aa r X i v : . [ m a t h . A P ] O c t I -PROPERNESS OF MABUCHI’S K -ENERGY KAI ZHENG
Abstract.
Over the space of K¨ahler metrics associated to a fixed K¨ahlerclass, we first prove the lower bound of the energy functional ˜ E β (1.7), thenwe provide the criterions of the geodesics rays to detect the lower bound of˜ J β -functional (1.3). They are used to obtain the properness of Mabuchi’s K -energy. The criterions are examined under (1.11) by showing the convergenceof the negative gradient flow of ˜ J β -functional. Contents
1. Introduction 12. Variational structure of ˜ J and ˜ E
63. Geodesics in the space of K¨aher potentials 94. A functional inequality of ˜ J β and ˜ E β
95. Proof of Theorem 1.1 106. Proof of Theorem 1.2 107. Proof of Theorem 1.3 128. Geodesic stability 139. Proof of Theorem 1.6 1410. Proof of Theorem 1.4 1411. Proof of Theorem 1.5 1511.1. Lower bound of the 2nd derivatives 1611.2. Upper bound of the 2nd derivatives 1711.3. Zero order estimate 20References 211.
Introduction
Let M be a compact K¨ahler manifold and Ω be an arbitrary K¨ahler class. Wechoose a K¨ahler metric ω in Ω and denote the space of K¨ahler potentials associatedto Ω by H Ω = { ϕ ∈ C ∞ ( M, R ) | ω ϕ = ω + √− ∂ ¯ ∂ϕ > } . Mabuchi’s K -energy [18] has the explicit formula (cf. [5] [25]) for any ϕ ∈ H Ω , ν ω ( ϕ ) = E ω ( ϕ ) + S · D ω ( ϕ ) + j ω ( Ric ( ω ) , ϕ ) . (1.1) Mathematics Subject Classification.
Primary 53C44; Secondary 58E30, 58B20.
In which, E ω ( ϕ ) = Z M log ω nϕ ω n ω nϕ ,D ω ( ϕ ) = 1 V Z M ϕω n − J ω ( ϕ ) ,J ω ( ϕ ) = √− V n − X i =0 i + 1 n + 1 Z M ∂ϕ ∧ ¯ ∂ϕ ∧ ω i ∧ ω n − − iϕ . and j ω ( Ric ( ω ) , ϕ )= − V n − X i =0 n !( i + 1)!( n − i − Z M ϕ · Ric ( ω ) ∧ ω n − − i ∧ ( √− ∂ ¯ ∂ϕ ) i . We also recall Aubin’s I -function, I ω ( ϕ ) = 1 V Z M ϕ ( ω n − ω nϕ ) = √− V n − X i =0 Z M ∂ϕ ∧ ¯ ∂ϕ ∧ ω i ∧ ω n − − iϕ . The properness of the K -energy ν ω ( ϕ ) is a kind of ”coercive” condition in thevariational theory. It was introduced in Tian [24], which states that there is anonnegative, non-decreasing function ρ ( t ) satisfying lim t →∞ ρ ( t ) = ∞ such that ν ω ( ϕ ) ≥ ρ ( I ω ( ϕ )) for all ϕ ∈ H Ω . It is conjectured to be equivalent to the existenceof the constant scalar curvature K¨ahler (cscK) metrics (Conjecture 7.12 in Tian[25]).When Ω = − C ( M ) or C ( M ) = 0, the function ρ is proved to be linear in Tian[25], Theorem 7.13, i.e. there are two positive constants A and B such that for all ϕ ∈ H Ω , ν ω ( ϕ ) ≥ AI ω ( ϕ ) − B. (1.2)In order to destine different notions of properness, in our paper, we say the K -energyis I -proper, if (1.2) holds.When Ω = C ( M ) > M , Phong-Song-Sturm-Weinkove [21] proved that Ding func-tional F ω ( ϕ ) (defined in Ding [8]) satisfies F ω ( ϕ ) ≥ AI ω ( ϕ ) − B. This inequality is a generalisation of the Moser-Trudinger inequalities on the sphere[20][19][26]. The I -properness of Ding functional also implies (1.2) by using the iden-tity between ν ω ( ϕ ) and F ω ( ϕ ) in Ding-Tian [9], we include the proof in Lemma 10.2for readers’ convenience.There are different notions of properness. In [7], Chen defined another proper-ness of the K -energy regarding to the entropy E ω ( ϕ ). The equivalent relationbetween the I -properness and the E -properness is discussed in [17]. Chen alsosuggest another properness which means that the K -energy bounds the geodesicdistance function. He furthermore conjectured that d -properness should be a nec-essary condition of the existence of the cscK or the general extremal K¨ahler metrics(see Conjecture/Question 2 in [5] and Conjecture/Question 6.1 in [6]). -PROPERNESS OF MABUCHI’S K -ENERGY 3 Let χ be a closed (1 , J -functional is defined to be the last two termsof the K -energy with Ric ( ω ) replaced by χ , J ω,χ ( ϕ ) = S · D ω ( ϕ ) + j ω ( χ, ϕ ) . We introduce a new parameter β within a range0 ≤ β < n + 1 n α. We then define a new functional to be˜ J βω,χ ( ϕ ) = J ω,χ ( ϕ ) + βJ ω ( ϕ ) . (1.3)Now we return back to the formula of the K -energy. With the notations aboveit is split into ν ω ( ϕ ) = E ω ( ϕ ) − βJ ω ( ϕ ) + ˜ J βω,Ric ( ω ) ( ϕ ) . (1.4)The lower bound of E ω ( ϕ ) is αI ω ( ϕ ) − C in Lemma 10.1. Inserting it into the K -energy, we arrive at the lower bound ν ω ( ϕ ) ≥ αI ω ( ϕ ) − C − βJ ω ( ϕ ) + inf ϕ ∈ H Ω ˜ J βω,Ric ( ω ) ( ϕ ) . Note that I -functional is equivalent to the J -functional,1 n + 1 I ω ( ϕ ) ≤ J ω ( ϕ ) ≤ nn + 1 I ω ( ϕ ) , then we have ν ω ( ϕ ) ≥ ( α − nβn + 1 ) I ω ( ϕ ) − C + inf ϕ ∈ H Ω ˜ J βω,Ric ( ω ) ( ϕ ) . (1.5)From this inequality, we observe that in order to prove the I -properness of the K -energy, it suffices to obtain the lower bound of the functional ˜ J βω,Ric ( ω ) .The critical points of ˜ J βω,χ satisfy a new fully nonlinear equation in H Ω , n · χ ∧ ω n − ϕ = c β · ω nϕ + βV ω n . (1.6)The constant c is a topological constant determined by c β = n [ χ ] · Ω n − Ω n − βV . We call such ω ϕ a ˜ J β -metric. We say that χ is semi-definiteif it is negative semi-definite or positive semi-definite.In these degenerate situation, (1.6) might have more than one solution. We firstprove the lower bound the ˜ J β -functional, when there is a ˜ J β -metric in Ω. Theorem 1.1.
