get functions |gS|2, Re such that,

|gS|2 4 log(|gS|2 + δ) ≥ mRe |gS|2 + δ

Z |gS|2 Z mRe (Ge − α)1+ ≤ 4 log(|gS|2 + δ)(Ge − α)1+ 2 β>G>αe |gS| + δ β>G>αe Z = log(|gS|2 + δ)4(Ge − α)1+ β>G>αe Z ∂ log(|gS|2 + δ) + (Ge − α)1+ Ge=β ∂n Z ∂Ge −(1 + ) log(|gS|2 + δ)(Ge − α) Ge=β ∂n

Z Z log(|gS|2 + δ)4(Ge − α)1+ ≤ sup log(|gS|2 + δ) 4(Ge − α)1+ β>G>αe G>αe β>G>αe Z ∂Ge = sup log(|gS|2 + δ) (1 + )(Ge − α) ∂n G>αe Ge=β

28 Letting → 0, we get

Z |gS|2 Z ∂Ge mRe (Ge − α) ≤ sup log(|gS|2 + δ) 2 ∂n β>G>αe |gS| + δ G>αe Ge=β Z ∂ log(|gS|2 + δ) + (Ge − α) Ge=β ∂n Z ∂Ge − log(|gS|2 + δ) Ge=β ∂n

c ∂Ge c0 We’ll prove later such that Ge ∼ d2n+2 , | ∂n | ∼ d2n+3 . By using these facts, we obtain,

Z ∂Ge cd2n+3 ∼ 2n+3 ∼ c Ge=β ∂n d Z cd2n+3 |Ge − α| → 2n+2 = cd → 0 Ge=β d

2 |S^(x0)| = 1

Letting β → +∞, then letting δ → 0(furthermore use Se = 0 has codimension 1)

Z mRe(Ge − α) ≤ c(n) sup log(|gS|2) G>αe G>αe

1 On Ge > 2α, we have (Ge − α) ≥ 2 Ge therefore Z mReGe ≤ c(n) sup log(|gS|2) G>e 2α G>αe

Goal: Change G(x0, −) level set coordinates back to regular geodesic ball. Tool: Green function estimate

c(n)−1d˜2(x, x ) c(n)d˜2(x, x ) 0 ≤ G˜(x, x ) ≤ 0 ˜ ˜ 0 ˜ ˜ V ol(B(x0, d(x, x0)) V ol(B(x0, d(x, x0))

29 From [42] Thm5.2, we have the following estimate for a space with nonnegative Ricci curvature:

Z ∞ Z ∞ −1 dt dt c(n) √ ≤ G(x, x0) ≤ c(n) √ . d2 V ol( t) d2 V ol( t)

Lower bound:

Z ∞ dt Z ∞ c(n)d2n+2dt c(n)d2n+2dt Z ∞ d2 √ ≥ √ = t−n−1 = c0 . 2 n+1 d2 V ol( t) d2 V ol( d ) · t V ol(d) d2 V (d)

√ √ V ol( d2) 2 Here we use Bishop-Gromov: √ ≥ ( √d )2n+2 when t > d2. V ol( t) t Upper bound: This is way back to observation (23) in Shi’s construction. Be- √ V ol( d2) cause we have a ﬂat factor which has accurate volume growth information, √ ≤ √ V ol( t) 2 C(n)( √d )4. t c Since the volume is locally are Euclidean, the above estimates imply Ge ∼ d2n+2 ∂Ge c0 when d → 0. By the Cheng-Yau gradient estimate | ∂n | ∼ d2n+3 . ˜ Let r(α) be the largest number such that Be(x0, r(α)) ⊂ {G > α}. Because of the Green function estimate,

˜ ˜ ˜ B(x0, r(α)) ⊂ {G > α} ⊂ B(x0, c(n)r(α)).

