The modified K\"ahler-Ricci flow and solitons

Abstract

We investigate the K\"ahler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a K\"ahler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.

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arXiv:0809.0941v1 [math.DG] 5 Sep 2008
THE MODIFIED K¨AHLER-RICCI FLOW AND
SOLITONS1
D.H. Phong∗, Jian Song∗∗, Jacob Sturm†and Ben Weinkove‡
Abstract
We investigate the K¨ ahler-Ricci flow modified by a holomorphic vector field. We
find equivalent analytic criteria for the convergence of the flow to a K¨ ahler-Ricci
soliton. In addition, we relate the asymptotic behavior of the scalar curvature along
the flow to the lower boundedness of the modified Mabuchi energy.
1Introduction
Let M be a compact K¨ ahler manifold of complex dimension n with c1(M) > 0. A K¨ ahler-
Ricci soliton on M is a K¨ ahler metric ω =i
together with a holomorphic vector field X such that
2g¯kjdzj∧ dzkin the cohomology class π c1(M)
Ric(ω) − ω = LXω.(1.1)
Alternatively, in coordinate notation, writing X¯k= g¯kℓXℓ,
R¯kj− g¯kj= ∇jX¯k.(1.2)
Let Φtbe the 1-parameter group of automorphisms of M generated by ReX. The family
of metrics g¯kj(t) ≡ Φ∗
−t(g¯kj) provides then a solution of the K¨ ahler-Ricci flow,
˙ g¯kj(t) = −R¯kj+ g¯kj
(1.3)
where the evolution in time is just by reparametrization.
If X is the zero vector field then (1.1) reduces to the K¨ ahler-Einstein equation. K¨ ahler-
Ricci solitons are in many ways similar to extremal metrics, which generalize constant
scalar curvature K¨ ahler metrics and are characterized by the condition that the vector
field ∇iR is holomorphic. Extremal metrics are the critical points of the Calabi functional
C(g¯kj) = ?R −¯R?2
of Yau [Y2] asserts that the existence of constant scalar curvature metrics in a given
integral K¨ ahler class should be equivalent to the stability of the polarization in the sense
of geometric invariant theory. Notions of K-stability for constant scalar curvature metrics
L2, where¯R is the average of the scalar curvature. A classic conjecture
1Research supported in part by National Science Foundation grants DMS-07-57372, DMS-06-04805,
DMS-05-14003, and DMS-08-48193.

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have been proposed by Tian [T] and Donaldson [D1], and extended to the case of extremal
metrics by Szekelyhidi [S1] (see also [M]). Similarly, the existence of K¨ ahler-Ricci solitons
is expected to be equivalent to a suitable notion of stability.
K¨ ahler-Ricci solitons can also be viewed as the stationary points of the modified K¨ ahler-
Ricci flow, that is,
˙ g¯kj= −R¯kj+ g¯kj+ ∇jX¯k
(1.4)
which is the flow (1.3) reparametrized by the automorphisms Φt generated by the real
part ReX of the holomorphic vector field X. Similar reparametrizations, in the context
of Hamilton’s original Ricci flow [H], had been introduced by DeTurck [DeT] to give a
simpler proof of the short-time existence of the flow.
The modified K¨ ahler-Ricci flow appears in the work of Tian-Zhu [TZ2] as part of their
study of the K¨ ahler-Ricci flow assuming a priori the presence of a K¨ ahler-Ricci soliton.
They make use of a Moser-Trudinger type inequality from [CTZ] to deduce convergence
of the flow in the sense of Cheeger-Gromov. (In the special case where there are no
nontrivial holomorphic vector fields, it is known by the work of [P2], [TZ2], [PSSW2]
that the existence of a K¨ ahler-Einstein metric implies the exponential convergence of the
K¨ ahler-Ricci flow to that metric.)
In this paper, we study the long-time behavior of the modified K¨ ahler-Ricci flow without
assuming the existence of a K¨ ahler-Ricci soliton. We give analytic conditions which are
both necessary and sufficient for the convergence of the flow to a K¨ ahler-Ricci soliton.
These conditions are analogous to the ones given in [PSSW1] for the convergence of the
K¨ ahler-Ricci flow. As explained in [PS1] and [PSSW1] they can be interpreted as stability
conditions in an infinite-dimensional geometric invariant theory, where the orbits are those
of the diffeomorphism group acting on the space of almost-complex structures.