Assume that χ is negative semi-definite (positive semi-definite) andthere is a ˜ J β -metric in Ω , then all ˜ J β -metrics has the same critical value and ˜ J β has lower (resp. upper) bound. There is another functional ˜ E β which is defined to be the square norm of thederivative of ˜ J β , ˜ E β ( ϕ ) = 1 V Z M ( c β − tr ω ϕ χ + β V ω n ω n ϕ ) ω n ϕ . (1.7) KAI ZHENG
The ˜ J β -function and the ˜ E β -functional play the roles as the K -energy and theCalabi energy in the study of extremal K¨ahler metrics. We next prove the lowerbound of ˜ E β .When χ is semi-definite, according to the 2nd variation formula of ˜ J β in (2.1),it is convex or concave along a C , geodesic ray ρ ( t ). Thus the limit of its firstderivative along ρ ( t ) exists F β ( ρ ) = lim t →∞ V Z M ∂ρ∂t ( c β − tr ω ϕ χ + β V ω n ω n ϕ ) ω n ϕ . (1.8)We require the following notions of the geodesic ray in the space of K¨ahlerpotentials. Definition 1.1.
We say a C , geodesic ray is • stable (semi-stable) if F β > F β ≥ • destabilising (semi-destabilising) if F β < F β ≤ • effective if lim sup t →∞ ˜ E β ( ρ ( t )) · t = 0 . Theorem 1.2.
Assume that χ is negative semi-definite. The following inequalityholds. inf ω ∈ Ω p ˜ E β ≥ sup ρ ( − F β ) . (1.9) The supreme is taking over all C , , effective, semi-destabilising geodesic ρ . We remark that when β = 0 and χ and ω are both algebraic, the lower boundof ˜ E was proved in Lejmi and Sz´ekelyhidi [15] in algebraic setting.We then prove the lower bound of ˜ J β without the existence of ˜ J β -metric. Theorem 1.3.
Suppose that χ is negative semi-definite. Assume that ˜ J β is boundedfrom below along a C , semi-destabilising geodesic ray and the infimum of theenergy ˜ E β is zero along this ray. Then ˜ J β is uniformly bounded from below in theentire K¨ahler class Ω . The tool we use here to obtain these lower bounds is based on Chen [7][6]. Theproof relies on the existence of the geodesic rays and the nonpositive curvatureproperty of the infinite dimensional space H Ω . In general, it is difficult to examinethe lower bound of functionals in an infinite dimensional space, however, Theo-rem 1.3 provides a method to examine it along only one geodesic ray.Furthermore, we apply Theorem 1.3 to the K -energy. When C ( M ) <
0, accord-ing to Aubin-Yau’s solution of the Calabi conjecture [29][1], there exists a uniqueK¨ahler metric ω such that Ric ( ω ) represents the first Chern class. So let χ = Ric ( ω )could be chosen to be <
0. We obtain the following criterion of the I -properness ofthe K -energy. Theorem 1.4.
Assume that there is a C , semi-destabilising geodesic ray ρ ( t ) such that along ρ ( t ) (1) ˜ J β is bounded from below,(2) the infimum of the energy ˜ E β is zero. -PROPERNESS OF MABUCHI’S K -ENERGY 5 Then the K -energy is I -proper. When Ω admits a ˜ J β -metric ϕ , the trivial geodesic ray ρ ( t ) = ϕ, ∀ t ≥ F β = 0, the first condition followsfrom Theorem 1.1 and the second one follows from Theorem 1.2.One way to obtain the critical metric of J -functional is its negative gradient flow.It was introduced in Chen [5] and also in Donaldson [10] from moment map picture.Theorem 1.1 in Song-Weinkove [22] showed that under the following condition ofa K¨ahler class Ω, that is, if there is a K¨ahler metric ω ∈ Ω such that − χ > − c · ω + ( n − χ ) ∧ ω n − >
0, the negative gradient flow of J -functionalconverges. Thus I -properness (1.2) holds when χ = Ric ( ω ) ∈ C ( M ) < − c · ω + ( n − Ric ( ω )) ∧ ω n − >
0. We extend their theorem to the negativegradient flow of ˜ J β -functional ∂ϕ∂t = − c β + nχ ∧ ω n − ϕ ω nϕ − βV ω n ω nϕ . (1.10)and prove its convergence in Proposition 11.2 under the condition, − χ > − c β · ω + ( n − χ ) ∧ ω n − > . The extra term involving β on the flow equation brings us trouble when we apply thesecond order estimate. In order to overcome this problem, we calculate a differentialinequality by using the linear elliptic operator L defined in (11.5) and apply themaximum principal.We remark that (1.6) and its flow have been generalised in different directions[14][13][12][16]... which is far from a complete list.Thus we verify the criterion in Theorem 1.4. Theorem 1.5.
Assume that there is a ω ∈ Ω such that ( − c β · ω + ( n − Ric ( ω )) ∧ ω n − > . (1.11) Then from any K¨ahler potential ϕ ∈ H Ω , there exists a C , semi-destabilisinggeodesic ray satisfying (1) and (2) . Thus the K -energy is I -proper in Ω . Paralleling to Donaldson’s conjecture of existence of the cscK metrics (Conjec-ture/Question 12 in [11]), we propose a notion called geodesic stability w.r.t to the˜ J β -functional (see Definition 8.1). We at last link the existence of ˜ J β -metric to thisgeodesic stability. Theorem 1.6.
Suppose that χ is negative semi-definite. Assume that Ω contains a ˜ J β -metric ϕ , then Ω is geodesic semi-stable at ϕ and moreover, it is weak geodesicsemi-stable. The criterion (8.1) means that along the geodesic ray, the first derivative ofthe ˜ J β -functional is strictly increase. The question 8.1 suggests that there is nosuch geodesic ray satisfying (8.1) implies the existence of ˜ J β -metric. Then fromTheorem 1.1 and (1.5), the K -energy is I -proper. So according to Tian’s conjecture(Conjecture 7.12 in [25]), there exists cscK metrics. In this sense, the question8.1 probably provides another possible point of view of Donaldson’s conjecture(Conjecture/Question 12 in [11]). KAI ZHENG
We further remark that with these theorems, it would be more interesting tofind the examples of K¨ahler class where the ˜ J β -functional has lower bound but the˜ J β -metric does not exist.2. Variational structure of ˜ J and ˜ E Recall our definition for any ϕ ∈ H Ω ,˜ J βω,χ ( ϕ ) = J ω,χ ( ϕ ) + β · J ω ( ϕ ) . Let ϕ ( t ) be a smooth family of K¨ahler potentials with ϕ (0) = ϕ . We denote δ = ddt | t =0 and ˙ ϕ = δϕ ( t ) . Lemma 2.1.
The 1st variation of ˜ J -functional is δ ˜ J β ( ˙ ϕ ) = 1 V Z M ˙ ϕ [ c β · ω nϕ − n · χ ∧ ω n − ϕ + βV ω n ] . Proof.
We compute δ ˜ J β ( ˙ ϕ ) = 1 V Z M ˙ ϕ ( c · ω nϕ − n · χ ∧ ω n − ϕ ) + βV Z M ˙ ϕ ( ω n − ω nϕ )= 1 V Z M ˙ ϕ [( c − βV ) · ω nϕ − n · χ ∧ ω n − ϕ + βV ω n ] . (cid:3) Lemma 2.2.
