Potential Theory on Berkovich Spaces Lecture 3: Harmonic …people.math.gatech.edu/~mbaker/pdf/AWS3.pdf · 2014-08-15 · Harmonic functions If f is an aﬃne function on M(Z), deﬁne

Potential Theory on Berkovich SpacesLecture 3: Harmonic functions

Matthew Baker

Georgia Institute of Technology

Arizona Winter School on p-adic GeometryMarch 2007

Matthew Baker Lecture 3: Introduction to Berkovich Curves

Goals

In this lecture, we will explore the notion of a harmonic function inthe context of M(Z) and P1

Berk.


Notation and terminology

K : an algebraically closed field which is complete with respectto a nontrivial non-archimedean absolute value (e.g. K = Cp)

A domain in a topological space X is a connected open subsetof X .


Notation and terminology

K : an algebraically closed field which is complete with respectto a nontrivial non-archimedean absolute value (e.g. K = Cp)

A domain in a topological space X is a connected open subsetof X .


Harmonic functions on M(Z)

We introduce the following notation for points of M(Z):

ζ∞,ε: the point of M(Z) corresponding to the archimedean norm| |∞,ε for 0 < ε < 1.

ζp,ε: the point of M(Z) corresponding to the p-adic norm | |p,ε for0 < ε < ∞.

ζ0: the point of M(Z) corresponding to the trivial norm | |0.ζp,∞: the point of M(Z) corresponding to the p-trivial seminorm

| |p,∞.







| |p,∞.







| |p,∞.






ζ0: the point of M(Z) corresponding to the trivial norm | |0.

ζp,∞: the point of M(Z) corresponding to the p-trivial seminorm| |p,∞.







| |p,∞.


A picture of M(Z)


Tangent directions in M(Z)

For x in M(Z), we define the set Tx of tangent directions atx to be the connected components of M(Z)\{x}.

When x = ζ0 is the point corresponding to the trivialseminorm | |0 on Z, there is a canonical bijection between Tx

and the set MQ of places of Q.

At all other points of M(Z), the space Tx has cardinality 1 or2.

For v ∈ MQ, we will refer to the segment

`v =

{{| |∞,ε}0≤ε≤1 v ↔∞{| |p,ε}0≤ε≤∞ v ↔ p.

as the branch emanating from ζ0 in the direction v .



For x in M(Z), we define the set Tx of tangent directions atx to be the connected components of M(Z)\{x}.When x = ζ0 is the point corresponding to the trivialseminorm | |0 on Z, there is a canonical bijection between Tx




`v =

{{| |∞,ε}0≤ε≤1 v ↔∞{| |p,ε}0≤ε≤∞ v ↔ p.








`v =

{{| |∞,ε}0≤ε≤1 v ↔∞{| |p,ε}0≤ε≤∞ v ↔ p.








`v =

{{| |∞,ε}0≤ε≤1 v ↔∞{| |p,ε}0≤ε≤∞ v ↔ p.



The metric space HZ

Recall that HZ denotes the complement of the points of typeζp,∞.

The points of HZ are precisely the ones at finite distance fromthe trivial point ζ0.

The space HZ is endowed with a metric topology which is notthe same as its subspace topology as a subset of M(Z).


The metric space HZ





The metric space HZ





Affine functions

Let f : M(Z) → R ∪ {±∞}. We say that f is affine onM(Z) if:

1 The restriction of f to each branch `v for v ∈ MQ is an affinefunction of the form t 7→ av t + bv for real constants av , bv

depending on v .2 f is constant (i.e, av = 0) on all but finitely many branches `v .


Affine functions



depending on v .

2 f is constant (i.e, av = 0) on all but finitely many branches `v .


Affine functions



depending on v .2 f is constant (i.e, av = 0) on all but finitely many branches `v .


Harmonic functions

If f is an affine function on M(Z), define

∆ζ0(f ) = −∑

v∈Tζ0

d~v f (x)

= −∑

v∈MQ

av .

Definition

An affine function f : M(Z) → R ∪ {±∞} is harmonic at ζ0 if∆ζ0(f ) = 0.


