Spectral Geometry and Asymptotically Conic Convergence

0.45 MB·English

by  J. M. Rowlett

Spectral Geometry and Asymptotically Conic Convergence
Julie Marie Rowlett ∗
7
0
0 February 2, 2008
2

n
a
J
3 Abstract
1
Inthispaperwedefineasymptoticallyconicconvergenceinwhichafamilyofsmooth

] Riemannianmetricsdegeneratestohaveanisolatedconicsingularity. Thisconvergence
P
is related to, and generalizes, the analytic surgery metric degeneration of [15]. For a
S
conic metric (M0,g0) and a scattering metric (Z,gz) we define a nonstandardblowup,
.
h the resolution blowup, in which the conic singularity in M0 is resolved by Z. Equiva-
t lently, the resolution blowup resolves the boundary of the scattering metric using the
a
m conic metric; the resolutionspaceis a smoothcompactmanifold. This blowupinduces
[ asmoothfamilyofmetrics gǫ onthecompactresolutionspaceM,andwesay(M,gǫ)
{ }
converges asymptotically conically to (M0,g0) as ǫ 0.
1 →
Let ∆ǫ and ∆0 be geometric Laplacianson (M,gǫ) and (M0,g0), respectively. Our
v
3 first result is convergenceof the spectrum of ∆ǫ to the spectrum of ∆0 as ǫ→0. Note
8 that this result implies spectral convergence for the k-form Laplacian under certain
3 geometric hypotheses. This theorem is provenusing rescaling arguments, standard el-
1 liptictechniques,andtheb-calculusof[26]. Oursecondresultistechnical: weconstruct
0
a parameter (ǫ) dependent heat operatorcalculus which contains,and hence describes
7
0 precisely, the heat kernelfor ∆ǫ as ǫ→0. The consequences of this result include: the
/ existence of a polyhomogeneous asymptotic expansion for Hǫ as ǫ 0, with uniform
h →
convergencefor alltime, and a precise descriptionof the behaviourof Hǫ uniformly as
t
a ǫ 0. To prove this result we construct several manifolds with corners (heat spaces)
m →
using both standard and non-standard blowups, on which we construct correspond-
: ing heat operator calculi. A parametrix construction modeled after the heat kernel
v
construction of [26] and a maximum principle type argument complete this proof.
i
X
r
a 1 Introduction
Cheeger andColdingwrote a series of three papersbetween 1997 and2000 on the Gromov-
Hausdorff limits of families of smooth, connected Riemannian manifolds with lower Ricci
curvaturebounds[5], [6], [7]. Theyproved Fukaya’s conjectureof 1987 [10]: onany pointed
∗Email: [email protected]. Department of Mathematics, South Hall Room 6607, University of
California, SantaBarbara, California 93106. AMS Classification primary: 58J50, secondary: 58J35.
1
Gromov-Hausdorff limit space of a family Mn of connected Riemannian manifolds with
{ i }
Ricci curvature bounded below, a self adjoint extension of the scalar Laplacian can be
defined, with discrete spectrum in the compact case, so that the eigenvalues and eigen-
functions of the scalar Laplacians on Mn behave continuously as i . An example,
i → ∞
provided by Perelman, showed that the results of Cheeger and Colding could not be ex-
tended to the k-form Laplacian or to more general geometric Laplacians. In 2002, Ding
proved convergence of the heat kernels and Green’s functions in the same setting [8]. The
estimates are uniform for time bounded strictly away from zero. These results are impres-
sive; the only hypothesis is a lower Ricci curvature bound. It would be useful to prove a
more general spectral convergence result for geometric Laplacians and to obtain uniform
estimates on the heat kernels for all time. In order to obtain such results, it becomes
necessary to impose more structure on the manifolds and the way in which they converge
to a singular limit space.
LetM beafixedcompact manifoldwithRiemannian metricandletH beanembedded
orientable hypersurface with defining function x and smooth metric g , and let
H
g := dx2+(ǫ2+x2)g ǫ [0,1).
