The Poincaré Conjecture Clay Research Conference Resolution of the Poincaré Conjecture Institut Henri Poincaré Paris, France, June 8–9, 2010 - bet 23
We have known that outside ﬁnitely many points, Y is a limit of spaces with
bounded curvature, so it has smooth points. In fact, it is shown in [NT08] that Y
is a quotient by a smooth manifold by a group action outside ﬁnitely many points.
We can say more when M is a K¨
ahler surface. Then
M(λ, χ, v) contains a
) of K¨
ahler–Einstein metrics with K¨
ahler class Ω, where c
denotes the ﬁrst and second Chern classes of M .
> 0 and
) is of positive dimension, then M is a Del-Pezzo
surface obtained by blowing up
at m points in general position, where 5
≤ 8. In [Ti89], it was proved that M(c
) can be compactiﬁed by adding
ahler–Einstein orbifolds, and furthermore, there are strong constraints on quotient
singularities. It was conjectured in [Ti89] that
) can be compactiﬁed by
ahler–Einstein orbifolds whose singularities are only rational double points
possibly with very few exceptional cases. Indeed, it is true when M is a blow-up of
at 5 point, see [MM93].
= 0, then M is either a complex 2-torus or a K3 surface. This is a
collapsing case and is related to problems on large complex limits in the Mirror
< 0, let (M, g
) be a sequence in
), let (M
) be one of its
limits as in Theorem 7.4, and let M
· · · , M
be its irreducible components.
We know that each (M
) is a complete K¨
ahler–Einstein orbifold. It should
be possible to identify these irreducible components more explicitly. For simplicity,
assume that M
is smooth, then we expect:
is of the form ¯
\D, where ¯
M is a projective surface
and D is a divisor with normal crossings, such that K
is positive outside D and each component of D has either pos-
itive genus, or at least two intersection points with components
Note that the main theorem in [TY86] implies: given ¯
M and D as above, there
is a K¨
ahler–Einstein metric on ¯
8. Complete Calabi-Yau 4-manifolds
To understand Y near those ﬁnitely many non-smooth points, we are led to
classifying all complete Ricci-ﬂat 4-manifolds (M, g) with ﬁnite L
-norm of curva-
ture. Almost all known examples of such complete Ricci-ﬂat manifolds are Calabi–
Yau ones, so we will focus on complete Calabi–Yau metrics. Non-ﬂat Calabi–Yau
metrics were ﬁrst constructed on a minimal resolution of the quotient of
GEOMETRIC ANALYSIS ON 4-MANIFOLDS
ﬁnite group in SU (2) by physicists, by Hitchin and Calabi explicitly or by using
the Twistor theory. Further complete Calabi–Yau 4-manifolds were constructed in
[Kr89], [TY90], [CK99], [CH05] and [He10] et al.. A natural question is to see
if they are all the complete Calabi–Yau 4-manifolds with L
The work of Cheeger and myself may shed a light on answering this question.
8.1. [CT06] If (M, g) is a complete Ricci-ﬂat manifold with L
bounded curvature, then its curvature decays quadratically.
A related question is the uniqueness of Calabi-Yau metrics on
Calabi a long time ago. There are indeed complete non-ﬂat Calabi-Yau metrics,
like the Taub–Nut metric. However, many years ago, I proved the following result
in an published note.
8.2. Any complete Calabi–Yau metric on
with maximal volume
growth must be ﬂat.
Its outlined proof appeared in [Ti06] and we refer the readers to there for more
8.3. In fact, by the same arguments, one can actually show that any
complete Calabi–Yau 4-manifolds with maximal volume growth must be a minimal
resolution of the quotient of
by a ﬁnite subgroup in SU (2).
I also conjectured many years ago that the same holds for higher dimensional
cases, that is, any complete Calabi-Yau metrics on
with maximal volume growth
9. Metrics of anti-self-dual type
Anti-self-dual metrics impose strong constraints on underlying 4-manifolds.
