aa r X i v : . [ m a t h . DG ] A p r A NOTE ON CHERN-YAMABE PROBLEM
SIMONE CALAMAI AND FANGYU ZOU
Abstract.
We propose a flow to study the Chern-Yamabe problemand discuss the long time existence of the flow. In the balanced case weshow that the Chern-Yamabe problem is the Euler-Lagrange equationof some functional. The monotonicity of the functional along the flow isderived. We also show that the functional is not bounded from below. Introduction
Let (
X, ω ) be a compact complex manifold of complex dimension n en-dowed with a Hermitian metric ω . The Chern scalar curvature of (
X, ω ) isthe scalar curvature with respect to the Chern connection associated to ω .The Chern scalar curvature can be succinctly expressed asS Ch ( ω ) = tr ω i ¯ ∂∂ log ω n , where ω n denotes the volume form. Under conformal transformation, theChern scalar curvature changes asS Ch (cid:0) exp(2 f /n ) ω (cid:1) = exp( − f /n ) (cid:16) S Ch ( ω ) − ∆ Ch ω f (cid:17) , where ∆ Ch ω is the Chern Laplacian operator with respect to ω , which isdefined as ∆ Ch ω f := ( ω, − dd c f ) ω = − ω i ¯ ∂∂f = ∆ d f − ( df, θ ) ω , where θ = θ ( ω ) is the torsion 1-form defined by dω n − = θ ∧ ω n − .In [1] the authors proposed the Chern-Yamabe problem of finding a con-formal metric in the conformal class of ω whose Chern scalar curvature isconstant. More specifically, it is to find a pair ( f, λ ) ∈ C ∞ ( X ; R ) × R solving(1) − ∆ Ch ω f + S Ch ( ω ) = λ exp(2 f /n ) . Let dµ be the volume form of the background metric. We can normalize f so that(2) Z X exp(2 f /n ) dµ = 1 . Then the constant λ is exactly the total Chern scalar curvature of the back-ground metric(3) λ = Z X S Ch ( ω ) dµ. The case λ ≤ λ > We use the analysts’ convention of Laplacian operator. more examples were recently found in [2]. This note serves as some partialefforts to close the gap.In [1, 5] two different flows were defined to approach the study of Hermit-ian metrics with constant Chern scalar curvature. Here we define a differentflow, in Section 2, which has the advantage of being monotone when theproblem is known to be variational. The main result of the present note is
Proposition 1.1.
The Chern-Yamabe flow exists as long as the maximumof Chern scalar curvature stays bounded.
Then, in Section 3 we prove that in the balanced case the functional F isdecreasing along the flow. Again in Section 3 we prove that the functional F is not bounded from below when the complex dimension of X is at least2. In section 4 we present more properties of the flow under additionalassumptions. Acknowledgments.
The authors are grateful to Professor Xiuxiong Chenfor his constant support and sharing his idea on the proof of the C estimate.Thanks to Haozhao Li for his interest in this project. Normalization.
A fundamental result by P. Gauduchon in [4] states that,on any compact complex manifold of complex dimension n ≥
2, every confor-mal class of Hermitian metrics contains a standard , also called
Gauduchon ,metric ω , such that dω n − = 0. Hence, we can take the Gauduchon met-ric in each conformal class as the background metric. Furthermore, we cannormalize the Gauduchon metric so that it has volume 1. From now on weassume the background metric ω in (1) is Gauduchon with unit volume.2. Chern-Yamabe Flow
Let f ( x ; t ) be a family of C ∞ functions on X parametrized by a realparameter t . Let S ( x ; t ) = S Ch (exp(2 f ( x ; t ) /n ) ω ). The Chern-Yamabe flow is the flow f ( x ; t ) defined by the following flow equation:(4) ∂f∂t = n (cid:0) λ − S (cid:1) = n − f /n ) (cid:16) ∆ Ch ω f − S Ch ( ω ) + λ exp(2 f /n ) (cid:17) with some initial value f satisfying the normalization constraint(5) Z X exp(2 f /n ) dµ = 1 . Under the flow some quantities are preserved.
Lemma 2.1.