The 2nd variation of ˜ J -functional is δ ˜ J β ( ˙ ϕ, ˙ ϕ ) = 1 V Z M ( ¨ ϕ − | ∂ ˙ ϕ | )( c β − tr ω ϕ χ ) ω n ϕ − Z M χ i¯j ˙ ϕ i ˙ ϕ ¯j ω n ϕ . (2.1) Proof.
We compute directly, δ ˜ J β ( ˙ ϕ, ˙ ϕ ) = ddt V Z M ˙ ϕ [( c β − g i ¯ jϕ χ i ¯ j ) ω nϕ + βV ω n ]= 1 V Z M ¨ ϕ ( c β − g i ¯ jϕ χ i ¯ j ) ω nϕ + 1 V Z M ˙ ϕ ˙ ϕ i ¯ j χ i ¯ j ω nϕ + 1 V Z M ˙ ϕ ( c β − g i ¯ jϕ χ i ¯ j ) △ ϕ ˙ ϕω nϕ . The second term becomes1 V Z M ˙ ϕ ˙ ϕ i ¯ j χ i ¯ j ω nϕ = − V Z M ˙ ϕ ¯ j ˙ ϕ i χ i ¯ j ω nϕ − V Z M ˙ ϕ ˙ ϕ i ( χ i ¯ j ) ¯ j ω nϕ = − V Z M ˙ ϕ ¯ j ˙ ϕ i χ i ¯ j ω nϕ − V Z M ˙ ϕ ˙ ϕ i ( χ j ¯ j ) i ω nϕ . The third term is1 V Z M ˙ ϕ ( c β − g i ¯ jϕ χ i ¯ j ) △ ϕ ˙ ϕω nϕ = − V Z M ( c β − g i ¯ jϕ χ i ¯ j ) | ∂ ˙ ϕ | ω nϕ + 1 V Z M ˙ ϕ ( g i ¯ jϕ χ i ¯ j ) ¯ l ˙ ϕ k g k ¯ lϕ ω nϕ . Then the lemma follows from adding them together. (cid:3) -PROPERNESS OF MABUCHI’S K -ENERGY 7 Therefore, when χ is strictly negative (positive), the ˜ J β -metric is local minimum(maximum). Proposition 2.3.
When χ is strictly negative or strictly positive, the ˜ J β -metric isunique up to a constant.Proof. Assume ϕ and ϕ are two ˜ J β -metrics. Then connecting them by the C , geodesic. Since all the computation above is well-defined along the C , geodesics,(2.1) implies that δ ˜ J ( ˙ ϕ, ˙ ϕ ) = − V Z M χ i ¯ j ( ω ) ˙ ϕ i ˙ ϕ ¯ j ω nϕ . Then integrating from 0 to 1, we have1 V Z Z M χ i ¯ j ( ω ) ˙ ϕ i ˙ ϕ ¯ j ω nϕ dt = δ ˜ J (1) − δ ˜ J (0) = 0 . Hence, ˙ ϕ is constant and ϕ and ϕ differ by a constant. (cid:3) We use the notion ˜ H = tr ω ϕ χ − c β − β V ω n ω n ϕ . The ˜ J β -metric is a K¨ahler metric satisfying˜ H = 0 . We define the energy ˜ E β as˜ E β ( ϕ ) = 1 V Z M (tr ω ϕ χ − c β − β V ω n ω n ϕ ) ω n ϕ . (2.2)Then we have δ ˜ H ( ˙ ϕ ) = − ˙ ϕ i ¯ j χ i ¯ j + βV △ ϕ ˙ ϕ ω n ω nϕ . Lemma 2.4.
The 1st derivative of the modified energy ˜ E is δ ˜ E β ( ˙ ϕ ) = 2 V Z M ˜ H ¯ j ˙ ϕ i χ i ¯ j ω nϕ − βV Z M ˜ H i ˙ ϕ i ω n . (2.3) Proof.
We calculate that δ ˜ E β ( ˙ ϕ ) = 2 V Z M ˜ H ( − ˙ ϕ i ¯ j χ i ¯ j + βV △ ϕ ˙ ϕ ω n ω nϕ ) ω nϕ + 1 V Z M ˜ H △ ϕ ˙ ϕω nϕ . The first term is2 V Z M ˜ H ¯ j ˙ ϕ i χ i ¯ j ω nϕ + 2 V Z M ˜ H ˙ ϕ i ( χ i ¯ j ) ¯ j ω nϕ = 2 V Z M ˜ H ¯ j ˙ ϕ i χ i ¯ j ω nϕ + 2 V Z M ˜ H ˙ ϕ i ( ˜ H + βV ω n ω nϕ ) i ω nϕ . While, the second term is2 V Z M ˜ H βV △ ϕ ˙ ϕ ω n ω nϕ ω nϕ = − V Z M ˜ H i βV ˙ ϕ i ω n ω nϕ ω nϕ − V Z M ˜ H βV ˙ ϕ i ( ω n ω nϕ ) i ω nϕ KAI ZHENG and the third term is − V Z M ˜ Hg i ¯ jϕ ˜ H ¯ j ˙ ϕ i ω nϕ which cancels the second component in the first term. (cid:3) The critical points of ˜ E satisfy that[ ˜ H ¯ j χ i ¯ j ω nϕ − βV ˜ H i ω n ω nϕ ] i = 0 . Lemma 2.5.
The 2nd derivative of the modified energy ˜ E β is δ ˜ E β ( u, v ) = 2 V Z M ( v p ¯ q χ p ¯ q )( u i ¯ j χ i ¯ j ) ω nϕ + 2 βV Z M g k ¯ jϕ ( △ ϕ v ω n ω nϕ ) k g i ¯ lϕ u ¯ l χ i ¯ j ω nϕ (2.4) − βV Z M g i ¯ jϕ ( − v p ¯ q χ p ¯ q + βV △ ϕ v ω n ω nϕ ) i u ¯ j ω n . Proof.