There have been many works on constructing anti-self-dual metrics by the gluing
method or the twistor method
(see [Fl91], [DoFr], [Kr89], [Le93]). In particu-
lar, Taubes proved that given any 4-manifold M , after making connected sum of
it with suﬃciently many copies of
, the resulting 4-manifold admits one anti-
self-dual metric ([Ta92]). A fundamental question remains open: how to deform a
metric on any given 4-manifold towards an anti-self-dual metric as we did for the
geometrization of 3-manifolds when the Ricci ﬂow is used. For this purpose, we
need to develop some analytic estimates for the anti-self-dual equation.
Consider the anti-self-dual equation:
= 0, S = const.
(9.1) is an elliptic equation modulo diﬀeomorphisms. Let us show why it is: re-
garding the curvature Rm as a symmetric tensor on Λ
M , the symbol σ(Rm) :
M of the linearized operator of Rm at (x, g) is given by
σ(Rm)(ξ, h)(e, e ) =
− 2 h(e(ξ), e (ξ)),
M , h
M and e, e
M . Here we have identiﬁed e, e
as endomorphisms of T
M through the metric g. It follows that the symbol σ(S)
There were also works on the moduli of anti-self-dual metrics, e.g., [AHS], [KK92]. In this
section, our emphasis on the moduli is diﬀerent and more towards compactifying the moduli.
of the scalar curvature is
σ(S)(ξ, h) =
} is an orthonormal basis of ΛT
M . Notice that
So one can deduce the symbol
)(ξ, h)(e, e ) =
− 2 h(e(ξ), e (ξ)) −
(e, e ),
} is an orthonormal basis of Λ
M . The ellipticity of (9.1) modulo
diﬀeomorphisms means that for any given unit ξ, h = 0 whenever if i
h = 0 and
)(ξ, h) = 0 and σ(S)(ξ, h) = 0. It follows directly from the above computa-
tions of symbols.
As for Einstein metrics, we need to study the following problem: given a se-
quence of an anti-self-dual metrics g
, or more general f -asd metrics, on M , what
are possible limits of g
as i tends to
By using the Index Formula for the signature, we have
τ (M ).
By the Gauss–Bonnet–Chern formula, we can deduce from this
τ (M )
χ(M ) +
If the scalar curvature S(g
) has uniformly bounded L
-norm, then we have a priori
-bound on curvature tensor Rm(g
Since the Weyl tensor is a conformal invariant, we can make conformal changes
. Recall the Yamabe constant:
) = inf
Using the Aubin–Schoen solution of the Yamabe conjecture, there is a u attaining
), so we simply take g
with volume 1 and such that the scalar curvature
) is Q(M, g
). Then we have
= Q(M, g
Therefore, if g
form a sequence of anti-self-dual metrics with bounded Yamabe
constant, then we may assume that their curvatures are uniformly L
that they have ﬁxed volume. One can ask two questions:
1. Given a compact 4-manifold, is there a uniform bound on the Yamabe
constant for anti-self-dual metrics?
2. What are possible limits of anti-self-dual metrics g
with uniformly bounded
As a corollary of Theorem 1.3 in [TV05], one has the following partial answer to
the second question:
GEOMETRIC ANALYSIS ON 4-MANIFOLDS
9.1. Let g
be a sequence of anti-self-dual metrics on M with bounded
Yamabe constant. We further assume that there is a uniform constant c such that
for any function f ,
Then by taking a subsequence if necessary, we have that g
converges to a multi-fold
) in the Cheeger–Gromov topology
9.2. A compactness result can be proved for K¨
ahler metrics with
constant scalar curvature by the same arguments.
It is possible to have an -regularity theorem similar to Theorem 7.1. The
following can be proved by extending the eﬀective transgression method in [CT06]
and bounding Sobolev constants for collapsing 4-manifolds with bounded curvature.
There exist uniform constants
> 0, c > 0, such that the
following holds: If g is an anti-self-dual metric or a K¨
ahler metric with constant
±12 or 0 and B
(p) is a geodesic ball of radius r
≤ 1 satisfying:
|Rm(g)| ≤ c · r
It is hoped that the moduli space of anti-self-dual metrics can be used for
constructing new diﬀerentiable invariants for 4-manifolds.