ǫ H
∈
Asǫ 0,g x2g +dx2,whichhasanisolated conicsingularityatx = 0. Geometrically,
ǫ H
→ →
M is pinched along the hypersurface H as ǫ 0, and the resulting metric has a conic
→
singularity as x 0. The study of this metric collapse is the content of the 1990 thesis of
→
McDonald [25].
In 1995, Mazzeo and Melrose developed pseudodifferential techniques to describe the
behavior of the spectral geometry under another specific type of metric collapse known
as analytic surgery similar to the conic collapse considered by McDonald. As ǫ 0 the
→
metrics
dx 2 dx 2
g = | | +h | | +h = g ;
ǫ x2+ǫ2 → x2 0
g is an exact b-metric on the compact manifold with boundary M¯ obtained by cutting M
0
alongH andcompactifying asamanifold withboundary,hencethename, analytic surgery.
Undercertain assumptionson theassociated Dirac operators,1 Mazzeo andMelrose proved
limη(∂ ) = η (∂ ),
ǫ b M¯
ǫ→0
where η (∂ ) is the b-version of the eta invariant introduced by Melrose. These results
b M¯
were proven by analyzing the resolvent family of the Dirac operators ∂ uniformly near
ǫ
zero. This led to a precise description of the behaviour of the small eigenvalues. Our goal
is to obtain precise analytic results like those of [15].
Theconvergenceconsideredhere,asymptotically conic (ac)convergence, ismorerestric-
tive than that of Cheeger, Coldingand Ding, butmoregeneral than theconic degeneration
1In a later collaboration with Hassel [15] these hypotheseswere removed.
2
of McDonald. We note, however, that ac convergence does not require Ricci curvature
bounds. The conic collapse of McDonald, the analogous smooth collapse of a higher codi-
mension submanifold and the collapse of an open neighborhood of the manifold with some
restrictions on the local geometry all fit this new definition. Before stating our spectral
convergence results for geometric Laplacians, we recall their definition.
Definition 1. Let(E, ) be aHermitian vectorbundle over aRiemannian manifold (M,g)
∇
with metric-compatible connection . A geometric Laplacian is an operator ∆ acting on
∇
sections of E which has the form
∆ = ∗ +R,
∇ ∇
where R is a non-negative self-adjoint endomorphism of E. By the Weitzenbo¨ck Theorem,
the Laplacian on k-forms is a geometric Laplacian, as is the Hodge Laplacian and the
conformal Laplacian; any geometric Laplace-type operator is a geometric Laplacian.
Our results are the following.
Spectral Convergence: Theorem 1. Let (M ,g ) be a compact Riemannian n-manifold
0 0
with isolated conic singularity, and let (Z,g ) be an asymptotically conic space, with n 3.
z
≥
Assume (M,g ) converges asymptotically conically to (M ,g ). Let (E , ) and (E , )
ǫ 0 0 0 0 z z
∇ ∇
be Hermitian vector bundles over (M ,g ) and (Z,g ), respectively, so that each of these
0 0 z
bundles in a neighborhood of the boundary is the pullback from a bundle over the cross
section (Y,h). Let ∆ , ∆ be the corresponding Friedrich’s extensions of geometric Lapla-
0 z
cians, and let ∆ be the induced geometric Laplacian on (M,g ). Assume ∆ has no 2
ǫ ǫ z
L
nullspace. Then the accumulation points of the spectrum of ∆ as ǫ 0 are precisely the
ǫ
→
points of the spectrum of ∆ , counting multiplicity.
0
Thesettingforournextresultistheaccheatspace,amanifoldwithcornersconstructed
in section 7.