We provide now a description of our results. We will assume always that M is a compact
K¨ ahler manifold with c1(M) > 0 and X is a holomorphic vector field whose imaginary
part ImX induces an S1action on M. Note that once a maximal compact subgroup G of
Aut0(M) is fixed, there is a natural choice of such a vector field X (for more details see
Section §2.1 below).
First, we define the notion of the Hamiltonian θX,ωand modified Ricci potential uX,ω.
Write KXfor the space of K¨ ahler metrics in πc1(M) which are invariant under ImX. Given
ω =i
2gkjdzj∧ dzk∈ KXwe define a real-valued function θX,ωby
Xjgkj= ∂kθX,ω,
?
MeθX,ωωn=
?
Mωn=: V.(1.5)
The Ricci potential f = f(ω) is given by
gkj− Rkj= ∂k∂jf,
?
Me−fωn= V, (1.6)
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and we define a modified Ricci potential uX,ωby
uX,ω= f + θX,ω. (1.7)
If M admits a K¨ ahler-Ricci soliton ω ∈ π c1(M) with respect to the vector field X, then
ω is necessarily in KXand uX,ω= 0. Let g¯kj(t) evolve by the modified K¨ ahler-Ricci flow
and set
YX(t) =
?
M|∇uX,ω|2eθX,ωωn.(1.8)
The modified K¨ ahler-Ricci flow starting at ω0∈ KXpreserves the K¨ ahler class, and can
be expressed as a flow of K¨ ahler potentials. Define
PX(M,ω0) = {ϕ ∈ C∞(M) | ω = ω0+i
2∂∂ϕ > 0, ImX(ϕ) = 0}, (1.9)
which, modulo constants, can be identified with KX. Let ϕ = ϕ(t) ∈ PX(M,ω0) be the
solution of the equation
˙ ϕ = logωn
ωn
0
ϕ(0) = c0.
+ ϕ + θX,ω+ f(ω0),
(1.10)
Then the K¨ ahler metrics ω = ω0+i
The initial constant c0can affect the growth of ϕ for large time, and has to be chosen with
some care. We choose it to be given by the value (2.34) described in Section §2 below.
2∂∂ϕ evolve by the modified K¨ ahler-Ricci flow (1.4).
Our first main theorem is a general characterization of the convergence of the modified
K¨ ahler-Ricci flow, which shows in particular that if convergence occurs, it always does so
at an exponential rate:
Theorem 1 Let ω0∈ KX, ω0:=
flow (1.4) with initial metric ω0. Then the following conditions are equivalent:
(i) The modified K¨ ahler-Ricci flow g¯kj(t) converges in C∞to a K¨ ahler-Ricci soliton
g¯kj(∞) with respect to X.
(ii) The function ?R − n − ∇jXj?C0 is integrable, i.e.,
?∞
(iii) Let the potential ϕ(t) evolve according to (1.10), with initial value c0as specified
in the equation (2.34) below. Then we have
i
2g0
¯kjdzj∧ d¯ zk, and consider the modified K¨ ahler-Ricci
0
?R − n − ∇jXj?C0 dt < ∞.(1.11)
supt≥0?ϕ(t)?C0 < ∞.(1.12)
(iv) Let the function YX(t) be defined by (1.8). Then there exist constants κ,C with κ
strictly positive so that
YX(t) ≤ C e−κt.(1.13)
(v) The modified K¨ ahler-Ricci flow g¯kj(t) converges exponentially fast in C∞to a
K¨ ahler-Ricci soliton g¯kj(∞) with respect to X.
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The preceding theorem shows that the growth of YX(t), or alternatively, of the function
?R−n−∇jXj?C0(t) is key to the problem of convergence of the modified K¨ ahler-Ricci flow.
Our next result addresses the behavior of these quantities under a stability assumption.
Following [TZ1], we define the modified Mabuchi K-energy µX: PX(M,ω0) → R by
δµX(ϕ) = −1
V
To clarify this definition, since R − n − ∇jXj− Xu = −(∆ + ReX)uX,ω, the integrand
is real and thus µX does indeed map into R. For a proof that µX(ϕ) is independent of
choice of path in PX(M,ω0), see [TZ1].
We consider the condition:
?
Mδϕ
?