Recall we have, Z mReGe ≤ c(n) sup log(|gS|2), G>α 1 e G>e 2 α

r(α)2 together with the lower bound G ≥ V ol(r(α)) inside B(r(α)),

Z Z mReGe ≥ mReGe G>αe Be(r(α)) r(α)2 Z ≥ m · C R.e V ol(r(α)) Be(r(α))

30 By Green function estimate, and the estimate for log(|gS|2),

2 2 0 sup log(|gS| ) ≤ sup log(|gS| ) ≤ c(n)(re+ c ). 1 B(c(n)r(α)) G>e 2 α

Therefore on Mf we have,

Z 1 re+ c 1 Re ≤ c 2 ≤ c . V ol(re) Be(r(α)) re re+ 1 The above estimates also hold for M since

1 1 B ( r) × B 2 ( r) ⊂ B(r) ⊂ B (r) × B 2 (r). M 2 C 2 e M C

31 Chapter 4

Uniqueness under constraints of the asymptotic geometry.

4.1 Background

Recall a steady Ricci soliton (M, g) is a Riemannian metric which satisfies the e- quation 2Ric = LX (g). For a steady Ricci soliton (M, g) if in addition the metric is Kählerand X is the gradient of some real valued function, then we call it steady gradient KählerRicci soliton. H.D. Cao constructed a family of steady gradient KählerRicci solitons on Cn with positive holomorphic bisectional curvature in [11]. This is the first noncompact (nontrivial) example of a steady Kähler Ricci soliton. There are also many important examples of KählerRicci solitons constructed by Koiso in [49], M. Feldman, T. Ilmanen, and D. Knopf in [43] and Akito Futaki and Mu-Tao Wang in [1]. In [11], Cao asked a question on symmetry of steady KählerRicci solitons with positive holomorphic bisectional curvature on Cn. O. Schnürer,and A. Chau’s result in [24] gave a partial answer to this question. Recently S. Brendle showed O(n)−symmetry of certain steady solitons in [57] [58]. His work solved an open problem proposed by Perelman in [51]. Otis Chodosh extended the argument to expanding solitons in [26]. Otis Chodosh and Frederick

32 Tsz-Ho Fong showed U(n)−symmetry of certain gradient expanding K¨ahlerRicci Solitons in [27]. In this chapter we will give a partial answer to the question proposed by Cao in [11] . The argument is similar to [27] which comes from [57] [58] [26]. We are going to compute norms, gradients and distance with respect to the model metric constructed by Cao in [11] in this Chapter.

4.2 Main Theorem

n Main Theorem For n ≥ 2. Let (C , gm,Xm) be a steady gradient K¨ahlerRicci n solitons constructed by Cao in [11]. Let (C , g,e Xe) be some steady gradient K¨ahler Ricci soliton with following properties.

2+ j j 1. r 2 |∇ (ge − gm)| = o(1) for j = 0, 1

2. ge has positive holomorphic bisectional curvature, here r is the geodesic distance to the origin with respect to gm. n ∗ Then there exists a point p ∈ C and a map Φp : z → z + p such that g = Φp(ge) satisﬁes standard U(n)− symmetry.

4.3 Preliminary

In this section we are going to present expression and basic properties of the metric construct by H.-D. Cao in [11]. Let (z1, z2, ..., zn) be standard holomorphic coordi- 2 nate on Cn. Let t = log(|z| ). Then the U(n)-symmetric steady K¨ahlerRicci soliton in [11] is given by

−t −2t 0 (gm)ij = e φ(t)δij + e zizj(φ (t) − φ(t)) (4.1)

δij 1 1 (g )ij = et + z z − (4.2) m φ(t) i j φ0(t) φ(t)

33 where φ(t) satisﬁes φn−1φ0eφ = ent after normalization. From computations in [11] we have following properties.

φ(t) → nt φ0(t) → n φ00(t) → 0 φ000(t) → 0 r(t) = O(t)

Here r is the distance to origin with respect to gm. We’ll always use r, t for above purpose.

4.4 Calculations

In the calculation part, we are going to analyze, under the asymptotic constraint, how much could various quantities diﬀer from the original model. From now on g is some steady gradient Kahler Ricci soliton metric satisﬁes assumptions 1,2. gm is the model metric constructed in [11].