Heat Kernel Convergence: Theorem 2. Let (M ,g ) be a compact Riemannian n-
0 0
manifold with isolated conic singularity, and let (Z,g ) be an asymptotically conic space,
z
with n 3. Assume (M,g ) converges asymptotically conically to (M ,g ). Let (E , )
ǫ 0 0 0 0
≥ ∇
and (E , ) be Hermitian vector bundles over (M ,g ) and (Z,g ), respectively, so that
z z 0 0 z
∇
each of these bundles in a neighborhood of the boundary is the pullback from a bundle over
the cross section (Y,h). Let ∆ , ∆ be the corresponding Friedrich’s extensions of geometric
0 z
Laplacians, and let ∆ be the induced geometric Laplacian on (M,g ). Then the associated
ǫ ǫ
heat kernels H have a full polyhomogeneous expansion as ǫ 0 on the asymptotically
ǫ
→
conic convergence (acc) heat space with the following leading terms at the ǫ = 0 faces:
H (z,z′,t) H (z,z′,t′) at F
ǫ 0 0101
→
H (z,z′,t) (ρ )2H (z,z′,τ) at F ,
ǫ 1010 b 1010
→
3
H (z,z′,t) = O (ρ )2(ρ )2 at F , F .
ǫ 1001 0110 1001 0110
Above, ρwxyz is the defining func(cid:0)tion for boundary(cid:1) face Fwxyz, H0 is the heat kernel for
∆ , H is the b-heat kernel for ∆ and t′ and τ are rescaled time variables with
0 b z
t t
t′ = , τ = .
(ρ )2 (ρ ρ )2
1010 1001 0110
This convergence is uniform in ǫ for all time and moreover, the error term is bounded by
C ǫtN, for any N N .
N 0
∈
Remark: The convergence of the heat kernels H (z,z′,t) on M M R+ to the
ǫ
× ×
heat kernel H (z,z′,t′) on M follows immediately from Theorem 2. The error term is
0 0
ǫ2(H (z,z′,τ) +O(1)) + O(ǫt∞) as ǫ,t 0 and ǫ2(H (z,z′,τ)+ O(1)) +O(ǫ) as ǫ 0
b b
→ →
for all t. However, Theorem 2 is stronger than this statement: the theorem specifies the
leading order behavior of H precisely at the ǫ = 0 faces of the acc heat space and gives
ǫ
the existence of a full polyhomogeneous expansion as ǫ 0.
→
These theorems are proven in sections 5 and 7, respectively. In section 2 we define the
resolution blowup and ac convergence. Sections 3 and 4 contain a brief review of geometric
and analytic results and terminology on manifolds with corners. In section 6 we construct
the heat spaces and heat operator calculi that will be used to prove the main theorem in
section 7.
This work is based on the author’s doctoral dissertation completed at Stanford Univer-
sity in June, 2006 under the supervision of Rafe Mazzeo. The author wishes to thank Rafe
Mazzeoforexcellentadvising,AndrasVasyformanyhelpfulconversations andsuggestions,
and Richard Melrose for insightful comments.
2 Asymptotically Conic Convergence
The definition of asymptotically conic convergence involves three geometries: a family of
smoothmetricsonacompactmanifold,aconicmetriconacompact(incomplete)manifold,
andan asymptotically conic or ac scattering metricon a compact (complete) manifold with
boundary.
First, we define ac scattering metric.
Definition 2. An ac scattering metric (Z¯,g ) is a smooth metric on a compact n-manifold
z
with boundary, ∂Z¯ = (Y,h) a smooth compact (n 1) manifold. Z¯ has a product decom-
−
position (0,r ) Y in a neighborhood of the boundary defined by r = 0, so that on this
1 r
×
neighborhood,
dr2 h(r)
g = + , h(r) h as r 0,
z r4 r2 → →
in other words, h extends to a ∞ tensor on [0,r ). Uniquely associated to the ac scattering
1
C
metric (Z¯,g ) is the complete non-compact manifold Z with asymptotically conic end; Z is
z
4
known as an ac space.2 Letting ρ= 1, there is a compact subset K Z so that
r z ⊂
Z K = (1/r , ) Y, g = dρ2+ρ2h(1/ρ).
− z ∼ 1 ∞ ρ× z|Z−Kz
A familiar example of an ac scattering metric is the standard metric on the radial
compactification of Rn with boundary Sn−1 at infinity.
