Local geodesics for plurisubharmonic functions Alexander Rashkovskii - - PowerPoint PPT Presentation

Local geodesics for plurisubharmonic functions

Alexander Rashkovskii

University of Stavanger, Norway

Alexander Rashkovskii (UiS) Local geodesics 1 / 28

Motivations

• 1. General goal: ’good’ transformations u0 → u1 of psh functions
• 2. Global setting: metrics on K¨

ahler manifolds (X, ω) ω > 0 K¨ ahler form another K¨ ahler form ω′ = ω + ddcϕ ∈ [ω], so ω′ ↔ ϕ: metrics Geodesics on the space of metrics: ϕt that minimize energy functional 1

• X

˙ ϕ2

t (ω + ddcϕt)n dt

(Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕt is a geodesic ⇔ (ω + ddcΦ)n+1 = 0 on X × S (n = dim X, Φ(z, ζ) = ϕlog |ζ|(z), and S is an annulus in C) Moreover, geodesics ϕt linearize Mabuchi functional t → M(ϕt). A curve ψt is subgeodesic if the corresponding function Ψ satisfies (ω + ddcΦ)n+1 ≥ 0. Mabuchi functional is convex on subgeodesics.

Alexander Rashkovskii (UiS) Local geodesics 2 / 28

Motivations

• 1. General goal: ’good’ transformations u0 → u1 of psh functions
• 2. Global setting: metrics on K¨

ahler manifolds (X, ω) ω > 0 K¨ ahler form another K¨ ahler form ω′ = ω + ddcϕ ∈ [ω], so ω′ ↔ ϕ: metrics Geodesics on the space of metrics: ϕt that minimize energy functional 1

• X

˙ ϕ2

t (ω + ddcϕt)n dt

(Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕt is a geodesic ⇔ (ω + ddcΦ)n+1 = 0 on X × S (n = dim X, Φ(z, ζ) = ϕlog |ζ|(z), and S is an annulus in C) Moreover, geodesics ϕt linearize Mabuchi functional t → M(ϕt). A curve ψt is subgeodesic if the corresponding function Ψ satisfies (ω + ddcΦ)n+1 ≥ 0. Mabuchi functional is convex on subgeodesics.

Alexander Rashkovskii (UiS) Local geodesics 2 / 28

Motivations

• 1. General goal: ’good’ transformations u0 → u1 of psh functions
• 2. Global setting: metrics on K¨

ahler manifolds (X, ω) ω > 0 K¨ ahler form another K¨ ahler form ω′ = ω + ddcϕ ∈ [ω], so ω′ ↔ ϕ: metrics Geodesics on the space of metrics: ϕt that minimize energy functional 1

• X

˙ ϕ2

t (ω + ddcϕt)n dt

(Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕt is a geodesic ⇔ (ω + ddcΦ)n+1 = 0 on X × S (n = dim X, Φ(z, ζ) = ϕlog |ζ|(z), and S is an annulus in C) Moreover, geodesics ϕt linearize Mabuchi functional t → M(ϕt). A curve ψt is subgeodesic if the corresponding function Ψ satisfies (ω + ddcΦ)n+1 ≥ 0. Mabuchi functional is convex on subgeodesics.

Alexander Rashkovskii (UiS) Local geodesics 2 / 28

Motivations

• 1. General goal: ’good’ transformations u0 → u1 of psh functions
• 2. Global setting: metrics on K¨

ahler manifolds (X, ω) ω > 0 K¨ ahler form another K¨ ahler form ω′ = ω + ddcϕ ∈ [ω], so ω′ ↔ ϕ: metrics Geodesics on the space of metrics: ϕt that minimize energy functional 1

• X

˙ ϕ2

t (ω + ddcϕt)n dt

(Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕt is a geodesic ⇔ (ω + ddcΦ)n+1 = 0 on X × S (n = dim X, Φ(z, ζ) = ϕlog |ζ|(z), and S is an annulus in C) Moreover, geodesics ϕt linearize Mabuchi functional t → M(ϕt). A curve ψt is subgeodesic if the corresponding function Ψ satisfies (ω + ddcΦ)n+1 ≥ 0. Mabuchi functional is convex on subgeodesics.