In the local coordinante, (2.3) is written as δ ˜ E β ( u ) = 2 V Z M g k ¯ jϕ ˜ H k g i ¯ lϕ u ¯ l χ i ¯ j ω nϕ − βV Z M g i ¯ jϕ ˜ H i u ¯ j ω n , we obtain that δ ˜ E β ( u, v )(2.5) = − V Z M v k ¯ j ˜ H k g i ¯ lϕ u ¯ l χ i ¯ j ω nϕ + 2 V Z M g k ¯ jϕ ( − v p ¯ q χ p ¯ q + βV △ ϕ v ω n ω nϕ ) k g i ¯ lϕ u ¯ l χ i ¯ j ω nϕ − V Z M g k ¯ jϕ ˜ H k v i ¯ l u ¯ l χ i ¯ j ω nϕ + 2 V Z M g k ¯ jϕ ˜ H k g i ¯ lϕ u ¯ l χ i ¯ j △ ϕ vω nϕ + 2 βV Z M v i ¯ j ˜ H i u ¯ j ω n − βV Z M g i ¯ jϕ ( − v p ¯ q χ p ¯ q + βV △ ϕ v ω n ω nϕ ) i u ¯ j ω n . The second term is further reduced to, − V Z M g k ¯ jϕ ( v p ¯ q χ p ¯ q ) k g i ¯ lϕ u ¯ l χ i ¯ j ω nϕ = − V Z M ( v p ¯ q χ p ¯ q ) ¯ j u i χ i ¯ j ω nϕ = 2 V Z M ( v p ¯ q χ p ¯ q )( u i ¯ j χ i ¯ j ) ω nϕ + 2 V Z M ( v p ¯ q χ p ¯ q ) u i ˜ H i ω nϕ . Thus the lemmas holds by inserting this formula into (2.5). (cid:3)
When β = 0, the variational structure of ˜ J ω,χ and ˜ E is studied in Chen [4]. Wedenote H = tr ω ϕ χ − c . The K¨ahler metric is called a J -metric if it satisfies H = 0. From (2.3), the 1stderivative of ˜ E -energy is δ ˜ E ( ˙ ϕ ) = 2 V Z M H ¯ j ˙ ϕ i χ i ¯ j ω nϕ . (2.6)From this formula, the critical metrics satisfy the equation[ H ¯ j χ i ¯ j ] i = 0 . (2.7) -PROPERNESS OF MABUCHI’S K -ENERGY 9 The critical metrics of the modified energy include the J -metrics. (2.4) shows that,at the critical point of J , δ ˜ E ( u, v ) = 2 V Z M ( v p ¯ q χ p ¯ q )( u i ¯ j χ i ¯ j ) ω nϕ . So the J -metric is local minimiser of ˜ E . However, it is not known whether all thecritical metrics of the energy ˜ E are minimisers. While, (2.6) suggests that when χ is strictly positive or negative, the critical metrics of the modified energy is the J -metric. 3. Geodesics in the space of K¨aher potentials
We recall the necessary progress of constructing the geodesic ray in this sectionfor the next several sections. the existence of the C , geodesic segment is proved inChen [ ? ]. In Calamai-Zheng [3], we improve the following existence of the geodesicsegment with slightly weaker boundary geometric conditions. Now we specify thegeometric conditions on the boundary metrics. Definition 3.1.
We label as H C ⊂ H Ω one of the following spaces; I = { ϕ ∈ H Ω such that sup Ric ( ω ϕ ) ≤ C } ; I = { ϕ ∈ H Ω such that inf Ric ( ω ϕ ) ≥ C } . Theorem 3.1. (Calamai-Zheng [3] ) Any two K¨ahler metrics in H C are connectedby a unique C , geodesic. More precisely, it is the limit under the C , -norm by asequence of C ∞ approximate geodesics. Due to Calabi-Chen [2], H has positive semi-definite curvature in the sense ofAleksandrov. Two geodesic ray ρ i are called paralleling if the geodesic distancebetween ρ ( t ) and ρ ( t ) is uniformly bounded. Lemma 3.2.
Given a geodesic ray ρ ( t ) in H C and a K¨ahler potential ϕ which isnot in ρ ( t ) . There is a C , geodesic ray starting from ϕ and paralleling to ρ ( t ) .Proof. According to Theorem 3.1 we could connect ϕ and ρ ( t ) by a C , geo-desic segment γ t ( s ) which have uniform C , norm. Thus after taking limit of theparameter t , we obtain a limit geodesic ray in W ,p , ∀ p ≥ C ,α , ∀ α < γ ( s ) = lim t →∞ γ t ( s ) . (cid:3) Remark . The condition of ρ ( t ) could be weakened to be the tamed conditionin Chen [7]. We only require that there is a ˜ ρ ( t ) ∈ H C and ˜ ρ ( t ) − ρ ( t ) is uniformlybounded. 4. A functional inequality of ˜ J β and ˜ E β We first prove a functional inequality.
Proposition 4.1.
Let ϕ and ϕ be two K¨ahler potentials then the following in-equality holds. ˜ J β ( ϕ ) − ˜ J β ( ϕ ) ≤ d ( ϕ , ϕ ) · q ˜ E β ( ϕ ) . Proof.
The functional inequality is proved by direct computation. Let ρ ( t ) be a C , geodesic segment connecting ϕ and ϕ .˜ J β ( ϕ ) − ˜ J β ( ϕ ) ≤ Z d ˜ J β ( ∂ρ∂t ) ϕ dt ≤ s V Z M ˜ H ω nϕ · sZ Z M ( ∂ρ∂t ) ω nϕ dt. Thus the resulting inequality follows from the H¨older inequality. (cid:3) Proof of Theorem 1.1
Proof.
Let ϕ be any K¨ahler potential in H Ω and ϕ be a ˜ J β -metric. Connecting ϕ and ϕ by a C , geodesic segment γ ( t ) and computing the expansion formulaalong γ ( t ) ˜ J β (1) − ˜ J β (0) = Z ∂ ˜ J β ∂t dt = Z ∂ ˜ J β ∂t ( t ) − ∂ ˜ J β ∂t (0) dt = Z Z t ∂ ˜ J β ∂t dsdt. In the second identify we use the assumption that ϕ is a ˜ J β -matric, so ∂ ˜ J β ∂t (0) = 0 . Applying the 2nd formula of the ˜ J β , Lemma 2.1, we see that(˜ J β ) ′′ ≥ γ ( t ). As a result, we obtain that˜ J β (1) ≥ ˜ J β (0) . Furthermore, assume that ϕ is another ˜ J β -metric when the solution is not unique,then we have ˜ J β (1) ≥ ˜ J β (0) . Switching the positions of ϕ and ϕ , we see that all ˜ J β -metrics has the same criticalvalue of ˜ J β . (cid:3) Proof of Theorem 1.2
Proof.