4.4.1 Killing vectors of the model metric

1 1 We are going to show |Ua| = O(r 2 ), |∇Ua| = O(1), |X| = O(r 2 ), |∇X| = O(1) by straightforward computations. Here Ua are killing vectors coming from the unitary symmetry of gm, X is the soliton vector of the model metric. r is the distance to origin with respect to gm. Notice that JX is Killing, therefore we only need to do explicit computation for Killing vector ﬁelds.

We pick following explicit R−basis of Killing vectors of gm.

1,0 ∂ 1. U = izk , k ∂zk

2. U 1,0 = z ∂ − z ∂ where u 6= v, u,v u ∂zv v ∂zu

34 3. U 1,0 = i(z ∂ + z ∂ ) where u 6= v. eu,v u ∂zv v ∂zu

1 Our goal is to show |Ua| = O(r 2 ), |∇Ua| = O(1). We can restrict the computa- n tion to the direction (z1, 0, ..., 0) ∈ C by symmetry.

1 •|Ua| = O(r 2 )

By expression 4.1 , at (z1, 0, ..., 0), metric looks like

−t 0 gij = e diag{φ (t), φ(t), ..., φ(t)} (4.3)

From the expression of Killing vectors above and the relationship between the Kählermetric and its associated Riemannian metric. It’s sufficient to calculate the ∂ ∂ length of z1 , z1 using the Kählermetric . ∂z1 ∂zk

∂ 2 −t 2 0 1 2 0 0 (a) |z1 | = e |z1| φ (t) = 2 |z1| φ (t) = φ (t) → n ∼ O(1) ∂z1 |z1|

∂ 2 −t 2 1 2 (b) |z1 | = e |z1| φ(t) = 2 |z1| φ(t) = φ(t) → nt ∼ O(r) ∂zk |z1|

Here we have used the asymptotic behaviour of φ(t) in Section 4.3.

•|∇Ua| = O(1)

Since all Ua are real holomorphic, when we calculate |∇Ua|, we can restrict all discussion to T 1,0. From 4.1, Christoﬀel symbol Γ i is C jk

gli h i z (−φ(t)+φ0(t))δ +e−t(2φ(t)−3φ0(t)+φ00(t))z z +(φ0(t)−φ(t))z δ (4.4) e2t j kl k l k jl

At (z, 0, ..., 0) we have the following 4 cases

i ji −2t 0 (A) j 6= 1 Γjk = g zke φ (t) − φ(t)

i ki −2t 0 (B) j = 1, k 6= 1 Γ1k = g z1e φ (t) − φ(t)

i (C) j = 1,k = 1,i 6= 1 Γ11 = 0

35 1 11 −2t 00 0 (D) j = 1,k = 1,i = 1 Γ11 = g z1e φ (t) − φ (t)

From the expression of a basis of Killing vectors we pick at the beginning of u ∂ 4.4.1, we only need to estimate the length of ∇(z ∂zv ) along (z, 0, ...0).

Case I u = v = k where k 6= 1

k ∂ I.1) ∇ ∂ (z ∂zk ) = 0 for l 6= k ∂zl

k ∂ ∂ I.2) ∇ ∂ (z ∂zk ) = ∂zk ∂zk Therefore |∇(zk ∂ )| = 1 | ∂ | = 1 ∂zk gm | ∂ | ∂zk ∂zk

Case II u = v = 1

∂ II.1) Taking the derivative along ∂zl where l 6= 1

1 ∂ 1 m ∂ 1 0 ∂ ∇ ∂ (z ∂z1 ) = z Γl1 ∂zm = φ(t) (φ (t) − φ(t)) ∂zl ∂zl

1 0 φ(t) (φ (t) − φ(t)) is bounded by the properties of φ(t) listed in Section 4.3.