Alexander Rashkovskii (UiS) Local geodesics 2 / 28

Motivations

• 1. General goal: ’good’ transformations u0 → u1 of psh functions
• 2. Global setting: metrics on K¨

ahler manifolds (X, ω) ω > 0 K¨ ahler form another K¨ ahler form ω′ = ω + ddcϕ ∈ [ω], so ω′ ↔ ϕ: metrics Geodesics on the space of metrics: ϕt that minimize energy functional 1

• X

˙ ϕ2

t (ω + ddcϕt)n dt

(Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕt is a geodesic ⇔ (ω + ddcΦ)n+1 = 0 on X × S (n = dim X, Φ(z, ζ) = ϕlog |ζ|(z), and S is an annulus in C) Moreover, geodesics ϕt linearize Mabuchi functional t → M(ϕt). A curve ψt is subgeodesic if the corresponding function Ψ satisfies (ω + ddcΦ)n+1 ≥ 0. Mabuchi functional is convex on subgeodesics.

Alexander Rashkovskii (UiS) Local geodesics 2 / 28

Motivations

• 1. General goal: ’good’ transformations u0 → u1 of psh functions
• 2. Global setting: metrics on K¨

ahler manifolds (X, ω) ω > 0 K¨ ahler form another K¨ ahler form ω′ = ω + ddcϕ ∈ [ω], so ω′ ↔ ϕ: metrics Geodesics on the space of metrics: ϕt that minimize energy functional 1

• X

˙ ϕ2

t (ω + ddcϕt)n dt

(Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕt is a geodesic ⇔ (ω + ddcΦ)n+1 = 0 on X × S (n = dim X, Φ(z, ζ) = ϕlog |ζ|(z), and S is an annulus in C) Moreover, geodesics ϕt linearize Mabuchi functional t → M(ϕt). A curve ψt is subgeodesic if the corresponding function Ψ satisfies (ω + ddcΦ)n+1 ≥ 0. Mabuchi functional is convex on subgeodesics.

Alexander Rashkovskii (UiS) Local geodesics 2 / 28

Motivations: cont’d

• 3. Further developments: other functionals, singular metrics, ...

( Berman, Berndtsson, Darvas, Guedj, Phong, Tian, Ross, Wytt Nystr¨

• m...)
• 4. Our aim: local counterpart of the theory for functions on open sets.

Especially: applications?

Alexander Rashkovskii (UiS) Local geodesics 3 / 28

Motivations: cont’d

• 3. Further developments: other functionals, singular metrics, ...

( Berman, Berndtsson, Darvas, Guedj, Phong, Tian, Ross, Wytt Nystr¨

• m...)
• 4. Our aim: local counterpart of the theory for functions on open sets.

Especially: applications?

Alexander Rashkovskii (UiS) Local geodesics 3 / 28

PSH

PSH(M): functions u : M → [−∞, ∞) plurisubharmonic on a complex manifold M, i.e.: (i) upper semicontinuous on M (ii) u ◦ φ subharmonic in the unit disk D for every holomorphic mapping φ : D → M. Basic examples:

• 1. u = c log |f | for any c > 0 and any holomorphic mapping f : M → Cn;
• 2. u = ψ(log |z1|, . . . , log |zn|) for a convex function ψ in S ⊂ Rn.

Basic properties:

• 1. uk ∈ PSH(M), 1 ≤ k ≤ N ⇒ u = maxk uk ∈ PSH(M);
• 2. uk ∈ PSH(M), uk ց u ⇒ u ∈ PSH(M);
• 3. uα ∈ PSH(M), uα < C ∀α ⇒ u = sup∗

α uα ∈ PSH(M).