Along the flow we have Z X exp(2 f /n ) dµ ≡ . Z X S exp(2 f /n ) dµ ≡ λ. NOTE ON CHERN-YAMABE PROBLEM 3
Proof.
1. Let φ ( t ) = Z X exp(2 f /n ) dµ. By the initial data (5) and the flow equation (4), we have φ (0) = 1 and φ ′ ( t ) = 2 n Z X exp(2 f /n ) ∂f∂t dµ = Z X (cid:16) ∆ Ch ω f − S Ch ( ω ) + λ exp(2 f /n ) (cid:17) dµ = λ (cid:0) φ ( t ) − (cid:1) . It is straightforward to show that φ ( t ) ≡ Z X S exp(2 f /n ) dµ = Z X (S Ch ( ω ) − ∆ Ch ω f ) dµ ≡ λ. (cid:3) Evolution of the Chern scalar curvature.
Under the Chern-Yamabeflow the Chern scalar curvature S ( x ; t ) = S Ch (exp(2 f /n ) ω ) evolves accord-ing to the following equation(6) ∂S∂t = n − f /n ) ∆ Ch ω S + S ( S − λ )with initial value S ( x ; 0) = S Ch (exp(2 f /n ) ω ).The following lemma gives a uniform lower bound of the Chern scalarcurvature. Lemma 2.2.
Let ( S ) min = min x ∈ X S ( x ; 0) . We have S ( x ; t ) ≥ min { ( S ) min , } , ∀ x ∈ X. Proof.
Let S min ( t ) = min x ∈ X S ( x ; t ). Applying maximum principle to (6)we obtain S min ′ ( t ) ≥ S min ( S min − λ ) ≥ − λS min . Hence, S ( x ; t ) ≥ S min ( t ) ≥ ( S ) min exp( − λt ) , ∀ x ∈ X. If ( S ) min ≥
0, then S ( x ; t ) ≥
0; otherwise S ( x ; t ) ≥ ( S ) min . Hence, S ( x ; t ) ≥ min { ( S ) min , } . (cid:3) Remark 2.3.
For Lemma 2.2, S ( x, t ) ≥ ( S ) min exp( − λt ) as long as theflow exists. In particular, if the initial Chern scalar curvature is strictlypositive, then the positiveness is preserved along the flow.We can always take a special initial f so that the initial Chern scalarcurvature is strictly positive. Let h ∈ C ∞ ( X ; R ) such that ∆ Ch ω h = S Ch ( ω ) − λ with Z X exp(2 h/n ) dµ = 1 . We have S Ch (exp(2 h/n ) ω ) = λ exp( − h/n ) > . Hence, the Chern-Yamabeflow with this specific initial f ( x ; 0) = h ( x ) has the positive Chern scalarcurvature as long as the flow exists. SIMONE CALAMAI AND FANGYU ZOU
Long time existence.
In this section we show that the Chern-Yamabeflow exists as long as the maximum of Chern scalar curvature stays bounded.The short time existence of the flow is straightforward as the principal sym-bol of the second-order operator of the right-hand side of the Chern-Yamabeflow is strictly positive definite. To obtain the long time existence, one needsto show the a priori C k estimatemax ≤ t Let h ∈ C ∞ ( X ; R ) such that∆ Ch ω h = S Ch ( ω ) − λ with Z X exp(2 h/n ) dµ = 1 . Such a function h exists because of (3). Similarly, by Lemma 2.1 there existssome v ( t ) ∈ C ∞ ( X × [0 , T ); R ) such that(7) ∆ Ch ω v = exp(2 f /n ) − . Differentiating (7) with respect to t , by the flow equation (4) we have ∂∂t (cid:16) ∆ Ch ω v (cid:17) = ∆ Ch ω f − S Ch ( ω ) + λ exp(2 f /n )= ∆ Ch ω f − (S Ch ( ω ) − λ ) + λ (exp(2 f /n ) − Ch ω f − ∆ Ch ω h + λ ∆ Ch ω v. Hence, ∆ Ch ω (cid:18) ∂v∂t − f + h − λv (cid:19) = 0 . We can normalize v ( x ; t ) (by adding some function depending only on t ifnecessary) so that(8) ∂v∂t − f + h − λv = 0with initial value v ( x ; 0) = v ( x ) for some v satisfying∆ Ch ω v = exp(2 f /n ) − Z X v dµ = 0 . Let w ( x ; t ) = ∂v/∂t . Differentiating (8) with respect to t , we have(9) ∂w∂t = ∂f∂t + λw = n − f /n ) ∆ Ch ω w + λw,w ( x ; 0) = f ( x ) − h ( x ) + λv ( x ) . NOTE ON CHERN-YAMABE PROBLEM 5 Let w max ( t ) = max x ∈ X w ( x ; t ) and w min ( t ) = min x ∈ X w ( x ; t ). By maximumprinciple, we have ddt w max ≤ λw max and ddt w min ≥ λw min . It follows that w min ( t ) ≥ w min (0) exp( λt ) and w max ( t ) ≤ w max (0) exp( λt ) . Hence, we have k w ( x ; t ) k C ( X ) ≤ K exp( λt ) with K = max (cid:0) | w min (0) | , | w max (0) | (cid:1) . It then follows that | v ( x ; t ) | = (cid:12)(cid:12)(cid:12)(cid:12) v ( x ) + Z t w ( x ; t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ k v k C ( X ) + Z t k w ( x ; t ) k C ( X ) dt ≤ k v k C ( X ) + Kλ exp( λt ) . By (8) we have f ( x ; t ) = w ( x ; t ) + h ( x ) − λv ( x ; t ). Hence, k f ( x ; t ) k C ( X ) ≤ k w ( x ; t ) k C ( X ) + k h k C ( X ) + λ k v ( x ; t ) k C ( X ) ≤ K exp( λt ) + k h k C ( X ) + λ (cid:18) k v k C ( X ) + Kλ exp( λt ) (cid:19) ≤ k h k C ( X ) + λ k v k C ( X ) + 2 K exp( λt ) := C ( T ) . Since the functions h , v and w are uniquely determined by ( X, ω ) and f ,the constant C ( T ) only depends on ( X, ω ) and f . (cid:3) Lemma 2.5. Suppose the Chern-Yamabe flow exists on Ω T = X × [0 , T ) for some T > . Moreover, suppose that sup ≤ t We first get the parabolic H¨older norm bound for f . For any p ≥ ≤ t < T , k f ( x ; t ) k W ,p ( X ) ≤ C p (cid:16) k f ( x ; t ) k L p ( X ) + k ∆ Ch ω f ( x ; t ) k L p ( X ) (cid:17) ≤ C (cid:16) sup ≤ t SIMONE CALAMAI AND FANGYU ZOU Let L be any differential operator in x and t . A simple calculation showsthat ∂∂t ( L f ) − n − f /n ) ∆ Ch ω ( L f ) + S ( L f ) = − n f /n )( L S Ch ( ω )) . By the interior Schauder estimate for parabolic equations (Theorem 4.9 in[6]), for any τ, τ ′ with 0 ≤ τ < τ ′ < T , we have kL f k C α ( X × ( τ ′ ,T )) ≤ C Sch ( kL f k C ( X × ( τ,T )) + kL S Ch ( ω ) k C α ( X × ( τ,T )) )where the constant C Sch depends on τ , τ ′ , k S k C ( X × ( τ,T )) and k f k C α ( X × ( τ,T )) .It then follows by the standard bootstrapping argument to obtain that forany τ > k ∈ N and 0 < α < 1, there exists constant C ( k, α, τ, T ) suchthat k f k C k + α ( X × ( τ,T )) ≤ C ( k, α, τ, T ) . Together with the short time existence near t = 0, we havesup ≤ t The Chern-Yamabe flow exists as long as the maximumof Chern scalar curvature stays bounded. We therefore put forward the following conjecture to fully resolve the longtime existence of the flow. Conjecture 2.7. Suppose the Chern-Yamabe flow exists on Ω T = X × [0 , T ) form some T > . Then there exists some constant C ( T ) depending on T such that S ( x ; t ) ≤ C ( T ) , ∀ ( x, t ) ∈ Ω T . Balanced Case A Hermitian metric on a compact complex manifold is called balanced , ifits torsion 1-form θ vanishes. A balanced metric is automatically Gaudu-chon, but