Harmonic functions


∆ζ0(f ) = −∑

v∈Tζ0

d~v f (x)

= −∑

v∈MQ

av .

Motivated by the classical notion of harmonic functions on aweighted graph, we make the following definition:

Definition



Harmonic functions


∆ζ0(f ) = −∑

v∈Tζ0

d~v f (x)

= −∑

v∈MQ

av .

Motivated by the classical notion of harmonic functions on aweighted graph, we make the following definition:

Definition



Example: log |n|

Lemma

If n ∈ Z is a nonzero integer, then the function x 7→ − log |n|x isaffine, and is harmonic at ζ0.

Proof.

Along the branch `v , the function − log |n|x is linear withslope equal to − log |n|v .

In particular, − log |n|x is constant along `v for all finite placesv corresponding to a prime p with p - n.

The fact that − log |n|x is harmonic at ζ0 is equivalent to theproduct formula for Q:

∆ζ0(− log |n|x) =∑

v∈MQ

log |n|v = 0.


Example: log |n|

Lemma


Proof.




∆ζ0(− log |n|x) =∑

v∈MQ

log |n|v = 0.


Example: log |n|

Lemma


Proof.




∆ζ0(− log |n|x) =∑

v∈MQ

log |n|v = 0.


Example: log |n|

Lemma


Proof.




∆ζ0(− log |n|x) =∑

v∈MQ

log |n|v = 0.


Harmonic functions on P1Berk

We now turn to what it means for an (extended) real-valuedfunction on P1

Berk to be harmonic. This is more complicatedthan the corresponding notion for M(Z), since the branchingbehavior of P1

Berk is much wilder.

Recall that if x ∈ P1Berk, the set Tx of tangent directions at x

is in one-to-one correspondence with the connectedcomponents of P1

Berk\{x}, and that:

|Tx | =

|P1(K )| x of type II2 x of type III1 x of type I or type IV.








Berk\{x}, and that:

|Tx | =









Berk\{x}, and that:

|Tx | =

|P1(K )| x of type II

2 x of type III1 x of type I or type IV.








Berk\{x}, and that:

|Tx | =

|P1(K )| x of type II2 x of type III

1 x of type I or type IV.








Berk\{x}, and that:

|Tx | =



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Continuous piecewise affine functions

Let U be a connected open subset of P1Berk (with respect to

the Berkovich topology), and let f : U → R ∪ {±∞} be anextended-real valued function which is finite-valued on HBerk.

We say that f is continuous piecewise affine on U, and writef ∈ CPA(U) if:

1 f is continuous.

2 The restriction of f to HBerk is piecewise-affine with respectto the path metric ρ.

(Concretely, this means that for each x ∈ U ∩HBerk and each~v ∈ Tx , for every sufficiently small path Λ~v representing ~v , therestriction of f to the segment Λ~v has the formt 7→ a~v t + b~v .)

3 For each x ∈ U ∩HBerk, we have a~v = 0 (i.e., f is constantalong sufficiently short paths emanating from x in thedirection ~v) for all but finitely many ~v ∈ Tx .






1 f is continuous.









1 f is continuous.









1 f is continuous.









1 f is continuous.

2 The restriction of f to HBerk is piecewise-affine with respectto the path metric ρ.(Concretely, this means that for each x ∈ U ∩HBerk and each~v ∈ Tx , for every sufficiently small path Λ~v representing ~v , therestriction of f to the segment Λ~v has the formt 7→ a~v t + b~v .)







1 f is continuous.

2 The restriction of f to HBerk is piecewise-affine with respectto the path metric ρ.(Concretely, this means that for each x ∈ U ∩HBerk and each~v ∈ Tx , for every sufficiently small path Λ~v representing ~v , therestriction of f to the segment Λ~v has the formt 7→ a~v t + b~v .)



Directional derivatives

If x ∈ U ∩HBerk and f ∈ CPA(U), then for each ~v ∈ Tx , thedirectional derivative d~v f (x) is well-defined.