∂ II.2) Taking the derivative along ∂z1

1 ∂ ∂ 1 m ∂ h 1 00 0 i ∂ ∇ ∂ (z 1 ) = 1 + z Γl1 m = 1 + 0 (φ (t) − φ (t)) 1 ∂z1 ∂z ∂z ∂z φ (t) ∂z

1 0 φ(t) (φ (t) − φ(t)) is bounded by Section 4.3.

Therefore |∇(z1 ∂ )| = O(1) ∂z1 gm

Case III u 6= v and u 6= 1

u ∂ ∂ ∇ ∂ (z ∂zv ) = δlu ∂zv ∂zl

u ∂ 1 ∂ ∂ Therefore |∇(z v )| ≤ ∂ C| u | ≤ C ( direction is longer by 4.3 and ∂z gm ∂z ∂zu | ∂zu | properties of φ(t) in Section 4.3 )

36 Case IV u 6= v and u = 1

∂ VI.1)Taking the derivative along ∂zl where l 6= 1

1 ∂ 1 0 ∂ ∇ ∂ (z ∂zv ) = δ1v φ(t) φ (t) − φ(t) ∂zl ∂zl

∂ VI.2)Taking the derivative along ∂z1

1 ∂ ∂ 1 s ∂ φ0(t) ∂ ∇ ∂ (z v ) = v + z Γlv s = v ∂z1 ∂z ∂z ∂z φ(t) ∂z From the expression of the metric in 4.3 and the properties of φ(t) in Section 4.3 we see that |∇(z1 ∂ )| is also O(1) ∂zv gm

4.4.2 Shifting preserves the assumption 1

n ∗ We are going to show for ge satisfies Assumption 1 , 2 , and any point p ∈ C ,Φp(ge) also satisfies Assumption 1 , 2. ∗ We just need to check for Assumption 1 still holds for Φp(g). This is equivalent 2+ j j ∗ to say that r 2 |∇ (Φ (g) − gm)| = o(1) for j = 0, 1. p e gm ∗ Directly pull back Assumption 1 by Φp, we get. 2+ j j 2+ j ∗ j ∗ ∗ r 2 |∇ (g − gm)| = o(1) =⇒ r 2 |Φ (∇) (Φ (g) − Φ (gm))| ∗ = o(1) e gm p p e p Φp(gm) After this, it’s sufficient to check following facts..

2 1. |Φ∗(g ) − g | = O( log(|z| ) ) p m m gm |z|

2 2 2. |∇(Φ∗(g ) − g )| = O( (log(|z| )) ) p m m gm |z|

2+ j ∗ j 2+ j j 3. r 2 |(Φ ∇) (k)| ∗ = o(1) for j = 0, 1 implies r 2 |∇ (k)| = o(1) for p Φpgm gm j = 0, 1

2 STEP 1 |Φ∗(g ) − g | = O( log(|z| ) ) p m m gm |z| ∗ Plug 4.1 into (Φp(gm) − gm)ij, we get

37 " 2 2 # ∗ φ(log(|z + p| )) φ(log(|z| )) (Φ (gm) − gm) = − p ij |z + p|2 |z|2 " # φ0(log(|z + p|2)) − φ(log(|z + p|2)) φ0(log(|z|2)) − φ(log(|z|2)) + (zi + pi)(zj + pj) − |z + p|4 |z|4 " # φ0(log(|z|2)) − φ(log(|z|2)) + (zi + pi)(zj + pj) − zizj |z|4 (4.5)

For the norm of the ﬁrst two terms, apply mean value theorem to real valued

φ(log(x2)) 2 2 0 function f(x) = 2 , we get f(|z+p| )−f(|z| ) ≤ |p| f (|z|+ξ) , ξ ∈ (−|p|, |p|) x φ0 log((|z|+ξ)2 −φ log((|z|+ξ)2) 2 f 0(|z| + ξ) = 2 ≤ C log(|z| ) . The last step uses the 3 |z|3 |z|+ξ asymptotic of φ in Section 4.3. Therefore the norms of the ﬁrst two terms is bounded 2 by C log(|z| ) . This bound works for other 2 coupled terms by similar arguments. |z|3 2 Together with the coarse estimate guv ≤ Cee t = Ce|z| from expression 4.2 , we 2 have |Φ∗(g ) − g | = O( log(|z| ) ) p m m gm |z|