the reverse is not necessarily true. If the background metric ω isbalanced, the Chern Laplacian identifies with the Hodge-de Rham Laplacian∆ Ch ω = ∆ d . In this section we assume the background metric ω is balanced .3.1. F functional. When the background metric is balanced, the partialdifferential equation (1) with normalization (2) is the Euler-Lagrange equa-tion for the following functional(10) F ( f ) := 12 Z X | df | ω dµ + Z X S Ch ( ω ) f dµ with constraint(11) Z X exp(2 f /n ) dµ = 1 . NOTE ON CHERN-YAMABE PROBLEM 7 To solve the partial differential equation (1) is then equivalent to find acritical point of the functional (10) with constraint (11). It would be nice ifthe functional could be bounded from below. However, this is not the casewhen the complex dimension n ≥ Proposition 3.1. For ( X, ω ) with complex dimension n ≥ , we have inf (cid:26) F ( f ) : f ∈ C ∞ ( X ) with Z X exp(2 f /n ) dµ = 1 (cid:27) = −∞ . Proof. We will construct a family of Lipschitz functions { f r } parameterizedby a positive real number r , each of which satisfies the constraint (11),yet lim r → F ( f r ) = −∞ . Choose an arbitrary point p ∈ X as the center.The function f r ( x ) is defined as constants both inside the geodesic ball B r ( p )and outside the larger ball B r ( p ), while interpolated linearly on the annulus B r ( p ) /B r ( p ), namely, f r ( x ) = c r , | x | ≤ r (log r − c r ) (cid:0) | x | /r − (cid:1) + c r , r ≤ | x | ≤ r log r, | x | ≥ r where | x | denotes the distance to the center of the geodesic ball and c r isa constant depending on r . Choose the radius r sufficiently small, then thegeodesic ball B r (0) is close to a Euclidean ball and log r < 0. The constant c r is determined so that Z X exp(2 f r /n ) dµ = 1 . We claim c r ≤ − n log r − n C for some dimensional constant C = C ( n ). To see this,1 = Z X exp(2 f r /n ) dµ ≥ Z B r ( p ) exp(2 c r /n ) dµ = exp(2 c r /n ) Vol( B r ( p )) . Hence, c r ≤ − n B r ( p )) = − n Cr n ) = − n log r − n C. Now we show that lim r → F [ f r ] = −∞ . First of all, we have F ( f r ) = Z X | df r | ω dµ + Z X S Ch ( ω ) f r dµ = Z B r ( p ) \ B r ( p ) | df r | ω dµ + Z B r ( p ) S Ch ( ω ) f r dµ + Z X \ B r ( p ) S Ch ( ω ) f r dµ ω . (12)By continuity there exists some r > Z B r ( p ) S Ch ( ω ) dµ ≤ λ , ∀ r ≤ r . SIMONE CALAMAI AND FANGYU ZOU Note that λ = R X S Ch ( ω ) dµ , hence, Z X \ B r ( p ) S Ch ( ω ) dµ ≥ λ , ∀ r ≤ r . Take r sufficiently small so that log r < 0. Then c r > Z X exp(2 f r /n ) dµ = 1 . It follows that F ( f r ) ≤ ( c r − log r ) r Vol (cid:0) B r ( p ) \ B r ( p ) (cid:1) + k S Ch ( ω ) k C ( X ) c r Vol( B r ( p )) + λ r ≤ C ( c r − log r ) r n − + Cr n c r + λ r = λ r + O ( r n − log r ) . (13)When n ≥ 2, we have lim r → r n − log r = 0. The leading term for F ( f r ) is λ log r . Therefore, lim r → F ( f r ) = −∞ . This finishes the proof. (cid:3) Monotonicity along the Chern-Yamabe flow. Let F ( t ) = F ( f ( · ; t )).We have the following