In the notation from the previous slide, if the restriction of fto some path representing ~v has the form t 7→ a~v t + b~v , then

d~v f (x) = a~v .

In particular, the quantity

∆x(f ) := −∑v∈Tx

d~v f (x)

is well-defined.





d~v f (x) = a~v .


∆x(f ) := −∑v∈Tx

d~v f (x)

is well-defined.





d~v f (x) = a~v .


∆x(f ) := −∑v∈Tx

d~v f (x)

is well-defined.


Definition of harmonic functions on P1Berk

Definition

Let U be a domain in P1Berk, and let h : U → R. We say that h is

harmonic on U if

1 h ∈ CPA(U), and2 h is harmonic at x for all x ∈ U, meaning that either:

x ∈ HBerk and ∆x(h) = 0 (i.e., the sum of the slopes of h inall tangent directions emanating from x is zero.); orx ∈ P1(K ) and h is constant on an open neighborhood of x .



Definition


harmonic on U if

1 h ∈ CPA(U), and

2 h is harmonic at x for all x ∈ U, meaning that either:




Definition


harmonic on U if





Definition


harmonic on U if


x ∈ HBerk and ∆x(h) = 0 (i.e., the sum of the slopes of h inall tangent directions emanating from x is zero.); or

x ∈ P1(K ) and h is constant on an open neighborhood of x .



Definition


harmonic on U if




Example: log+ |x |

Example

Consider the function G : P1Berk → R ∪ {+∞} defined by

G (x) =

{+∞ x = ∞logv max(|T |x , 1) x ∈ A1

Berk.

The restriction of G to K is log+v |x | = logv max(|x |, 1).

G is harmonic on P1Berk\{ζGauss,∞}.


Example: log+ |x |

Example


G (x) =

{+∞ x = ∞logv max(|T |x , 1) x ∈ A1

Berk.




Example: log+ |x |

Example


G (x) =

{+∞ x = ∞logv max(|T |x , 1) x ∈ A1

Berk.




Example: log+ |x | (continued)

Indeed, let Λ denote the closed path from ζGauss to ∞ in P1Berk,

and let rΛ : P1Berk � Λ be the natural retraction map from P1

Berk

onto Λ. Then:

1 G (x) is linear with slope 1 along Λ, i.e., G (x) = ρ(ζGauss, x).

2 G (x) is locally constant off Λ, i.e., for all x ∈ P1Berk, we have

G (x) = G (rΛ(x)).

Remark

1 G is not harmonic at ζGauss: the sum of the slopes of G in alldirections emanating from ζGauss is 1.

2 G is also not harmonic at ∞.





Berk

onto Λ. Then:



G (x) = G (rΛ(x)).

Remark







Berk

onto Λ. Then:



G (x) = G (rΛ(x)).

Remark







Berk

onto Λ. Then:



G (x) = G (rΛ(x)).

Remark







Berk

onto Λ. Then:



G (x) = G (rΛ(x)).

Remark







Berk

onto Λ. Then:



G (x) = G (rΛ(x)).

Remark




Example: log |f | for f analytic and nowhere zero

Example

Let V = M(AV ) be an affinoid subdomain of P1Berk, and let U be

a connected open subset of V . If f ∈ AV is a nowhere zeroanalytic function on V , then the function logv |f |x is harmonic onU.


Example: log |f | for f analytic and nowhere zero

Example


a connected open subset of V . If f ∈ AV is a nowhere zeroanalytic function on V , then the function logv |f |x is harmonic onU.

This generalizes the well-known classical fact that if f is a nowherezero analytic function on an open subset U of the complex plane,then log |f | is harmonic on U.


The maximum principle

The following result is the Berkovich space analogue of the classicalmaximum principle for harmonic functions on domains in C:

Theorem (Maximum Principle)

1 If h is a nonconstant harmonic function on a domainU ⊂ P1

Berk, then h does not achieve a maximum or aminimum value on U.

2 If h is a harmonic function on a domain U ⊂ P1Berk which

extends continuously to the closure U of U, then h achievesboth its minimum and maximum values on the boundary ∂Uof U.


