2 2 STEP 2 |∇(Φ∗(g ) − g )| = O( (log(|z| )) ) p m m gm |z| 2 Let δg = Φ∗(g ) − g . Then for any i, j,(δg) = O( log(|z| ) ) by STEP 1. p m m ij |z|3 ∂ ∂ ∂ (∇mδg) = (δg ) − (δg)(∇m , ) ij ∂zm ij ∂zi ∂zj 1. ∂ (δg ) ∂zm ij

Expression of δgmt has been separated into three lines in 4.5 When ∂ hits the ﬁrst line of 4.5 ∂zm

38 " # ∂ φ(log(|z + p|2)) φ(log(|z|2)) 2 − 2 ∂zm |z + p| |z| " # (φ0 − φ)(log(|z + p|2)) (φ0 − φ)(log(|z|2)) = (zm + pm) − |z + p|4 |z|4 (φ0 − φ)(log(|z|2))h i + zm + pm − zm |z|4

φ0(log(x2))−φ(log(x2)) By applying the mean value theorem to f(x) = x4 and asymptotic 2 of φ(x), we get the order of the sum of these two terms is O( log(|z| ) ). |z|4 ∂ log(|z|2) When hits the second and third line of 4.5, the order is also O( 4 ) by ∂zm |z| the asymptotic of φ(t) up to third order and a similar discussion. ∂ ∂ (log(|z|2))2 2. (δg)(∇m , ) = O( 4 ) This is done by using the result from STEP ∂zi ∂zj |z| 2 r log(|z| ) 1 and the coarse bound |Γpq| = O( |z| ) which is directly from its expression 4.4 2 2 Therefore |∇(Φ∗(g ) − g )| = O( (log(|z| )) ) p m m gm |z|

2+ j ∗ j 2+ j j STEP 3 r 2 |(Φ ∇) (k)| ∗ = o(1) j = 0, 1 implies r 2 |∇ (k)| = o(1) p Φpgm gm ∗ For j = 0, this comes from the equivalence of Φpgm and gm by STEP 1. 2 For j = 1, it’s suﬃcient to show |(Φ∗∇ − ∇)k| = O( log(|z| ) )) p gm |z| (Φ∗∇ − ∇ )k = k (Γs − Γs ) = k gsv∂ g − gsv∂ g p i i jl sl fij ij sl e iejv i jv 1.As a complex number, |k | = O( 1 ) by properties of k and g. sl |z|2 2 2.|gsv∂ g − gsv∂ g | = |(gsv − gsv)∂ g − gsv∂ (g − g )| = O( log(|z| ) ) e iejv i jv e iejv i ejv jv |z|2 2 |(Φ∗∇ − ∇)k| = O( log(|z| ) )) holds by 1,2 together with |guv| ≤ |z|2. p gm |z|

4.4.3 Rigidity of the soliton vector

In this subsection we are going to see the soliton vector Xe of g must satisﬁes Xe1,0 = (λzi + bi) ∂ where Re(λ) 6= 0. Therefore there exists an shifting map (Φ∗(X))1,0 = ∂zi p e λzi ∂ . In other words, the soliton vector ﬁeld is rigid suppose we know information ∂zi about the asymptotic geometry.

39 Write X1,0 as ui(z) ∂ . Then ui(z) is a holomorphic function on n. e ∂zi C 2 By [12] |Xe|g + R(g) is constant. R(g) > 0 by Assumption 2 . Therefore we have |X|2 < ∞. By Assumption 1 , we see that |X|2 < ∞ e g e gm From properties of φ(t) and φ0(t) in section 4.3 . We see that there exists a 0 positive C(t0) s.t. φ(t), φ (t) is greater than C for all t > t0. Recall the expression 0 of g at (z, 0, ...0) is diag{ φ (t) , φ(t) ... φ(t) }. Therefore |X|2 is ﬁnite implies for m et et et e gm |ui(z)|2 2 |z| > et0 , +∞ > g ui(z)uj(z) ≥ P C . Therefore |ui(z)| is at most linear ij i |z|2 i i j i i ∂ growth. u (z) = a z + b . Let XL = b Then |XL| → 0 uniformly. j f ∂zi f gm