Next, we define the conic metric on a compact manifold.
Definition 3. Let M be a compact metric space with Riemannian metric g. Then, (M ,g)
c c
has an isolated conic singularity at the point p and g is called a conic metricif the following
hold.
1. (M p ,g) is a smooth, open manifold.
c
−{ }
2. There is a neighborhood N of p and a function x: N p (0,x ] for some x > 0,
1 1
−{ } →
such that N p is diffeomorphic to (0,x ] Y with metric g = dx2 + x2h(x)
1 x
− { } ×
where (Y,h) is a compact, smooth n 1 manifold and h(x) is a smooth family of
− { }
metrics on Y converging to h as x 0, in other words, h extends to a ∞ tensor on
→ C
[0,x ) Y.
1 x
×
Associated to a manifold with isolated conic singularity is the manifold with boundary
obtained by blowing up the cone point, adding a copy of (Y,h) at this point. Then, x is a
boundary defining function for the boundary, (Y,h). We use M0 = M p to denote the
0 c−{ }
smooth incomplete conic manifold, M to denote the metric space closure of M0, and M
c 0 0
to denote the associated manifold with boundary. We work almost exclusively with M .
0
The resolution blowup is a nonstandard blowup that resolves an isolated conic singu-
larity in a compact manifold using an asymptotically conic space. In the definition of
resolution blowup we use the notation M N for a smooth manifold constructed from the
φ
∪
smooth manifolds M and N with a diffeomorphism φ from V N to U M that gives
⊂ ⊂
the equivalence relation, V p φ(p) U. M N is the disjoint union of M and N
φ
∋ ∼ ∈ ∪
modulo the equivalence relation of φ. The smooth structure on M N and the topology
φ
∪
is induced by that of M and N.
Definition 4. Let (M ,g ) be a compact n manifold with isolated conic singularity and let
0 0
(Z,g ) be an asymptotically conic space of dimension n, so that ∂M = ∂Z¯ = (Y,h). Then,
z 0
M0 = K V ,
0 0∪ 0
where V = (0,x ) Y, and K is compact. With this diffeomorphism
0 ∼ 1 x 0
×
g = dx2+x2h˜(x) on (0,x ) Y.
0 1
×
2Notethatasymptoticallyconicspacesaresometimescalled“asymptoticallylocallyEuclidean,”orALE.
However,thattermisoftenusedforthemorerestrictiveclassofspacesthatareasymptoticatinfinitytoa
cone over a quotient of the sphere by a finite group, so to avoid confusion, we use theterm asymptotically
conic.
5
We may assume the boundary of K in M0 is of the form ∂K = x = x = Y, and we
0 0 0 { 1} ∼
may extend x smoothly to K so that 2x > x x on K . Similarly,
0 1 1 0
≥
Z = K V ,
z z
∪
where V = (ρ , ) Y, and K is compact. With this diffeomorphism
z ∼ 1 ∞ ρ× z
g = dρ2+ρ2h(ρ,y) on (ρ , ) Y.
z 1
∞ ×
We may assume that ∂K is of the form ∂K = ρ = ρ = Y, and we similarly extend ρ
z z { 1} ∼
smoothly to K so that ρ ρ > ρ /2 on K .
z 1 1 z
≥
Let δ = min x ,1/ρ . Then for 0< ǫ < δ, and R > ρ , let
1 1 1
{ }
M = (x,y) M :x > ǫ , Z = (ρ,y) Z : ρ < R .
0,ǫ 0 R
{ ∈ } { ∈ }
The resolution blowup of (M ,g ) by (Z,g ) is,
0 0 z
M := M Z ,
ǫ 0,ǫ∪φǫ 1/ǫ
where the joining map φ is defined for each ǫ by
x
φ :M M Z Z , φ (x,y) = ,y .