A consequence of the maximum principle

Recall that a simple domain in P1Berk is a connected open set

V ⊆ P1Berk whose boundary is a finite subset of HR

Berk.

Lemma

Every harmonic function on a simple domain V in P1Berk extends

continuously to V .

Corollary

If U = P1Berk or U is an open Berkovich disk, then every harmonic

function on U is constant.





Berk.

Lemma


continuously to V .

Corollary







Berk.

Lemma


continuously to V .

If V = P1Berk, then then ∂V is empty, and if V is a Berkovich open

disk, then ∂V consists of a single point. By the second part of themaximum principle, we conclude:

Corollary







Berk.

Lemma


continuously to V .

If V = P1Berk, then then ∂V is empty, and if V is a Berkovich open

disk, then ∂V consists of a single point. By the second part of themaximum principle, we conclude:

Corollary




The main dendrite of a domain

An important observation is that the behavior of a harmonicfunction on a domain U ⊆ P1

Berk is controlled by its behavior on acertain special subset.

Definition

If U is a domain in P1Berk, the main dendrite D(U) ⊂ U is set of all

points x ∈ U belonging to paths between boundary points of U.

If U is a Berkovich open disk, then D(U) is empty.

If U = B(a,R)−\B(a, r) is a Berkovich open annulus, thenD(U) is the open segment joining the two boundary pointsζa,r and ζa,R of U.





Definition









Definition



For example:







Definition



For example:







Definition



For example:




A Berkovich open annulus


The main dendrite of a domain (continued)

The most important topological fact about the main dendrite is:

Lemma

If the main dendrite of a domain U is nonempty, then it is finitelybranched at each point of HBerk.

Example

If K = Cp and U = P1Berk\P1(Qp), then the main dendrite D(U) is

a locally finite real tree in which the set of branch points isdiscrete, and every branch point has degree p + 1. In fact, D(U)can be identified with the (geometric realization of the)Bruhat-Tits tree for PGL(2, Qp).




Lemma


Example






Lemma


Example




Harmonic functions and the main dendrite

Theorem

Let U be a domain in P1Berk, and let h be harmonic on U. If the

main dendrite is empty, then h is constant; otherwise, h is constanton all branches leading away from the main dendrite.


The Poisson Formula (classical case)

In the classical theory of harmonic functions in the complexplane, if f is harmonic on an open disk V then it has acontinuous extension to the closure of V , and the PoissonFormula expresses the values of f on V in terms of its valueson the boundary of V .

Specifically, if V ⊆ C is an open disk of radius r centered atz0, and if f is harmonic in V , then f extends continuously toV and f (z0) =

∫∂V f dµV , where µV is the uniform

probability measure dθ/2π on the boundary circle ∂V .

More generally, for any z ∈ V there is a measure µz,V

depending on z and V , called the Jensen-Poisson measure,such that

f (z) =

∫∂V

f dµz,V

for every harmonic function f on V .









f (z) =

∫∂V

f dµz,V










f (z) =

∫∂V

f dµz,V










f (z) =

∫∂V

f dµz,V



The Poisson Formula for P1Berk

In P1Berk, the basic open neigbhorhoods are the simple

domains, which have only a finite number of boundary points.

Every harmonic function f on a simple domain V has acontinuous extension to its closure.

There is an analogue of the Jensen-Poisson measure whichyields an explicit formula for f on V in terms of its values onthe boundary of V . In other words, one can explicitly solvethe Berkovich space analogue of the Dirichlet problem on anysimple domain.














The Gromov product

Definition

For x , y , z ∈ HBerk, define the Gromov product (x |y)z by

(x |y)z = ρ(w , z),

where w is the first point where the unique paths from x to zand y to z intersect.

Alternatively, one checks easily that

(x |y)z =1

2(ρ(x , z) + ρ(y , z)− ρ(x , y)) .

Remark

This definition plays an important role in Gromov’s theory ofδ-hyperbolic spaces, with HBerk being an example of a0-hyperbolic space.