Let ρ ( t ) be a geodesic ray parameterized by the arc length and satisfy theassumption in the theorem. Let ϕ be a K¨ahler potential outside ρ ( t ) and connect-ing ϕ and ρ ( t ) by a C , geodesic γ t ( s ) which is also parameterized by the arclength. Let θ be the angle expanding by −−−−−→ ρ ( t ) ρ (0) and −−−−−→ ρ ( t ) ϕ (0).Since H Ω is nonpositive curve, we obtain d ( ϕ , ρ (0)) ≥ d by comparing the cosine formulae in the Euclidean space d = d ( ϕ , ρ ( t )) + d ( ρ (0) , ρ ( t )) − d ( ϕ , ρ ( t )) d ( ρ (0) , ρ ( t )) cos θ. -PROPERNESS OF MABUCHI’S K -ENERGY 11 Then knowing that d ( ρ (0) , ρ ( t )) = t, and letting d t = d ( ϕ , ρ ( t )) be the distance between ϕ and ρ ( t ), we have d ≥ d t + t − d t · t · cos θ = d t + t − d t · t + 2 d t · t − d t · t · cos θ ≥ d t · t · (1 − cos θ ) . Thus the cosine formula implies2(1 − cos θ ) ≤ d t · d t . While, the triangle inequality implies that t − d ≤ d t ≤ t + d . When t is sufficient large, we further have d ≤ t . Thus 0 ≤ − ( ∂ρ∂t , ∂γ∂s )) ρ ( t ) (6.1) = 2(1 − cos θ ) ≤ d t · d t ≤ d t · ( t − d ) ≤ d t . Applying the H¨older inequality to d ˜ J β ( ∂γ∂s ) ρ ( t ) ≤ d ˜ J β ( ∂γ∂s − ∂ρ∂t ) ρ ( t ) + d ˜ J β ( ∂ρ∂t ) ρ ( t ) , then using (6.1), we obtain d ˜ J β ( ∂γ∂s ) ρ ( t ) ≤ q ˜ E β ( ρ ( t )) r − ∂γ∂s , ∂ρ∂t ) ρ ( t ) + d ˜ J β ( ∂ρ∂t ) ρ ( t ) ≤ q ˜ E β ( ρ ( t )) √ · d t + d ˜ J β ( ∂ρ∂t ) ρ ( t ) . (6.2)Since ρ ( t ) is effective ˜ E β ( ρ ( t )) = o ( t ) t , the first term becomes o ( t ). Then d ˜ J β ( ∂γ∂s ) ρ ( t ) ≤ o ( t ) + d ˜ J β ( ∂ρ∂t ) ρ ( t ) . (6.3)On the other hand, note that (˜ J β ) ′ and (˜ J β ) ′′ are well-defined along C , geodeisc.When χ is negative semi-definite, from Lemma 2.1,(˜ J β ) ′′ ( γ ( s )) ≥ . So d ˜ J β ( ∂γ∂s ) ϕ (0) ≤ d ˜ J β ( ∂γ∂s ) ρ ( t ) . Thus combining (6.3), we have d ˜ J β ( ∂γ∂s ) ϕ (0) ≤ o ( t ) + d ˜ J β ( ∂ρ∂t ) ρ ( t ) . Inverting this inequality, − o ( t ) − d ˜ J β ( ∂ρ∂t ) ρ ( t ) ≤ − d ˜ J β ( ∂γ∂s ) ϕ (0) . (6.4)The right hand side is controlled by the H¨older inequality again q ˜ E β ( ϕ ) · ( Z M ( ∂γ∂s ) | s =0 ω nϕ ) = q ˜ E β ( ϕ ) . The inequality follows from choosing the unit arc-length of γ . Taking t → ∞ onboth sides of (6.4), − F β ( ρ ) ≤ q ˜ E β ( ϕ ) . Thus the theorems follows. (cid:3) Proof of Theorem 1.3
Proof.
Since when χ is negative semi-definite, (˜ J β ) ′′ ≥ γ t ( s ), ∂ ˜ J β ∂s is non-decreasing. Then letting τ ( t ) be the length of the γ t ( s ), we have˜ J β ( ρ ( t )) − ˜ J β ( ϕ ) = Z τ ( t )0 d ˜ J β ( ∂γ∂s ) ds ≤ Z τ ( t )0 d ˜ J β ( ∂γ∂s ) ρ ( t ) ds. From (6.2) in the proof above, we obtain that d ˜ J β ( ∂γ∂s ) ρ ( t ) ≤ q ˜ E β ( ρ ( t )) r − ∂γ∂s , ∂ρ∂t ) ρ ( t ) + d ˜ J β ( ∂ρ∂t ) ρ ( t ) ≤ q ˜ E β ( ρ ( t )) √ · d t + d ˜ J β ( ∂ρ∂t ) ρ ( t ) . (7.1)From the assumption that ρ ( t ) is semi-destabilising, so d ˜ J β ( ∂ρ∂t ) ρ ( t ) ≤ . Putting the inequalities above together, we arrive at˜ J β ( ρ ( t )) − ˜ J β ( ϕ ) ≤ q ˜ E β ( ρ ( t )) C · d ( ϕ , ρ (0)) t τ ( t ) . Taking limit of t , since τ ( t ) = O ( t )and from assumption in Theorem 1.3 along ρ ( t ),lim t →∞ q ˜ E β ( ρ ( t )) = 0 , we have ˜ J β ( ϕ ) ≥ lim t →∞ ˜ J β ( ρ ( t )) . -PROPERNESS OF MABUCHI’S K -ENERGY 13 Thus the theorem follows from the assumption that ˜ J β is bounded below along ρ ( t ). (cid:3) Geodesic stability
Inspired from the geodesic conjecture of the extremal metrics in Donaldson [11],we proposal a counterpart of ˜ J β -metric. Conjecture/Question 8.1.
The following are equivalent:(1) There is no ˜ J β -metric in H Ω .(2) There is infinite geodesic ray ϕ ( t ) , t ∈ [0 , ∞ ) , in H Ω such that V Z M ∂ϕ∂t ( c β − tr ω ϕ χ + β V ω n ω n ϕ ) ω n ϕ > for all t ∈ [0 , ∞ ) .(3) For any point ϕ ∈ H Ω , there is a geodesic ray in (2) starting at ϕ . We need some definitions.
Definition 8.1.
A K¨ahler class is called • geodesic semi-stable at a point ϕ if every non-trivial C , geodesic raystarting from ϕ is semi-stable. • geodesic semi-stable if every non-trivial C , geodesic ray is semi-stable. • weak geodesic semi-stable if every non-trivial geodesic ray with uniform C , bound is semi-stable.We say a C , geodesic ray is trivial if it is just a point. Proposition 8.2.
Suppose that χ is negative semi-definite. We assume that thereis a C , geodesic ray ρ ( t ) staying in H C and the ˜ J β -functional is non-increasingalong ρ ( t ) . If there is a ˜ J β -metric, then ρ ( t ) converges to the ˜ J β -metric.Proof. Let ϕ be a ˜ J β -metric. We first connect ϕ and ρ ( t ) by a C , geodesicsegment γ t ( s ), this follows from Theorem 3.1 since ρ ( t ) ∈ H C . Moreover, since the C , norm is uniform, after taking limit on t , we obtain a C , geodesic ray γ ( s )starting at ϕ . Thus, ˜ J β strongly converges and is well-defined along γ ( s ).Since the ˜ J β is non-increasing along ρ ( t ), so ˜ J β has upper bound along γ ( s ).While, Theorem 1.1 implies that when Ω has a ˜ J β -metric, then ˜ J β has lower bound.Meanwhile, when χ is negative semi-definite, from Lemma 2.1, ˜ J β is convex alongthe geodesic ray γ ( s ). Moreover, ˜ J β obtains its lower bound at s = 0. So, we claimthat ˜ J β ( s ) ≡ min ˜ J β along γ ( s ). I.e. γ ( s ) are constituted of ˜ J β -metrics.We prove this claim by the contradiction method. Since along γ ( s ), the firstderivative (˜ J β ) ′ is non-negative, we assume that s is the first finite time such that(˜ J β ) ′ is strictly positive, otherwise, the claim is proved. Since along γ ( s ), (˜ J β ) ′′ isalso non-negative, so (˜ J β ) ′ is strictly positive for any s ≥ s . This is a contradictionto lim s →∞ (˜ J β ) ′ ( s ) = 0 which follows from that ˜ J β is bounded and monotonic. (cid:3) Remark . When χ is strictly negative, using Lemma 2.1 again, we see that V R M χ i ¯ j ˙ γ i ˙ γ ¯ j ω nγ = 0. This implies γ ( s ) is just a point which coincides with ϕ .Therefore ρ ( t ) will converges to ϕ . Remark . If a C , geodesic ray γ ( t ) is destabilizing, then the ˜ J β -functional isnon-increasing when t is large enough. Proof of Theorem 1.6
Proof.