1. ai = δi aj. To see this we use |X − X | is ﬁnite and make a calculation at j j e e fL gm (0, ..., z, ..., 0) where the j-th place is not 0. By 4.1 , the metric is 1 diag{φ, ..., φ0, ..., φ} |z|2 at this point.

D ∂ ∂ E ∞ > |X − X | = aqzp , aszr L gm p r ∂zq ∂zs q s j 2 = aj aj|z | gqs X q 2 j 2 0 = φ(t)|aj | + |aj| φ (t) q6=j

q From the properties of φ(t) in Section 4.3. We must have aj = 0 for q 6= j

2. a1 = a2 = ... = an We still use |X − X | is ﬁnite and make a calculation at e e e e fL gm (0, ..., z, ..., z, ..., 0). Here the nonzero places, the ith and jth ones, are equal. By 4.1, at this point, we have

−t −2t 2 0 (gm)pp = (gm)qq = e φ(t) + e |z| (φ (t) − φ(t)) −2t 2 0 (gm)pq = (gm)qp = e |z| (φ (t) − φ(t))

Therefore ∞ > |X −X | = 1 |ai|2 +|aj|2 φ0(t)+ 1 |ai −aj|2φ(t). From Section e fL gm 4 e e 4 e e 4.3, we see that aei = aej, ∀i, j

40 3. Re(λ) 6= 0 Suppose λ is purely imaginary. Then there exists a closed circle as integral curve for vector ﬁeld Xe. This contradicts with Xe being the gradient of some real function.

4.4.4 Decay rate of Ricci of the model metric

C We are going to show there exists a C > 0 such that Ricm − t2 gm ≥ 0 by straightforward computations. Since both Ricm and gm are U(n)−symmetric, we can restrict our discussion along (z, 0, ..., 0). Along this direction, we have

−t 0 gij = e diag {φ (t), φ(t), ..., φ(t)} −t 00 0 0 Rij = e diag {v (t), v (t), ..., v (t)}

0 where v(t) = nt − (n − 1) log(φ(t)) − log(φ (t)). As we assume gm have the same normalization as in [11]. The equation of φ is φn−1φ0eφ = ent. v00(t) v0(t) Let’s consider λ1(t) = φ0(t) , λk(t) = φ(t) where k ≥ 2

1. λk, k ≥ 2

φ0(t) φ00(t) n − (n − 1) − 0 C λ = φ(t) φ (t) > k k φ(t) t

2. λ1

1 (φ0(t))2 − φ00(t)φ(t) (φ00(t))2 − φ000(t)φ0(t) λ = (n − 1) + 1 φ0(t) (φ(t))2 (φ0(t))2

41 We have the following identities from Cao’s paper [11]

(φ0)2 φ00 =nφ0 − (φ0)2 − (n − 1) φ (φ0)2 φ000 =n2φ0 − 3n(φ0)2 + 2(φ0)3 − 3n(n − 1) φ (φ0)3 (φ0)2 + 4(n − 1) + (n − 1)(2n − 1) φ φ2

Plugging them into expression of λ1, we have

1 (2n − 1)(n − 1)φ0 λ = φ00 + (φ0 − 1) 1 φ0 φ2

(2n−1)(n−1)φ0 0 By the asymptotic of φ in Section 4.3, we see that φ2 (φ − 1) ∼ n(2n−1)(n−1)2 t2 Therefore we only need to show φ00 > 0 if n > 1. Since φ0 > 0, we can write t as a function of φ. From the soliton equation, we 0 ent 00 have φ = φn−1eφ as a function of φ. Plug it into the expression of φ , we have

