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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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
A picture of M(Z)
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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).
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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).
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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).
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
Harmonic functions
If f is an affine function on M(Z), define
∆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
An affine function f : M(Z) → R ∪ {±∞} is harmonic at ζ0 if∆ζ0(f ) = 0.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
Harmonic functions
If f is an affine function on M(Z), define
∆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
An affine function f : M(Z) → R ∪ {±∞} is harmonic at ζ0 if∆ζ0(f ) = 0.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 II
2 x of type III1 x of type I or type IV.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 III
1 x of type I or type IV.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
The Berkovich projective line
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Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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), and
2 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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.); or
x ∈ P1(K ) and h is constant on an open neighborhood of x .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞}.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞}.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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,∞}.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 ∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 ∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 ∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 ∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 ∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 ∞.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
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
If U = P1Berk or U is an open Berkovich disk, then every harmonic
function on U is constant.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
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
If U = P1Berk or U is an open Berkovich disk, then every harmonic
function on U is constant.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
For example:
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
For example:
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
For example:
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
A Berkovich open annulus
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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).
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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).
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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).
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
One can give an explicit formula for h1(z), . . . , hm(z) usingCramer’s rule.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
The Poisson formula
Theorem (Poisson formula)
Let V be a simple domain in P1Berk with boundary points
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
The Poisson formula
Theorem (Poisson formula)
Let V be a simple domain in P1Berk with boundary points
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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 .
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
Examples of subharmonic functions
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 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}.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
Examples of subharmonic functions
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 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}.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
Maximum principle for subharmonic functions
Theorem (Maximum Principle)
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
Maximum principle for subharmonic functions
Theorem (Maximum Principle)
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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.
Matthew Baker Lecture 3: Introduction to Berkovich Curves
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