If such an invariant
exists, one can compute it and use it to establish existence of anti-self-dual metrics
on a 4-manifold. However, there are two major diﬃculties to be overcome in order
to deﬁne the invariant: compactness and transversality. We have discussed the
compactness. For transversality, the readers can ﬁnd some discussions in [Ti06].
M. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds. J.
Amer. Math. Soc. 2 (1989), no. 3, 455–490. MR999661 (90g:53052)
M. Atiyah, N. Hitchin and I. Singer, Self-duality in four-dimensional Riemannian ge-
ometry. Proc. Roy. Soc. London Ser. A 362 (1978), 425–461. MR506229 (80d:53023)
T. Aubin, Some nonlinear problems in Riemannian geometry. Springer Monographs in
Mathematics, Sringer, 1998. MR1636569 (99i:58001)
J.G. Cao and J. Ge, Perelman’s collapsing theorem for 3-manifolds. To appear in The
Journal of Geometric Analysis, arXiv:0908.3229.
X.X.Chen, P. Lu and G. Tian, A note on uniformization of Riemann surfaces by Ricci
ﬂow. Proc. Amer. Math. Soc. 134 (2006), no. 11, 3391–3393 MR2231924 (2007d:53109)
X.X.Chen, C. Lebrun and B. Weber, On conformally K¨
ahler, Einstein manifolds. J.
Amer. Math. Soc. 21 (2008), no. 4, 1137–1168. MR2425183 (2010h:53054)
B. Chow, The Ricci ﬂow on the 22-sphere. J. Diﬀerential Geom. 33 (1991), no. 2,
325–334. MR1094458 (92d:53036)
T. Colding and W. Minicozzi, Width and ﬁnite extinction time of Ricci ﬂow. Geom.
Topol. 12 (2008), no. 5, 2537–2586. MR2460871 (2009k:53166)
A multi-fold is a connected sum of ﬁnitely many orbifolds, see [TV05] for deﬁnition.
This was announced in some of my lectures in early 2010, but a written proof has not
appeared yet and is in preparation.
J. Cheeger and G. Tian, Curvature and injectivity radius estimates for Einstein 4-
manifolds. J. Amer. Math. Soc. 19 (2006), no. 2, 487–525. MR2188134 (2006i:53042)
S. Cherkis and N. Hitchin, Gravitational instantons of type D
. Comm. Math. Phys.
260 (2005), no. 2, 299–317. MR2177322 (2007a:53095)
D. Cooper, C. Hodgson, S. Kerchoﬀ, 3-dimensional orbifolds and cone-manifolds. Math.
Soc. Japan Memoirs, Vol. 5, Tokyo, 2000.
S. Cherkis and A. Kapustin, Singular monopoles and gravitational instantons. Comm.
Math. Phys. 203 (1999), no. 3, 713–728. MR1700937 (2000m:53062)
B-L Chen and X-P Zhu, Ricci ﬂow with surgery on four-manifolds with positive
isotropic curvature. J. Diﬀerential Geom. 74 (2006), no. 2, 177–264.
auser, 1992, Boston-Basel-Berlin. MR1138207 (92i:53001)
S. Donaldson, The geometry of 4-manifolds. Proceedings of the International Con-
gress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 43–54, Amer. Math. Soc.,
Providence, RI, 1987. MR934214 (89d:57002)
S. Donaldson and R. Friedman, Connected sums of self-dual manifolds and deforma-
tions of singular spaces. Nonlinearity 2 (1989), no. 2, 197–239. MR994091 (90e:32027)
A. Floer, Self-dual conformal structures on
. J. Diﬀ. Geom. 33 (1991), no. 2,
551–573. MR1094469 (92e:53049)
M. Freedman, The topology of four-dimensional manifolds. J. Diﬀ. Geom. 17 (1982),
no. 3, 357–453. MR679066 (84b:57006)
R. Hamilton, Three-manifolds with positive Ricci curvature. J. Diﬀerential Geom. 17
(1982), no. 2, 255–306. MR664497 (84a:53050)
R. Hamilton, Four-manifolds with positive curvature operator. J. Diﬀerential Geom.
24 (1986), no. 2, 153–179. MR862046 (87m:53055)