Due to Theorem 1.1, ϕ is a global minimiser. So ˜ J β is non-decreasing alongany C , geodesic ray ρ ( t ). So the first statement holds. For the second statement,we consider the sign of F β and prove by contradiction method. Assume that ρ ( t )is a geodesic ray with uniform C , bund and F β is strictly negative along it. Soaccording to the definition of F β (1.8), when t is large enough, d ˜ J β ( ∂ρ∂t ) ρ ( t ) < . According to Proposition 8.2, ρ ( t ) will converges to a ˜ J β -metric and F β = 0. Con-tradiction! So the theorem follows. (cid:3) Proof of Theorem 1.4
Recall the entropy E ω ( ϕ ) = 1 V Z M log ω nϕ ω n ω nϕ . The proof of Theorem 1.4 follows from the following lemma and (1.4).
Lemma 10.1. (Tian [25] ) There is a uniform constant C = C ( ω ) > , E ω ( ϕ ) ≥ αI ω ( ϕ ) − C, ∀ ϕ ∈ H . (10.1) Proof.
The α -invariant was introduced by Tian [23]: α ([ ω ]) = sup { α > |∃ C > , s.t. Z M e − α ( ϕ − sup M ϕ ) ω n ≤ C holds for all ϕ ∈ H } > . From the definition of the α -invariant Z M e − α ( ϕ − V R M ϕω n ) − h ω nϕ = Z M e − α ( ϕ − V R M ϕω n ) ω n ≤ Z M e − α ( ϕ − sup M ϕ ) ω n and then the Jensen inequality Z M [ α ( − ϕ + 1 V Z M ϕω n ) − log ω nϕ ω n ] ω nϕ ≤ C, we obtain the lower bound of the entropy. (cid:3) Lemma 10.2. I -properness of Ding functional implies I -properness of Mabuchi K -energy.Proof. From assumption, in Ω = C ( M ), there are two positive constants A and A such that for all ϕ ∈ H Ω , F ω ( ϕ ) ≥ A I ω ( ϕ ) − A . (10.2)Let f be the scalar potential which is defined to be the solution of the equation △ ϕ f = S − S with the normalisation condition Z M e f ω nϕ = V. -PROPERNESS OF MABUCHI’S K -ENERGY 15 Ding-Tian [9] introduced the following energy functional A ( ϕ ) = 1 V Z M f ω nϕ . Let H be the space of K¨ahler potential ϕ under the normalization condition Z M e − ϕ + h ω ω n = V. In H , the relation between Mabuchi K -energy and Ding F -functional is F ω ( ϕ ) = ν ω ( ϕ ) + A ( ϕ ) − A (0) . Applying the Jensen inequality to the normalization condition of f , we have A ( ϕ ) ≤
0. Thus the I -properness of Mabuchi K -energy is achieved by another positiveconstant A from (10.2), ν ω ( ϕ ) ≥ A I ω ( ϕ ) − A . (cid:3) Proof of Theorem 1.5
We construct the required geodesic ray by using the ˜ J β -flow. Proposition 11.1.
Assume that the ˜ J β -flow converges to a ˜ J β -metric. From anyK¨ahler potential ψ , there exists a semi-destabilising C , -geodesic ray such that(1) ˜ J β is bounded from below,(2) the infimum of the energy ˜ E β is zero.Proof. We connect ψ to the ˜ J β -flow ϕ ( t ) with the C , -geodesic ϕ t ( s ). Then wedefine ρ ( s ) = lim t →∞ ϕ t ( s ). Since the ˜ J β -flow ϕ ( t ) satisfies two conclusions in thisproposition and the end-points of each ρ t ( s ) are all in ϕ ( t ), so ρ ( s ) also satisfiesthese two conclusion automatically. The semi-destabilising is proved as following. F β ( ρ ) = lim s →∞ δ ˜ J β ( ∂ρ∂s ) ρ ( s ) ≤ lim s →∞ lim t →∞ d ˜ J β ( ∂ρ∂s − ∂ϕ∂t ) ρ t ( s ) + d ˜ J β ( ∂ϕ∂t ) ρ t ( s ) = lim s →∞ lim t →∞ d ˜ J β ( ∂ρ∂s − ∂ϕ∂t ) ϕ ( t ) + d ˜ J β ( ∂ϕ∂t ) ϕ ( t ) ≤ lim s →∞ lim t →∞ d ˜ J β ( ∂ρ∂s − ∂ϕ∂t ) ϕ ( t ) . From (6.1), we further have the right hand side is bounded by ≤ lim s →∞ lim t →∞ q ˜ E β ( ϕ ( t )) r − ∂ρ∂s , ∂ϕ∂t ) ϕ ( t ) ≤ lim s →∞ lim t →∞ q ˜ E β ( ϕ ( t )) C · d ( ϕ , ρ (0)) t = 0 . Thus, the proposition holds. (cid:3)
Now we prove the convergence of the negative gradient flow ˜ J β -functional. As-sume that there is a ω ∈ Ω such that( − nc β · ω + ( n − χ ) ∧ ω n − > . (11.1)and − χ > . (11.2) Proposition 11.2.
The conditions (11.1) and (11.2) is equivalent to convergenceof the ˜ J β -flow to a ˜ J β -metric. The shot tome existence from the fact that the linearisation operator L is elliptic.In the following, we prove the a priori estimates. As long as we have the secondorder estimate and the zero estimate, the C ,α estimate follows from the Evans-Krylov estimate. The higher order estimates is obtained by the bootstrap method.Recall the ˜ J β -flow, ˙ ϕ = − c β + nχ ∧ ω n − ϕ ω nϕ − βV ω n ω nϕ . (11.3)We take derivative ∂ t on the both sides,¨ ϕ = − ˙ ϕ i ¯ j χ i ¯ j + βV △ ϕ ˙ ϕ ω n ω nϕ (11.4) = ˙ ϕ i ¯ j [ − g k ¯ jϕ g i ¯ lϕ χ k ¯ l + βV ω n ω nϕ g i ¯ jϕ ] . We denote L = [ − g k ¯ jϕ g i ¯ lϕ χ k ¯ l + βV ω n ω nϕ g i ¯ jϕ ] ∂ i ∂ ¯ j . (11.5)From (11.2), we see that on the short time interval, L is an elliptic operator, i.e. − χ + βV ω n ω nϕ ω ϕ > . (11.6)From the maximum principle, we havemin M ˙ ϕ (0) ≤ ˙ ϕ ( t ) ≤ max M ˙ ϕ (0) . (11.7)11.1. Lower bound of the 2nd derivatives.