ent φ00 = φneφn − entφ − (n − 1)ent . φ2n−2e2φφ

dk n φ nt nt Let fk(φ) = dφk φ e n − e φ − (n − 1)e . By expansion of φ at zero from [11], d nt n−1 φ limφ→0f0(φ) = 0. Now take the derivative on f0 using dφ e = nφ e

d f (φ) = f = nφn−1eφ − ent. 1 dφ 0

Then we have limφ→0f1(φ) = 0. Take the derivative again

d f (φ) = f = nφn−2eφ(n − 1) > 0. 2 dφ 1

00 C Therefore we have φ > 0. Hence ∃ C1 > 0 s.t. λ1 > t2 .

42 4.4.5 Main Argument

Proof We can assume that gm has the same normalization(α = 1, β = 1) as in [11], since scaling on metric and dilation on coordinates does not aﬀect our assumption. n2 Pick an R–basis of Killing vectors of gm, {Ua}a=1. n We have seen in 4.4.3 that there exist a p ∈ C s.t. Xfp = 0. For this speciﬁc p, ∗ ∗ let g = Φp(ge), X = Φp(Xe). We’ll show Ua is also Killing for g.

Now let h = LUa g, Z = 4RUa + Ric(Ua) = 0, then by Proposition 2.3.7 in[53]

4L(h) = −2LUa (Ric) + LZ (g)

= −LUa (LX (g))

= −LX (LUa (g))

= −LX (h)

Here we use a fact in 4.4.3 that X is a radial vector. Therefore [X,Ua] = 0.

Furthermore Ua is real holomorphic, hence Z = 4RUa + Ric(Ua) = 0 In 4.4.1 , we have seen that |X| = O(1), |∇X| = O(1). Since assumption 1 is −2 preserved by shifting by 4.4.2, we get |LX (g − gm)| = o(r ). By the argument in 1,0 1,0 −2 4.4.3, X = λXm where Re(λ) 6= 0. Hence we have |Ricg − Re(λ)Ricgm | = o(r ). c In 4.4.4 we saw that Ricgm ≥ r2 gm for n ≥ 2. Together with positivity of Ricg, we ec get Ricg ≥ r2 gm. 1 By 4.4.1 , |Ua| = O(r 2 ), |∇Ua| = O(1). These combined with assumption 1 −2 give us |h| = o(r ). Therefore for suﬃcient large θ, θ(Ricg) > h. The following argument is quite similar to the analysis in Prop 4.4 in [27].

Consider θ0 = inf{ θ | θ(Ricg) > h} . And let w = θ0Ricg − h. ec We’ll see that θ0 > 0 leads to a contradiction. If θ0 > 0, by Ricg ≥ r2 gm, −2 |h| = o(r ) and the positivity of Ricg, ∃ p ∈ M, e1 ∈ TpM s.t. w(e1, e1) = 0.

Parallel translating e1 in a neighbourhood of p, then we have (4w)(e1, e1) > 0,

(DX w)(e1, e1) = 0 at p. Now the discussion goes to the complexiﬁed tangent bundle. Extend w by C−linearity. The discussion separates into two parts. The ﬁrst part is to show

43 T r w = 0 at p. The second part is to show T r w satisﬁes 4(T r w) + DX (T r w) ≤ 0. For the ﬁrst part, let η = 1 (e − iJe ) ∈ T 1,0, then w(η , η ) = 0. Together with 1 2 1 1 p,C 1 1 w ≥ 0, this implies we can extend η into unitary basis η ... η ∈ T 1,0 such that 1 1 n q,C w(ηi, ηj) is diagonal. Also we can parallel extend ηi like η1. Since 4 w + L w = 0, plug in η ,η ∈ T 1,0M (Extend J-invariant (0,2)-tensor L X i i p,C w by C-linearity.)