The Gromov product

Definition


(x |y)z = ρ(w , z),



(x |y)z =1

2(ρ(x , z) + ρ(y , z)− ρ(x , y)) .

Remark



The Gromov product

Definition


(x |y)z = ρ(w , z),



(x |y)z =1

2(ρ(x , z) + ρ(y , z)− ρ(x , y)) .

Remark



Some linear algebra

Let V be a simple domain in P1Berk with boundary points

x1, . . . , xm ∈ HRBerk.

A probability vector on Rm is a vector [p1, . . . , pm] ∈ Rm suchthat pi ≥ 0 for 1 ≤ i ≤ m and p1 + · · ·+ pm = 1.

Lemma

For each z ∈ V ∩HBerk, there is a unique probability vector[h1(z), . . . , hm(z)] for which the quantity

h1(z)(xi |x1)z + · · ·+ hm(z)(xi |xm)z

is independent of i .


Some linear algebra


x1, . . . , xm ∈ HRBerk.


Lemma


h1(z)(xi |x1)z + · · ·+ hm(z)(xi |xm)z



Some linear algebra


x1, . . . , xm ∈ HRBerk.


Lemma


h1(z)(xi |x1)z + · · ·+ hm(z)(xi |xm)z



Some linear algebra


x1, . . . , xm ∈ HRBerk.


Lemma


h1(z)(xi |x1)z + · · ·+ hm(z)(xi |xm)z


One can give an explicit formula for h1(z), . . . , hm(z) usingCramer’s rule.


Harmonic measures

For each i , the function z 7→ hi (z), defined originally forz ∈ V ∩HBerk, extends by continuity to a map hi : V → R,called the i th harmonic measure with respect to V .

hi is a harmonic function on V .

The values of hi on ∂V are given by the formula hi (xj) = δij .

By construction, we have 0 ≤ hi (z) ≤ 1 for all z ∈ V andh1 + · · ·+ hm ≡ 1 on V .


Harmonic measures






Harmonic measures






Harmonic measures






The Poisson formula

Theorem (Poisson formula)


x1, . . . , xm. Then each harmonic function f on V has a continuousextension to V , and there is a unique such function with aprescribed set of boundary values

Moreover, f can be computedfrom its boundary values using the formula

f (z) =m∑

i=1

f (xi ) · hi (z),

valid for all z ∈ V .


The Poisson formula

Theorem (Poisson formula)


x1, . . . , xm. Then each harmonic function f on V has a continuousextension to V , and there is a unique such function with aprescribed set of boundary values Moreover, f can be computedfrom its boundary values using the formula

f (z) =m∑

i=1

f (xi ) · hi (z),

valid for all z ∈ V .


The Jensen-Poisson measure

For z ∈ V , define the Jensen-Poisson measure µz,V on Vrelative to the point z by

µz,V =m∑

i=1

hi (z)δxi .

The Poisson formula can be reformulated as follows:

Corollary

If V is a simple domain in P1Berk, then a continuous function

f : V → R is harmonic in V if and only if

f (z) =

∫∂V

f dµz,V

for all z ∈ V .




µz,V =m∑

i=1

hi (z)δxi .


Corollary



f (z) =

∫∂V

f dµz,V

for all z ∈ V .




µz,V =m∑

i=1

hi (z)δxi .


Corollary



f (z) =

∫∂V

f dµz,V

for all z ∈ V .


Limits of harmonic functions

The Poisson formula can be used to prove that any limit of asequence of harmonic functions is harmonic, under a much weakercondition than is required classically.

Theorem

Let U be a domain in P1Berk. Suppose f1, f2, . . . are harmonic in U

and converge pointwise to a function f : U → R. Then f (z) isharmonic in U, and the fi (z) converge uniformly to f (z) oncompact subsets of U.