Using the flow equation, we havemin M ˙ ϕ (0) ≤ ˙ ϕ ( t ) = − c β + nχ ∧ ω n − ϕ ω nϕ − βV ω n ω nϕ = − c β + g i ¯ jϕ χ i ¯ j − βV ω n ω nϕ ≤ − c β + g i ¯ jϕ χ i ¯ j . In the following, we always use the normal coordinate diagonalize ω and ω ϕ suchthat their eigenvalues are 1 and λ i for 1 ≤ i ≤ n respectively. Denote the diagonalof χ by µ i .Thus for any 1 ≤ i ≤ n , − µ i λ i ≤ min M ˙ ϕ (0) − c β , -PROPERNESS OF MABUCHI’S K -ENERGY 17 or λ i ≥ − µ i min M ˙ ϕ (0) − c β . Upper bound of the 2nd derivatives.
Let A = χ i ¯ j g ϕi ¯ j . When we work on the second order estimate, the extra term in the equation causethe trouble, we overcome it by using the linearisation operator L as the ellipticoperator. Then we compute ( ∂ t − L )(log A − Cϕ ) . Let B = g p ¯ qϕ χ p ¯ q . We have B i ¯ j = [ g p ¯ qϕ χ p ¯ q ] i ¯ j = − ( g r ¯ qϕ g p ¯ sϕ ( g ϕr ¯ s ) i ) ¯ j χ p ¯ q − g p ¯ qϕ R p ¯ qi ¯ j ( χ )(11.8) = [ − g r ¯ qϕ g p ¯ sϕ ( g ϕr ¯ s ) i ¯ j + g r ¯ qϕ g p ¯ bϕ g a ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i + g r ¯ bϕ g a ¯ qϕ g p ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i ] χ p ¯ q − g p ¯ qϕ R p ¯ qi ¯ j ( χ ) . So using the flow equation, ∂ t A = χ i ¯ j ˙ ϕ i ¯ j (11.9) = χ i ¯ j [ − c β + g p ¯ qϕ χ p ¯ q − βV ω n ω nϕ ] i ¯ j = χ i ¯ j [ − g r ¯ qϕ g p ¯ sϕ ( g ϕr ¯ s ) i ¯ j + g r ¯ qϕ g p ¯ bϕ g a ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i + g r ¯ bϕ g a ¯ qϕ g p ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i ] χ p ¯ q − g p ¯ qϕ R p ¯ q ( χ ) − βV ( ω n ω nϕ ) i ¯ j χ i ¯ j . Then computing under normal coordinate of ω ,( ω n ω nϕ ) i ¯ j = [ g k ¯ l ( g k ¯ l ) i ω n ( ω nϕ ) − − ω n ( ω nϕ ) − g k ¯ lϕ ( g ϕk ¯ l ) i ] ¯ j (11.10) = − g k ¯ l R k ¯ li ¯ j ( ω ) ω n ( ω nϕ ) − + ω n ( ω nϕ ) − g p ¯ qϕ ( g ϕp ¯ q ) ¯ j g k ¯ lϕ ( g ϕk ¯ l ) i + ω n ( ω nϕ ) − g k ¯ qϕ g p ¯ lϕ ( g ϕp ¯ q ) ¯ j ( g ϕk ¯ l ) i − ω n ( ω nϕ ) − g k ¯ lϕ ( g ϕk ¯ l ) i ¯ j . Again, A k ¯ l = [ χ p ¯ q g ϕp ¯ q ] k ¯ l (11.11) = R p ¯ qk ¯ l ( χ ) g ϕp ¯ q + χ p ¯ q ( g ϕp ¯ q ) k ¯ l . Furthermore, from the flow equation,( ∂ t − L ) ϕ (11.12) = − c β + g i ¯ jϕ χ i ¯ j − βV ω n ω nϕ + [ g k ¯ jϕ g i ¯ lϕ χ i ¯ j − βV ω n ω nϕ g k ¯ lϕ ] ϕ k ¯ l = − c β + 2 g i ¯ jϕ χ i ¯ j − g k ¯ jϕ g i ¯ lϕ χ i ¯ j g k ¯ l − βV ω n ω nϕ ( n + 1) + βV ω n ω nϕ g k ¯ lϕ g k ¯ l . Putting them together, we obtain( ∂ t − L )[log A − Cϕ ](11.13) = 1 A ∂ t A + g k ¯ jϕ g i ¯ lϕ χ i ¯ j ( A k ¯ l A − A k A ¯ l A ) − βV ω n ω nϕ g k ¯ lϕ ( A k ¯ l A − A k A ¯ l A ) − C [( ∂ t − L ) ϕ ]= ∂ t A + g k ¯ jϕ g i ¯ lϕ χ i ¯ j A k ¯ l A − g k ¯ jϕ g i ¯ lϕ χ i ¯ j A k A ¯ l A − βV ω n ω nϕ g k ¯ lϕ A k ¯ l A + βV ω n ω nϕ g k ¯ lϕ A k A ¯ l A − C [ − c β + 2 g i ¯ jϕ χ i ¯ j − g k ¯ jϕ g i ¯ lϕ χ i ¯ j g k ¯ l − βV ω n ω nϕ ( n + 1) + βV ω n ω nϕ g k ¯ lϕ g k ¯ l ] . The first line in the last identity is, ∂ t A + g k ¯ jϕ g i ¯ lϕ χ i ¯ j A k ¯ l A − g k ¯ jϕ g i ¯ lϕ χ i ¯ j A k A ¯ l A = 1 A [ − χ i ¯ j g r ¯ qϕ g p ¯ sϕ ( g ϕr ¯ s ) i ¯ j χ p ¯ q + χ i ¯ j g r ¯ qϕ g p ¯ bϕ g a ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i χ p ¯ q + χ i ¯ j g r ¯ bϕ g a ¯ qϕ g p ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i χ p ¯ q − g p ¯ qϕ R p ¯ q ( χ ) − βV ( ω n ω nϕ ) i ¯ j χ i ¯ j ]+ 1 A g k ¯ jϕ g i ¯ lϕ χ i ¯ j [ R p ¯ qk ¯ l ( χ ) g ϕp ¯ q + χ p ¯ q ( g ϕp ¯ q ) k ¯ l ] − g k ¯ jϕ g i ¯ lϕ χ i ¯ j A k A ¯ l A = 1 A { χ i ¯ j g r ¯ qϕ g p ¯ bϕ g a ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i χ p ¯ q + χ i ¯ j g r ¯ bϕ g a ¯ qϕ g p ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i χ p ¯ q (11.14) − g p ¯ qϕ R p ¯ q ( χ ) + βV χ i ¯ j g k ¯ l R k ¯ li ¯ j ( ω ) ω n ω nϕ − βV χ i ¯ j ω n ω nϕ g p ¯ qϕ ( g ϕp ¯ q ) ¯ j g k ¯ lϕ ( g ϕk ¯ l ) i − βV χ i ¯ j ω n ω nϕ g k ¯ qϕ g p ¯ lϕ ( g ϕp ¯ q ) ¯ j ( g ϕk ¯ l ) i (11.15) + βV χ i ¯ j ω n ω nϕ g k ¯ lϕ ( g ϕk ¯ l ) i ¯ j } + 1 A g k ¯ jϕ g i ¯ lϕ χ i ¯ j R p ¯ qk ¯ l ( χ ) g ϕp ¯ q − g k ¯ jϕ g i ¯ lϕ χ i ¯ j A k A ¯ l A . (11.16)Here we use the identity to cancel the first term in the 2rd line and the second termin the 4th line,( g ϕp ¯ q ) k ¯ l = R p ¯ qk ¯ l + ∂ ∂z p ∂z ¯ q ∂z k ∂z ¯ l ϕ = R k ¯ lp ¯ q + ∂ ∂z p ∂z ¯ q ∂z k ∂z ¯ l ϕ = ( g ϕk ¯ l ) p ¯ q . The second line in the last identity in (11.13) is − βV ω n ω nϕ g k ¯ lϕ [ R p ¯ qk ¯ l ( χ ) g ϕp ¯ q + χ p ¯ q ( g ϕp ¯ q ) k ¯ l ] A + βV ω n ω nϕ g k ¯ lϕ A k A ¯ l A . In order to annihilate the 2nd term with 2nd term in (11.15) and 2nd term in(11.14) with (11.16), we need the lemma, -PROPERNESS OF MABUCHI’S K -ENERGY 19 Lemma 11.3.