ǫ 0,ǫ 0,δ 1/ǫ 1/δ ǫ
− → − δǫ
(cid:16) (cid:17)
For δ > ǫ > ǫ′ > 0, the manifolds M and M are diffeomorphic, and so the resolution
ǫ ǫ′
blowup of M by Z which we call M is unique up to diffeomorphism.
0
Remark: The resolution blowup resolves the singularity in a conic manifold using an
acscatteringspace. Insteadof resolvingthesingularity inM usingZ,wemayequivalently
0
define the resolution blowup to resolve the boundary of Z¯ using M .3
0
The ac single space, analogous to the analytic surgery single space in [24], is the setting
for the definition of ac convergence.
Definition 5. Let (M ,g ) be a conic metric and let (Z¯,g ) be a scattering metric; assume
0 0 z
both are dimension n with the same cross section (Y,h) at the boundary, and assume
δ = 1 (definition 4). Then, M = ((0,1) Y) K and Z¯ = ((0,1) Y) K with
0 ∼ x × ∪ 0 ∼ r × ∪ z
∂K = Y = ∂K . The asymptotically conic convergence (acc) single space is defined as
0 ∼ ∼ z
S
follows,
:= [0,1) [0,1) Y (K x = 1,r = 1 ) (K r =1,x = 1 ).
x r 0 z
S × × ∪ ×{ 6 } ∪ ×{ 6 }
3Let r=1/ρ be thedefiningfunction for ∂Z¯. The resolution blowup of Z¯ byM0 is
ǫδ
M0∪ψǫ Z¯, ψǫ(x,y)=„x,y«,
where the patching map ψǫ is defined on M0 −K0 with image (0,δ)r ×Y a neighborhood of ∂Z¯. The
resulting smooth compact resolution space is diffeomorphic to M.
6
The smooth structure of is induced by that of M and Z¯. Namely, smooth functions
0
S
on are functions which are smooth jointly in x and r on (0,1) (0,1) Y, smoothly
x r
S × ×
extend to a smooth function on K at x = 1, on K at r = 1, on M at r = 0, and on Z¯
0 z 0
at x= 0. To give a precise description of the metrics considered here, we define below the
acc tensor, a smooth, polyhomogeneous, symmetric 2-cotensor on the acc single space.
Definition 6. Let be the acc single space associated to M and Z¯ as in definition 5.
0
S
Let ǫ(p) = x(p)r(p) : [0,1), where x,r are extended to K and K respectively to be
0 z
S →
identically 1. We define the acc tensor as follows:
G
dx2+x2h(x)+(xr)2 dr2 + h(r) x,r [0,1)
r4 r2 ∈
=  dx2+x2h(x)+(xr)2((cid:16)g ) (cid:17) r = 1, x [0,1)
G  g +(xr)2 dr2 + h(rz)|Kz x= 1, r ∈ [0,1)
0|K0 r4 r2 ∈
 (cid:16) (cid:17)

For 0 < ǫ < 1,let M = xr = ǫ ; note that this M is diffeomorphic to the
ǫ ǫ
{ } ⊂ S
resolution blowup M of M by Z.4 The family of metrics g = on M is said to
0 { ǫ G|Mǫ}
converge asymptotically conically to (M ,g ).
0 0
Remarks
1. Theaccsinglespacehastwoboundaryhypersurfacesatǫ = 0;thesearediffeomorphic
toM atr = 0andZ¯ atx= 0,andtheymeetinacodimension2cornerdiffeomorphic
0
to Y. There are also boundary hypersurfaces at ǫ = 1 which we ignore since we are
interested in ǫ 0.
→
2. Since g is a smooth metric on M = ((0,1) Y) K and g is a smooth metric
0 0 ∼ x 0 z
× ∪
on Z¯ = ((0,1) Y) K , is a smooth symmetric 2-cotensor on which extends
∼ r z
× ∪ G S
smoothly across neighborhoods where it is piecewise defined.
3. At r = 0, restricts to = g ; as x 0, vanishes to order 2.
{r=0} 0
G G| → G
4. On M where 0 < r(p),x(p) < 1,
ǫ
⊂ S
ǫ ǫ2
r = = dr2 = dx2,
x ⇒ x4
so
dr2 h(r)
g = dx2+x2h(x) = ǫ2 + .