lemma showing the monotonicity of the F functionalalong the flow. Lemma 3.2. ddt F ( t ) = − Z X ( S − λ ) exp(2 f /n ) dµ. Proof. First, by Lemma 2.1, we have Z X ∂f∂t exp(2 f /n ) dµ = 0 . Hence, ddt F ( t ) = Z X ∂f∂t ( − ∆ d f + S Ch ( ω )) dµ = Z X ∂f∂t ( − ∆ d f + S Ch ( ω ) − λ exp(2 f /n )) dµ = − Z X ( S − λ ) exp(2 f /n ) dµ ≤ . The proof is finished. (cid:3) Second variation.Lemma 3.3. The second variation of F functional is given by (14) δ F ( u, v ) | f = Z X (cid:18) ( du, dv ) ω − λn exp(2 f /n ) uv (cid:19) dµ for any u and v in the tangent space of f , namely Z X exp(2 f /n ) udµ = 0 and Z X exp(2 f /n ) vdµ = 0 . NOTE ON CHERN-YAMABE PROBLEM 9 Proof. Note that the unconstrained functional is˜ F ( f ) = 12 Z X | df | ω dµ + Z X S Ch ( ω ) f dµ − nλ (cid:18)Z X exp(2 f /n ) dµ − (cid:19) . The second variation follows by simple calculation. (cid:3) Given some specific direction v , we have the second variation at v as δ F ( v ) = Z X (cid:18) | dv | − λn exp(2 f /n ) v (cid:19) dµ. It’s interesting that the positivity of the second variation may have somerelation with the Rayleigh quotient, or the first principal eigenvalue of theLaplacian operator λ . In the special case when the background Gauduchonmetric is itself a constant Chern-Scalar curvature metric, we have f = 0 isa critical point.If λ ≥ λ/n , then δ F ( v ) ≥ ( λ − λ/n ) Z X v dµ ≥ , ∀ v with Z X vdµ = 0shows that f = 0 is a local minimum.If λ < λ/n , then we can take some non-zero eigenvector v with R X v dµ =0 and δ F ( v ) ≤ ( λ − λ/n ) Z X v dµ < . Hence, f = 0 is a saddle point and unstable.To construct concrete example for the above argument, one can consider P × θ P with P and P both endowed with the standard Fubini-Studymetrics. For such family of complex manifolds, the background Fubini Studymetrics ω θ are constant Chern scalar curvature metrics; so we write down thefunctional F with respect to the reference metric ω θ , and f = 0 representsa constant scalar Chern curvature metric with F (0) = 0. By adjusting thescaling parameter θ , it is not hard to adjust λ and the total Chern scalarcurvature λ such that − λ + λ < 0; this makes possible to find a sequence ofconformal factors f k that are arbitrarily close to f = 0, and with F ( f k ) < F , then the flow starting at f k willnot converge to f = 0. The conclusion we can draw is that saddle points arepossible and we should not expect only local minima in general. Togetherwith the fact, proved in Lemma 3.2, that the F functional always is notbounded from below, the techniques for only minima is not enough.4. Additional results under assumptions We have already shown in Lemma 3.1 that the functional F is unboundedfrom below. So it is impossible to find a global minimum. Yet it is stillpossible that the functional is bounded from below along the flow for somespecific initial value. In particular, if the flow finally converges to a solution,one of the necessary conditions is that the functional