R. Hamilton, The Harnack estimate for the Ricci ﬂow. J. Diﬀ. Geom. 37 (1993), no.
1, 225–243. MR1198607 (93k:58052)
R. Hamilton, The Ricci ﬂow on surfaces. Mathematics and general relativity. 237–262,
Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988. MR954419 (89i:53029)
R. Hamilton, Four-manifolds with positive isotropic curvature. Comm. Anal. Geom. 5
(1997), no. 1, 1–92. MR1456308 (99e:53049)
R. Hamilton, Nonsingular solutions of the Ricci ﬂow on three-manifolds. Comm. Anal.
Geom. 7 (1999), 695–729. MR1714939 (2000g:53034)
Hans-Joachim Hein, Complete Calabi-Yau metrics from
N. Hitchin, Compact four-dimensional Einstein manifolds. J. Diﬀerential Geom. 9
(1974), 435–441. MR0350657 (50:3149)
N. Hitchin, Poisson modules and generalized geometry. Geometry and analysis. No. 1,
403–417, Adv. Lect. Math. (ALM), 17, Int. Press, 2011. MR2882431
M. Ishida, Einstein metrics and exotic smooth structures. Paciﬁc J. Math. 258 (2012),
no. 2, 327–348. MR2981957
V. Kapovitch, Perelman’s stability theorem. Surveys in diﬀerential geometry. Vol. XI,
103-136. MR2408265 (2009g:53057)
A. King and D. Kotschick, The deformation theory of anti-self-dual conformal struc-
tures. Math. Ann. 294 (1992), no. 4, 591–609. MR1190446 (93j:58021)
B. Kleiner and J. Lott, Locally collapsed 3-manifolds. arXiv:1005.5106.
H. Kneser, Geschlossene Fl¨
achen in drei-dimesnionalen Mannigfaltigkeiten. Jahresber.
Deutsch. Math. Verbein 38 (1929), 248–260.
P. Kronheimer, The construction of ALE spaces as hyper-K¨
ahler quotients. J. Diﬀer-
ential Geom. 29 (1989), no. 3, 665–683. MR992334 (90d:53055)
C. LeBrun, Self-dual manifolds and hyperbolic geometry. Einstein metrics and Yang-
Mills connections (Sanda, 1990), 99–131, Lecture Notes in Pure and Appl. Math., 145,
Dekker, New York, 1993. MR1215284 (94h:53060)
C. LeBrun, Einstein metrics and Mostow rigidity. Math. Res. Lett. 2 (1995), 1–8.
G. La Nave and G. Tian, Soliton-type metrics and K¨
ahler-Ricci ﬂow on symplectic
GEOMETRIC ANALYSIS ON 4-MANIFOLDS
P. Li and S.T. Yau, On the parabolic kernel of the Schr¨
odinger operator. Acta Math.
156 (1986), no. 3-4, 153–201. MR834612 (87f:58156)
T. Mabuchi and S. Mukai, Stability and Einstein–K¨
ahler metric of a quartic del Pezzo
surface. Einstein metrics and Yang-Mills connections, 133–160, Lecture Notes in Pure
and Appl. Math., 145, Dekker, New York, 1993. MR1215285 (94m:32043)
J. Morgan and G. Tian, Ricci ﬂow and the Poincar´
e conjecture. Clay Mathematics
Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics
Institute, Cambridge, MA, 2007, 521 pp. MR2334563 (2008d:57020)
J. Morgan and G. Tian, Completion of the proof of the Geometrization Conjecture.
H. Nakajima, Hausdorﬀ convergence of Einstein 4-manifolds. J. Fac. Sci. Univ. Tokyo
Sect. IA Math. 35 (1988), no. 2, 411–424. MR945886 (90e:53063)
A. Naber and G. Tian, Geometric Structures of Collapsing Riemannian Manifolds I.
T. Oliynyk, V. Suneeta and E. Woolgar, A gradient ﬂow for worldsheet nonlinear sigma
models, Nuclear Phys. B 739 (2006), no. 3, 441–458. MR2214659 (2006m:81185)
G. Perelman, The entropy formula for the Ricci ﬂow and its geometric applications.
G. Perelman, Ricci ﬂow with surgery on three-manifolds. arXiv:math/0303109.
G. Perelman, Finite extinction time for the solutions to the Ricci ﬂow on certain three-
N. Sesum, Curvature tensor under the Ricci ﬂow. Amer. J. Math. 127 (2005), no. 6,
1315–1324. MR2183526 (2006f:53097)
J. Song and G. Tian, The K¨
ahler–Ricci ﬂow on surfaces of positive Kodaira dimension.