ǫ r4 r2
(cid:18) (cid:19)
4In the definition of the acc single space we have assumed δ from the definition of resolution blowup is
1. This is to simplify our calculations. On {xr = ǫ} we then have r = ǫ, equivalently x = ǫ, so letting
x r
δ = 1 in the definition of resolution blowup the identification of {xr = ǫ} ⊂ S and the resolution blowup
M follows immediately.
ǫ
7
5. On M for x ǫ, r 1 and this subset is diffeomorphic to K with metric g =
ǫ z ǫ
≡ ≡
ǫ2g . On M for x 1, r ǫ and this subset is diffeomorphic to K with metric
z|Kz ǫ ≡ ≡ 0
g = g +O(ǫ)2.
ǫ 0
The following lemma is useful both for visualizing ac convergence and for proving
spectral convergence.
Lemma 1. Let (M ,g ) and (Z,g ) be as in definitions 4, 5, 6 and let (M,g ) converge
0 0 z ǫ
asymptotically conically to (M ,g ). Then, there exists a family of diffeomorphisms φ
0 0 ǫ
{ }
from a fixed open proper subset U M to increasing neighborhoods Z Z such that
1/ǫ
⊂ ⊂
g = ǫ2(φ )∗g . Moreover, on M U, g g smoothly as ǫ 0 and any
ǫ|U ∼ ǫ |Z1/ǫ |U − ǫ → 0 →
K M(cid:16)0 is diffeomo(cid:17)rphic to some fixed K′ M so that g g smoothly and uniformly
⊂⊂ 0 ⊂ ǫ → 0
on K′.
Proof
The existence of φ and U M follows immediately from the definition of resolution
ǫ
⊂
blowup and the diffeomorphism between the resolution blowup M and xr = ǫ . By
{ } ⊂ S
the above remarks, on the neighborhood U M where this diffeomorphism is defined,
⊂
g =(ǫ2)(φ )∗g .
ǫ|U ǫ z|Z1/ǫ
Sinceg = g + (ǫ2),thesmoothconvergence ofg tog onM U follows immediately.
ǫ 0 ǫ 0
O −
Any compact subset K M is contained in M for some ǫ > 0 and so is diffeomorphic
0 0,ǫ
⊂⊂
to K M and to K′ (M U). Conversely, any K (M U) is diffeomorphic to
ǫ ǫ
⊂⊂ ⊂⊂ − ⊂⊂ −
K M and to K′ M M .
ǫ ǫ 0,ǫ 0
⊂ ⊂⊂ ⊂
♥
3 Geometric Preliminaries
This section is a brief review of the theory and terminology of manifolds with corners, b
maps, and blowups. A complete reference is [26], see also [23].
3.1 Manifolds with Corners
Let X beamanifold with corners. Thismeans that near any of its points, X is modeled on
a product [0, )k Rn−k where k depends on the point and is the maximal codimension
∞ ×
of the boundary face containing that point. We also assume that all boundary faces of
X are embedded so they too are manifolds with corners. The space (X) of all smooth
V
vector fields on X is a Lie algebra under the standard bracket operation. It contains the
Lie subalgebra
(X) := V (X); V is tangent to each boundary face of X. (1)
b
V { ∈ V }
8
Then (X) is itself the space of all smooth sections of a vector bundle,
b
V
(X) = ∞(X; bTX),
b
V C
wherebTX isthebundledefinedsothattheabove holdsandiscalled theb-tangent bundle.