0 = (4w)(ηi, ηi) + 2ΣkRm(ηi, ηk, ηi, ηk)w(ηk, ηk) − 2w(Ric(ηi), ηi)

+(DX w)(ηi, ηi) + w(Dηi X, ηi) + w(ηi,Dηi X)

w(Dηi X, ηi) = ηi(ηif)w(ηi, ηi) = ηi(ηif)w(ηi, ηi) = w(ηi,Dηi X)

From the soliton equation, we have Ric(ηi) = Dηi X, therefore

0 = (4w)(ηi, ηi) + 2ΣkRm(ηi, ηk, ηi, ηk)w(ηk, ηk) + (DX w)(ηi, ηi). (4.6)

Now take i = 1 ,we see that 0 ≥ 2ΣkRm(η1, ηk, η1, ηk)w(ηk, ηk). That g has positive holomorphic bisectional curvature implies Rm(η1, ηk, η1, ηk) > 0. Therefore w(ηk, ηk) = 0 for any k at p. The nonnegative function T r w = 0 at p.

Equation (4.6) only uses 4Lw + LX w = 0, the soliton equation, w(ηi, ηj) is diagonal and the extension is parallel. Now sum (4.6) for all i at q give us 4(T r w)+

DX (T r w) ≤ 0. By Hopf’s strong maximum principle, T r w = 0. Therefore w is 0. This violates the asymptotic of Ricg and h. Therefore θ0 = 0, h ≤ 0. Now apply a similar argument to −h implies h = 0. Therefore g is U(n)−symmetric. Therefore it must be in the family of steady solitons in [11].

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[64] Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J. 25(1976), 659-670.

51 Xin Cui Department of Mathematics, Lehigh University Christmas-Saucon Hall, 14 East Packer Avenue, Bethlehem, PA 18015 Phone: 610-888-8731

Education

• B.S. in Mathematics, Zhejiang University, Hangzhou, China, 2010.

• Ph.D. in Mathematics, Lehigh University, PA, May 2016 (expected).

Academic Honors

• First Prize, 9th East China Math Modeling Competition, 2007.

• Second Prize, Contemporary Undergraduate Mathematical Contest in Model- ing, 2007.

• Certiﬁcate for Chu Kochen Honors Program, Zhejiang University, 2010.

• University Fellowship, Lehigh University, 2010-2011.

• Strohl Summer Research Fellowship, Lehigh University, 2013.

• Strohl Dissertation Fellowship, Lehigh University, 2015-2016.

Research Interests

• Diﬀerential Geometry, Ricci Flow and Ricci solitons, Complex Geometry.

Publication

• Huai-Dong, Cao and Xin Cui. “Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons.” arXiv preprint arXiv:1411.3631 (2014).

• Xin Cui. “A note on complete steady Kahler-Ricci solitons with positive Ricci curvature”, in preparation.

52 Talk

• Graduate Student Intercollegiate Mathematics Seminar Talk, Lehigh Univer- sity, Fall 2013.

Conference Attendance

• Conference on Geometric Analysis in honor of Peter Lis 60th Birthday, UC Irvine, 2012.

• Lehigh Geometry/Topology conference, Lehigh University, 2012.

• Yamabe Memorial Symposium, University of Minnesota: Twin Cities, 2012.

• Great Lakes Geometry Conference, Northeastern University, 2013.

• Complex Geometry, and Mathematical Physics: A Conference in Honor of Duong H. Phong, Columbia University,2013.

• Graduate Workshop on Kahler Geometry, Stony Brook University, 2013.

• Summer school in complex geometry, Rutgers University, 2013.

• RU-CUNY Symposium on Geometric Analysis, Rutgers University, 2013.

• Journal of Diﬀerential Geometry Conference, Harvard University, 2014.

Teaching Experience Teaching Assistant:

• Math 22, Calculus 2, Fall 2011.

• Math 22, Calculus 2, Fall 2012.

• Math 22, Calculus 2, Spring 2013.

• Math 23, Calculus 3, Fall 2013.

• Math 22, Calculus 2, Fall 2014.

53 • Math 23, Calculus 2, Spring 2015.

Grader:

• Math 205, Linear Algebra, Spring 2012.

• Math 401, Real Analysis (Graduate course), Fall 2013.