is bounded under theflow.In this section we assume the flow exists on [0 , ∞ ) and(15) lim t →∞ F ( t ) ≥ C > −∞ . What can we say about the flow?Since the functional is decreasing and bounded from below, we can finda sequence of time slices { t k } , so that ddt F ( t k ) → 0. Let f k = f ( t k ) and S k = S (cid:0) exp(2 f k /n ) ω (cid:1) . Note that by Lemma 3.2, ddt F ( t ) = − Z X ( S − λ ) exp(2 f /n ) dµ = λ − Z X S exp(2 f /n ) dµ. On the other hand, by Lemma 2.2, we have S ( x ; t ) > − C . Hence, we have(16) Z X exp(2 f k /n ) dµ = 1 , Z X S k exp(2 f k /n ) dµ → λ and S k > − C, |F ( f k ) | ≤ C. Assuming uniform upper bound. In this section we assume thatthere exists uniform upper bound for the sequence { f k } in (16). We showthat there exists a smooth solution to the Chern-Yamabe equation 1. Inwhat follows the constant C may vary from line to line. Lemma 4.1. Suppose there is a sequence { f k } satisfying (16) . Supposeadditionally there exists some constant C such that max x ∈ X f k ( x ) ≤ C , ∀ k. Then k f k k H ≤ C. Proof. First of all, we have Z X ( S k exp(2 f k /n )) dµ ≤ exp(2 C /n ) Z X S k exp(2 f k /n ) dµ ≤ C. Note that S k = exp( − f k /n )(S Ch ( ω ) − ∆ f k ), namely, ∆ f k = S Ch ( ω ) − S k exp(2 f k /n ).Hence, we have k ∆ f k k L ( X ) ≤ C . Claim. Let ¯ f k = R X f k dµ . There exists some constant C > such that − C ≤ ¯ f k ≤ .Proof of the Claim. First of all, since Vol( X ) = 1, we haveexp(2 ¯ f k /n ) = exp (cid:16) Z X (2 f k /n ) dµ (cid:17) ≤ Z X exp(2 f k /n ) dµ = 1 . Hence, ¯ f k ≤ Z X S k exp(2 f k /n ) f k dµ = Z X ( − ∆ f k + S Ch ( ω )) f k dµ = 2 F ( f k ) − Z X (S Ch ( ω ) − λ ) f k − λ ¯ f k = 2 F ( f k ) − Z X ∆ hf k − λ ¯ f k = 2 F ( f k ) − Z X h ∆ f k − λ ¯ f k ≥ F ( f k ) − k h k L ( X ) k ∆ f k k L ( X ) − λ ¯ f k ≥ C − λ ¯ f k . (17) NOTE ON CHERN-YAMABE PROBLEM 11 On the other hand, since S k > − C , we have Z X S k exp(2 f k /n ) f k dµ = Z X ( S k + C ) exp(2 f k /n ) f k dµ − C Z X exp(2 f k /n ) f k dµ ≤ C exp( C /n ) Z X S k exp( f k /n ) dµ + C C + C · n e ≤ C exp( C /n ) (cid:16) Z X S k exp(2 f k /n ) dµ (cid:17) / + C ≤ C. (18)By (17) and (18), we obtain that ¯ f k ≥ − C . This finishes the proof of theClaim. (cid:3) We continue our proof for the Lemma. By Poincare inequality, thereexists some constant C p so that Z X ( f k − ¯ f k ) dµ ≤ C p Z X |∇ f k | dµ. On the other hand, Z X |∇ f k | dµ = Z X ( − f k ∆ f k ) dµ ≤ C p Z X f k dµ + C p Z X (∆ f k ) dµ. Hence, Z X f k dµ − ¯ f k ≤ Z X f k dµ + C p Z X (∆ f k ) dµ. Hence,(19) Z X f k dµ ≤ f k + C p Z X (∆ f k ) dµ. It then follows by Sobolev estimate that k f k k H ( X ) ≤ C (cid:16) k f k k L ( X ) + k ∆ f k k L ( X ) (cid:17) ≤ C (cid:16) | ¯ f k | + k ∆ f k k L ( X ) (cid:17) ≤ C. This finishes the proof. (cid:3) Proposition 4.2. Suppose there is a sequence { f k } satisfying (16) . Supposeadditionally there exists some constant C such that max x ∈ X f k ( x ) ≤ C , ∀ k. Then there exists a function f ∞ ∈ C ∞ ( X ) which solves the differential equa-tion (1) .Proof. By Lemma 4.1, we have k f k k H ( X ) ≤ C . Hence, by passing to asubsequence if necessary, we have f k ⇀ f ∞ weakly in H ( X ) for some f ∞ .It follows that f k → f ∞ strongly in L ( X ) and ∆ f k ⇀ ∆ f ∞ weakly in L ( X ). As a result of the strong convergence in L ( X ), by passing to asubsequence if necessary, we have f k → f ∞ dµ -a.e.. Then by Egonov’s theorem, for any δ > 0, there exists a subset Ω δ ⊂ X with Vol( X \ Ω δ ) < δ ,such that f k → f ∞ uniformly on Ω δ . We have Z Ω δ (∆ f k − S Ch ( ω )) exp( − f k /n ) dµ = Z Ω δ (∆ f k − S Ch ( ω )) exp( − f ∞ /n ) dµ + Z Ω δ (∆ f k − S Ch ( ω )) (cid:0) e − f k /n − e − f ∞ /n (cid:1) dµ ≥ Z Ω δ (∆ f k − S Ch ( ω )) exp( − f ∞ /n ) dµ − C k e − f k /n − e − f ∞ /n k L ∞ (Ω δ ) . Hence, lim inf k →∞ Z Ω δ (∆ f k − S Ch ( ω )) exp( − f k /n ) dµ ≥ lim inf k →∞ Z Ω δ (∆ f k − S Ch ( ω )) exp( − f ∞ /n ) dµ ≥ Z Ω δ (∆ f ∞ − S Ch ( ω )) exp( − f ∞ /n ) dµ. Notice that Z X (∆ f k − S Ch ( ω )) exp( − f k /n ) dµ = Z X S k exp(2 f k /n ) dµ → λ , as k → ∞ . Hence, Z Ω δ (∆ f ∞ − S Ch ( ω )) exp( − f ∞ /n ) dµ ≤ λ . Let δ → 0, we obtain that(20) Z X (∆ f ∞ − S Ch ( ω )) exp( − f ∞ /n ) dµ ≤ λ . Note that f k → f ∞ dµ -a.e., and f k ≤ C by assumption, we have f ∞ ≤ C dµ -a.e.. Then by Dominance Convergence Theorem, we have(21) Z X exp(2 f ∞ /n ) dµ = lim k →∞ Z X exp(2 f k /n ) dµ = 1 . By (20) and (21), we have Z X (cid:16) ∆ f ∞ − S Ch ( ω ) + λ exp(2 f ∞ /n ) (cid:17) exp( − f ∞ /n ) dµ ≤ . It follows that the equality holds and(22) ∆ f ∞ − S Ch ( ω ) + λ exp(2 f ∞ /n ) = 0 , dµ − a.e. . Since f ∞ ≤ C dµ -a.e., we have ∆ f ∞ = S Ch ( ω ) − λ exp(2 f ∞ /n ) ∈ L ∞ ( X ).Hence, f ∞ ∈ W ,p ( X ) for any p > 1. By Sobolev embedding theorem,this implies that f ∞ ∈ C ,α ( X ). Then ∆ f ∞ ∈ C ,α ( X ). By the standardbootstrapping argument, we eventually have f ∞ ∈ C ∞ ( X ). This finishesthe proof. (cid:3) NOTE ON CHERN-YAMABE PROBLEM 13 References [1] Daniele Angella, Simone Calamai, Cristiano Spotti. On Chern-Yamabe problem . Math.Res. Lett. 24 (2017), no. 3, 645-677[2] Daniele Angella, Simone Calamai, Cristiano Spotti. Remarks on Chern-Einstein Her-mitian metrics . ArXiv:1901.04309[3] Simone Calamai, Positive projectively flat manifolds are locally conformally flat-K¨ahlerHopf manifolds . ArXiv:1711.00929.[4] P. Gauduchon. Le th´eor`eme de l’excentricit´e nulle . C. R. Acad. Sci. Paris S´er. A-B 285(1977), no. 5, A387– A390.[5] Mehdi Lejmi, Ali Maalaoui, On the Chern-Yamabe flow . ArXiv:1706.04917[6] Gary M. Lieberman. Second Order Parabolic Differential Equations . World ScientificPress. ISBN 981-02-2883-X.(Simone Calamai) Dipartimento di Matematica e Informatica “Ulisse Dini”,Universit`a di Firenze, via Morgagni 67/A, 50134 Firenze, Italy E-mail address : [email protected] E-mail address : [email protected] (Fangyu Zou) Mathematics Department, Stony Brook University, Stony BrookNY, 11794-3651 USA E-mail address ::