3.1.1 Blowing Up
An embedded codimension k submanifold Y of a manifold with corners X is called a p-
submanifold (p for product) if near each point of Y there are local product coordinates
so that Y is defined by the vanishing of some subset of them. In other words, X and Y
must have consistent local product decompositions. Then one can define a new manifold
with corners [X;Y] to be the normal blowup of X around Y. This is obtained by replacing
Y by its inward-pointing spherical normal bundle. The union of this normal bundle and
X Y has a unique minimal differential structure as a manifold with corners so that the
−
lifts of smooth functions on X and polar coordinates around Y are smooth. One can also
consider iterated blowups, written [[X;Y ];Y ]], where Y is a p-submanifold of X and Y
1 2 1 2
is a p-submanifold of Y . We may consider any finite sequence of such blowups. If we have
1
such a sequence of embedded p-submanifolds,
X Y Y Y ... Y
1 2 3 n
⊃ ⊃ ⊃ ⊃
then the iterated blowup
[[X;Y ];Y ];...;Y ]
1 2 n
can be performed in any order with the same result [24]. Blowups may also be defined
using equivalence classes of curves [26]. Let r be a defining function for the p submanifold
and consider the family of curves γ(t) = (r(t),y(t)) such that
γ(t) Y t = 0,
∈ ⇐⇒
r(t)= O(t).
Let E be the set of equivalence classes of all such curves with
γ γ′ (y y′)(t) = O(t), (r r′)(t) = O(t2).
∼ ⇐⇒ − −
There is a natural R+ action on E given by
R+ a: γ(t) γ(at).
∋ →
Then E modulo this equivalence relation is naturally diffeomorphic to N+(Y), the inward
pointing spherical normal bundle of Y, so we can define [X;Y] by
[X;Y] = (X Y) E/(R+ 0 ).
− ∪ −{ }
9
We can also define parabolic blowups in certain contexts [9]. Let Y be a p-submanifold of
codimension k so that there exist local coordinates (r,y) = (r ,...,r ,y ,...,y ) in a
1 k 1 n−k
neighborhood of Y with r vanishing precisely at Y, and so that dr induces a sub-budle
i 1
of the tangent bundle TX. Instead of the above equivalence classes of curves we consider
γ(t) such that
γ(t) = (r (t),...,r (t),y (t),...y (t)) Y t = 0,
1 k 1 n−k
∈ ⇐⇒
r (t) = O(t), i = 1, r (t) = O(t2).
i 1
6
Two such curves are equivalent if
γ γ′ (y y′)(t) = O(t),(r r′)(t) = O(t2)i = 1, (r (t) r′(t)) = O(t3).
∼ ⇐⇒ j − j i− i 6 1 − 1
Since dr is a sub-bundle of TX there is a natural R+ action on the set of equivalence
1
classes E of all such curves,
2
R+ a: γ(t) (r (a2t),r (at),...,y (at)).
1 i j
∋ →
The set of equivalence classes of all such curves modulo this R+ action is naturally diffeo-
morphic to the inward pointing r -parabolic normal bundle of Y,
1
E /(R+ 0 ) = PN+(Y).
2 −{ } ∼ r1
We define the r -parabolic blowup of X around Y as the union of X Y and this inward
1
−
pointing r -parabolic bundle,
1
[X;Y,dr ] := (X Y) PN+(Y).
1 − ∪ r1
The union of this r -parabolic bundle and X Y again has a unique minimal differential
1
−
structure as a manifold with corners so that the lifts of smooth functions on X and r -
1
parabolic coordinates around Y are smooth. By r -parabolic coordinates around Y, we
1
mean the coordinates,
ρ= (r12+r24+...+rk4)14, θ = (θ1,...θn) ∈Sn−1,
with local coordinates (r ,...r ,y ,...,y ) in a neighborhood of Y satisfying
1 k 1 n−k
r = ρθ , i = 1, r = ρ2θ , y = ρθ .
i i 1 1 j j
6
Foranyparabolicorsphericalblowupthereisanaturalblow-downmapβ : [X;Y] X
∗
→
and corresponding blow-up map β∗ : X [X;Y], so that the image of Y under β∗
→
is a boundary hypersurface of [X;Y] diffeomorphic to the inward pointing spherical (or
parabolic) normal bundle of Y. As such,
[X;Y]= (X Y) β∗(Y).
− ∪
10