Harnack’s principle

There is also a Berkovich space analogue of Harnack’s principle:

Theorem (Harnack’s Principle)

Let U be a domain in P1Berk, and suppose f1, f2, . . . are harmonic in

U, with 0 ≤ f1 ≤ f2 ≤ · · · . Then either

1 limi→∞ fi ≡ +∞; or

2 f (z) = limi→∞ fi (z) is finite for all z, the fi (z) convergeuniformly to f (z) on compact subsets of U, and f (z) isharmonic in U.


























Subharmonic functions

Definition

Let U ⊂ P1Berk be a domain. A function f : U → [−∞,∞) with

f (x) 6≡ −∞ is called subharmonic on U if

(SH1) f is upper semicontinuous. (This means that f −1([−∞, b)) isopen for each b ∈ R.)

(SH2) For each simple subdomain V of U (i.e., a simple domain Vwhose closure is contained in U), we have

f (z) ≤∫

∂Vf dµz,V

for all z ∈ V .

f is called superharmonic on U if −f is subharmonic on U.



Definition





f (z) ≤∫

∂Vf dµz,V

for all z ∈ V .




Definition





f (z) ≤∫

∂Vf dµz,V

for all z ∈ V .




Definition





f (z) ≤∫

∂Vf dµz,V

for all z ∈ V .



Subharmonic functions (continued)

Remark

1 By the Poisson formula, condition (SH2) can be replaced bythe condition that for each simple subdomain V ⊂ U andeach harmonic function h on V , if f (x) ≤ h(x) on ∂V thenf (x) ≤ h(x) on V .

2 f is harmonic on U if and only if it is both subharmonic andsuperharmonic on U.


Subharmonic functions (continued)

Remark

1 By the Poisson formula, condition (SH2) can be replaced bythe condition that for each simple subdomain V ⊂ U andeach harmonic function h on V , if f (x) ≤ h(x) on ∂V thenf (x) ≤ h(x) on V .

2 f is harmonic on U if and only if it is both subharmonic andsuperharmonic on U.


Examples of subharmonic functions

Example


a connected open subset of V . If f ∈ AV is analytic on V , thenthe function logv |f |x is subharmonic on U.

Example

For fixed y , z ∈ HBerk, the function f (x) = (x |y)z issuperharmonic in P1

Berk\{z}, and subharmonic in P1Berk\{y}.


Examples of subharmonic functions

Example


a connected open subset of V . If f ∈ AV is analytic on V , thenthe function logv |f |x is subharmonic on U.

Example

For fixed y , z ∈ HBerk, the function f (x) = (x |y)z issuperharmonic in P1

Berk\{z}, and subharmonic in P1Berk\{y}.


Maximum principle for subharmonic functions


1 If f is a nonconstant subharmonic function on a domainU ⊂ P1

Berk, then f does not achieve a global maximum on U.

2 If f is a subharmonic function on a domain U ⊂ P1Berk which

extends continuously to U, then f achieves its maximum valueon ∂U.


Maximum principle for subharmonic functions


1 If f is a nonconstant subharmonic function on a domainU ⊂ P1

Berk, then f does not achieve a global maximum on U.

2 If f is a subharmonic function on a domain U ⊂ P1Berk which

extends continuously to U, then f achieves its maximum valueon ∂U.


Subharmonic functions and the main dendrite

The following result shows that at any given point, there are onlyfinitely many tangent directions in which a subharmonic functioncan be increasing:

Theorem

Let f be subharmonic on a domain U. Then f is non-increasing onpaths leading away from the main dendrite of U. If U is a disk,then f is non-increasing on paths leading away from the uniqueboundary point of U.


Subharmonic functions and the main dendrite

The following result shows that at any given point, there are onlyfinitely many tangent directions in which a subharmonic functioncan be increasing:

Theorem

Let f be subharmonic on a domain U. Then f is non-increasing onpaths leading away from the main dendrite of U. If U is a disk,then f is non-increasing on paths leading away from the uniqueboundary point of U.


Documents

Potential Theory on Berkovich Spaces Lecture 3: Harmonic …people.math.gatech.edu/~mbaker/pdf/AWS3.pdf · 2014-08-15 · Harmonic functions If f is an aﬃne function on M(Z), deﬁne