The following lemma holds. [ χ i ¯ j g k ¯ qϕ g p ¯ lϕ ( g ϕp ¯ q ) ¯ j ( g ϕk ¯ l ) i ] A ≥ g k ¯ lϕ A k A ¯ l , [ χ i ¯ j g r ¯ bϕ g a ¯ qϕ g p ¯ sϕ ( g ϕa ¯ b ) ¯ j ( g ϕr ¯ s ) i χ p ¯ q ] A ≥ g k ¯ jϕ g i ¯ lϕ χ i ¯ j A k A ¯ l . Proof.
Under the normal chordate of χ which is negative-defined, and ω χ is diago-nalized, the first inequality becomes,[ g k ¯ qϕ g p ¯ lϕ X i ( g ϕp ¯ q ) i ( g ϕk ¯ l ) i ] X i g ϕi ¯ i ≥ g k ¯ kϕ ( X i g ϕi ¯ i ) k ( X i g ϕi ¯ i ) ¯ k . This follows from the H¨older’s inequality. The second inequality is proved in Lemma3.2 in [27]. (cid:3)
Thus (11.13) becomes( ∂ t − L )[log A − Cϕ ](11.17) = 1 A {− g p ¯ qϕ R p ¯ q ( χ ) + βV χ i ¯ j g k ¯ l R k ¯ li ¯ j ( ω ) ω n ω nϕ } + 1 A g k ¯ jϕ g i ¯ lϕ χ i ¯ j R p ¯ qk ¯ l ( χ ) g ϕp ¯ q − βV ω n ω nϕ g k ¯ lϕ R p ¯ qk ¯ l ( χ ) g ϕp ¯ q A − C [ − c β + 2 g i ¯ jϕ χ i ¯ j − g k ¯ jϕ g i ¯ lϕ χ i ¯ j g k ¯ l − βV ω n ω nϕ ( n + 1) + βV ω n ω nϕ g k ¯ lϕ g k ¯ l ] . Since ω ϕ has lower bound from Subsection 11.1, the first four terms and the 4thterm in the last line are bounded above by constant C , thus at the maximum point p of log A − Cϕ , 0 ≤ C − C [ − c β + 2 g i ¯ jϕ χ i ¯ j − g k ¯ jϕ g i ¯ lϕ χ i ¯ j g k ¯ l ] . Written in the normal co-ordinate where χ has negative diagonal µ i , it becomes0 ≤ C − C [ − c β + 2 n X i =1 µ i λ i − n X i =1 µ i λ i ] . (11.18)From the condition, ( − nc β · ω + ( n − χ ) ∧ ω n − > , We have there is positive constant δ such that( − nc β · ω + ( n − χ ) ∧ ω n − ≥ δω n − , then − c β + n X i =1 ,i = k µ i ≥ δ. From (11.18), we have for large C , − c β + 2 n X i =1 µ i λ i − n X i =1 µ i λ i ≤ C C ≤ . δ. We choose 1 ≤ k ≤ n and consider,0 ≥ n X i =1 ,i = k µ i ( 1 λ i − + µ k λ k = c β − n X i =1 µ i λ i + n X i =1 µ i λ i − [ c β − n X i =1 ,i = k µ i − µ k λ k ] ≥ − . δ + δ + 2 µ k λ k . Thus, λ k ≤ − µ k δ , or at p , ω ϕ ≤ − δ χ. Therefore, we obtain that at any x ∈ M log A ( x ) − Cϕ ( x ) ≤ log A ( p ) − Cϕ ( p ) , then, log A ( x ) ≤ log 4 nδ − C · ( ϕ − inf ϕ ) . Therefore, there is constant C such that ω ϕ ≤ e C · ( ϕ − inf ϕ ) . (11.19)11.3. Zero order estimate.
It suffices to obtain the iteration formula. Letting C = max { , − ˙ ϕ − c β + 1 } from (11.3), we have ω nϕ ≤ ( ˙ ϕ + c β + C ) ω nϕ = nω n − ϕ ∧ χ − βV ω n + C ω nϕ . We compute that ω nϕ − ω n − ϕ ∧ ω (11.20) ≤ ( ˙ ϕ + c β + C ) ω nϕ − ω n − ϕ ∧ ω = nω n − ϕ ∧ χ − βV ω n + C ω nϕ − ω n − ϕ ∧ ω. -PROPERNESS OF MABUCHI’S K -ENERGY 21 Then we let φ = ϕ − inf ϕ and u = e − C φ , we multiply (11.20) with u and itegrateover M . The right hand side becomes, Z M u [ ω nϕ − ω n − ϕ ∧ ω ]= Z M e − C φ [ ω nϕ − ω n − ϕ ∧ ω ]= C Z M e − C φ ∂ϕ ∧ ¯ ∂ϕ ∧ ω n − ϕ = C Z M e − C φ ∂ϕ ∧ e − C φ ¯ ∂ϕ ∧ ω n − ϕ = 4 C Z M ∂u ∧ ¯ ∂u ∧ ω n − ϕ ≥ C C Z M | ∂u | ω ω n . In the last inequality we used the lower bound of ω ϕ . While, the right hand side is Z M u [ nω n − ϕ ∧ χ − βV ω n + C ω nϕ − ω n − ϕ ∧ ω ] ≤ C Z M uω nϕ ≤ C Z M e − C φ e C · ( ϕ − inf ϕ ) ω n ≤ C Z M e − C φ e C · φ e − C · inf φ ω n ≤ C || u || C C Z M e C ( − C C ) · φ ω n . We apply (11.19) in the second inequality. Let v = e − C φ . We choose C = pC and C C = 1 − δ , we thus obtain Z M | ∂v p | ω ω n ≤ pC || v || − δ Z M e C ( − p +1 − δ ) φ ω n ≤ pC || v || − δ Z M v C ( p − δ ) ω n . Thus the zero order estimate follows from the iteration Lemma 3.3 in [28].
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