R − n − ∇jXj− XuX,ω
?
eθX,ωωn, µX(0) = 0. (1.14)
(AX)µXis bounded from below on PX(M,ω0)
In [TZ1] it is shown that (AX) is a necessary condition for the existence of a K¨ ahler-
Ricci soliton ω with respect to X. Here we shall establish the following theorem:
Theorem 2 Assume that Condition (AX) holds, and let ω0∈ KX. Then we have, along
the modified K¨ ahler-Ricci flow (1.4) starting at ω0,
YX(t) → 0and?R − n − ∇jXj?C0 → 0, as t → ∞. (1.15)
Furthermore, for any p > 2, we have
?∞
0
?R − n − ∇jXj?p
C0dt < ∞. (1.16)
Note that a metric ω ∈ KXsatisfies R−n−∇jXj= 0 if and only if ω is a K¨ ahler-Ricci
soliton with respect to X. However, the convergence of ?R − n − ∇jXj?C0 to zero is
of course weaker than the convergence of the metrics g¯kj(t) themselves to a K¨ ahler-Ricci
soliton, and this is to be expected, since the condition (AX) is only a semi-stability type
of condition.
Next, we describe another consequence of the condition (AX). Associated to the mod-
ified Mabuchi K-energy is the modified Futaki invariant FX(see [TZ1]), given by
FX(Z) = −
?
M(ZuX,ω)eθX,ωωn, (1.17)
for a holomorphic vector field Z. The modified Futaki invariant FXis independent of the
choice of ω ∈ KX. It follows immediately that FX ≡ 0 is a necessary condition for the
existence of a K¨ ahler-Ricci soliton in KX.
In the unmodified case, corresponding to X = 0, the condition (AX) reduces to the
condition (A) of lower boundedness of the Mabuchi K-energy. It is then easy to show that
(A) implies that the unmodified Futaki invariant FX=0(Z) vanishes for all holomorphic
vector fields Z ∈ H0(M,T1,0), by differentiating the functional along the integral paths of
Z. We show how to rework this argument to prove the analogous statement when X ?= 0
(to our knowledge, this result is not in the literature).
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Proposition 1 If (AX) holds then FX(Z) = 0 for all holomorphic vector fields Z.
Our third theorem shows that (AX) together with one additional assumption give
necessary and sufficient conditions for the convergence of the metrics g¯kj(t) themselves.
Let λ(t) be the first positive eigenvalue of the operator −gjk∇j∇kacting on smooth T1,0
vector fields. Namely,
λ(t) = infV ⊥H0(M,T1,0)?¯∂V ?2
?V ?2, (1.18)
where H0(M,T1,0) is the space of holomorphic vector fields on M and we are using the
natural L2inner product induced by gkj(t) on the spaces T1,0and T1,−1. This quantity
was first introduced in the context of the K¨ ahler-Ricci flow in [PS1]. Recall the following
condition from [PSSW1]:
(S) inft≥0λ(t) > 0.
Then we have:
Theorem 3 The modified K¨ ahler-Ricci flow (1.4), starting at an arbitrary metric ω0∈
KX, converges exponentially fast in C∞to a K¨ ahler-Ricci soliton with respect to the holo-
morphic vector field X if and only if the conditions (AX) and (S) are satisfied.
Since condition (S) is invariant under automorphisms, an immediate consequence of
Theorem 3 is that convergence modulo automorphisms implies full convergence. More
precisely, suppose that g¯kj(t) is a solution of the modified K¨ ahler-Ricci flow starting at
ω0∈ KXand assume there exists a family of automorphisms {Ψt}t∈[0,∞)such that Ψ∗
converges to a K¨ ahler-Ricci soliton with respect to X. Then g¯kj(t) converges exponentially
fast to a K¨ ahler-Ricci soliton with respect to X.
t(g¯kj)
Finally we discuss in more detail the behavior of YX(t) which, as can be seen from
Theorem 1, is key to the convergence of the K¨ ahler-Ricci flow. The next result provides
information on the growth of YX(t) for the completely general modified K¨ ahler-Ricci flow
and brings to light the obstructions to the convergence of the flow.
It is convenient to introduce a quantity λXwhich is uniformly equivalent to the eigen-
value λ described above (see Lemma 12 below). Equip the spaces T1,0and T1,−1with the
norms
?V ?2
θ=
?
?
Mg¯kjVjVkeθX,ωωn,
?W?2
θ=
Mg¯kjWj
¯ pWk
¯ qgq¯ peθX,ωωn,(1.19)
with respect to which they can be completed into Hilbert spaces. Define the eigenvalue
λX(t) by
λX(t) = infV ⊥H0(M,T1,0)?¯∂V ?2
θ
?V ?2
θ
, (1.20)
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