Invent. Math. 170 (2007), no. 3, 609–653. MR2357504 (2008m:32044)
J. Song and G. Tian, The K¨
ahler–Ricci ﬂow through singularities. arXiv:0909.4898.
J. Streets and G. Tian, A parabolic ﬂow of pluriclosed metrics. Int. Math. Res. Not.
IMRN 2010, no. 16, 3101–3133. MR2673720 (2011h:53091)
J. Streets and G. Tian, Symplectic curvature ﬂow. arXiv:1012.2104.
J. Streets and G. Tian, Regularity theory for pluriclosed ﬂow. C. R. Math. Acad. Sci.
Paris 349 (2011), no. 1-2, 1–4. MR2755684 (2012a:32025)
J. Song and B. Weinkove, Contracting exceptional divisors by the K¨
Duke Math. J. 162 (2013), no. 2, 367–415. MR3018957
T. Shioya and T. Yamaguchi, Collapsing three-manifolds under a lower curvature
bound. J. Diﬀ. Geom., 56 (2000), no. 1, 1–66. MR1863020 (2002k:53074)
T. Shioya and T. Yamaguchi, Volume collapsed three-manifolds with a lower curvature
bound. Math. Ann. 333 (2005), no. 1, 131–155. MR2169831 (2006j:53050)
C. Taubes, The existence of anti-self-dual conformal structures. J. Diﬀerential Geom.
36 (1992), no. 1, 163–253. MR1168984 (93j:53063)
A. Teleman, Instantons and curves on class VII surfaces. Ann. of Math. (2) 172 (2010),
no. 3, 1749–1804. MR2726099 (2011h:32020)
W. Thurston, Three-dimensional geometry and topology, vol. 1, Princeton Math. Ser.,
vol. 35, Princeton Univ. Press, 1997. MR1435975 (97m:57016)
G. Tian, On Calabi’s conjecture for complex surfaces with positive ﬁrst Chern class.
Invent. Math. 101 (1990), no. 1, 101–172. MR1055713 (91d:32042)
G. Tian, Aspects of metric geometry of four manifolds. Inspired by S. S. Chern, 381–
397, Nankai Tracts Math., 11, World Sci. Publ., 2006. MR2313343 (2008i:53044)
G. Tian and J. Viaclovsky, Moduli spaces of critical Riemannian metrics in dimension
four. Adv. Math. 196 (2005), no. 2, 346–372. MR2166311 (2006i:53051)
G. Tian and S.T. Yau, Existence of K¨
ahler–Einstein metrics on complete K¨
ifolds and their applications to algebraic geometry. Mathematical aspects of string
theory, 574–628, World Sci. Publishing, 1987. MR915840
G. Tian and S.T. Yau, K¨
ahler–Einstein metrics on complex surfaces with C
Comm. Math. Phys. 112 (1987), no. 1, 175–203. MR904143 (88k:32070)
G. Tian and S.T. Yau, Complete K¨
ahler manifolds with zero Ricci curvature. I. J.
Amer. Math. Soc. 3 (1990), no. 3, 579–609. MR1040196 (91a:53096)
G. Tian and Z. Zhang, On the K¨
ahler–Ricci ﬂow on projective manifolds of general
type. Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179–192. MR2243679 (2007c:32029)
E. Witten, Monopoles and four-manifolds. Math. Res. Lett. 1 (1994), no. 6, 769–796.
S.T. Yau, Calabi’s conjecture and some new results in algebraic geometry. Proceedings
of the National Academy of Sciences of the USA, 74 (1977), 1798–1799. MR0451180
Beijing University and Princeton University
Do'stlaringiz bilan baham:
©2018 Учебные документы
Рады что Вы стали частью нашего образовательного сообщества.