Alexander Rashkovskii (UiS) Local geodesics 4 / 28

Energy functional on Cegrell classes

M = D ⊂ Cn: bounded hyperconvex domain. Cegrell’s class E0(D): bounded plurisubharmonic functions u in D, u|∂D = 0 with finite total Monge-Amp` ere mass

• D(ddcu)n < ∞.

Energy functional on E0: E(u) =

• D

u(ddcu)n. Identity: E(u) − E(v) =

• D

(u − v)

n

• k=0

(ddcu)k ∧ (ddcv)n−k. Corollary: If u, v ∈ E0 satisfy u ≤ v, then E(u) ≤ E(v). If, in addition, E(u) = E(v), then u = v on D.

Alexander Rashkovskii (UiS) Local geodesics 5 / 28

Energy functional on Cegrell classes

M = D ⊂ Cn: bounded hyperconvex domain. Cegrell’s class E0(D): bounded plurisubharmonic functions u in D, u|∂D = 0 with finite total Monge-Amp` ere mass

• D(ddcu)n < ∞.

Energy functional on E0: E(u) =

• D

u(ddcu)n. Identity: E(u) − E(v) =

• D

(u − v)

n

• k=0

(ddcu)k ∧ (ddcv)n−k. Corollary: If u, v ∈ E0 satisfy u ≤ v, then E(u) ≤ E(v). If, in addition, E(u) = E(v), then u = v on D.

Alexander Rashkovskii (UiS) Local geodesics 5 / 28

Energy functional on Cegrell classes

M = D ⊂ Cn: bounded hyperconvex domain. Cegrell’s class E0(D): bounded plurisubharmonic functions u in D, u|∂D = 0 with finite total Monge-Amp` ere mass

• D(ddcu)n < ∞.

Energy functional on E0: E(u) =

• D

u(ddcu)n. Identity: E(u) − E(v) =

• D

(u − v)

n

• k=0

(ddcu)k ∧ (ddcv)n−k. Corollary: If u, v ∈ E0 satisfy u ≤ v, then E(u) ≤ E(v). If, in addition, E(u) = E(v), then u = v on D.

Alexander Rashkovskii (UiS) Local geodesics 5 / 28

Energy functional on Cegrell classes

M = D ⊂ Cn: bounded hyperconvex domain. Cegrell’s class E0(D): bounded plurisubharmonic functions u in D, u|∂D = 0 with finite total Monge-Amp` ere mass

• D(ddcu)n < ∞.

Energy functional on E0: E(u) =

• D

u(ddcu)n. Identity: E(u) − E(v) =

• D

(u − v)

n

• k=0

(ddcu)k ∧ (ddcv)n−k. Corollary: If u, v ∈ E0 satisfy u ≤ v, then E(u) ≤ E(v). If, in addition, E(u) = E(v), then u = v on D.

Alexander Rashkovskii (UiS) Local geodesics 5 / 28

Geodesics for E0

S = {0 < log |ζ| < 1} ⊂ C, Sj = {log |ζ| = j}, log |S| = (0, 1) Given u0, u1 ∈ E0(D), denote W (u0, u1) = {u ∈ PSH−(D × S) : lim sup

ζ→Sj

u(·, ζ) ≤ uj(·), j = 0, 1}.

• Definition. vt is a subgeodesic for u0, u1 if vlog |ζ| ∈ W (u0, u1).

The largest subgeodesic, ut, is called geodesic: ut(z) = u(u, et), where

• u = sup{u ∈ W (u1, u2)} ∈ PSH−(D × S).

Alexander Rashkovskii (UiS) Local geodesics 6 / 28

Geodesics for E0

S = {0 < log |ζ| < 1} ⊂ C, Sj = {log |ζ| = j}, log |S| = (0, 1) Given u0, u1 ∈ E0(D), denote W (u0, u1) = {u ∈ PSH−(D × S) : lim sup

ζ→Sj

u(·, ζ) ≤ uj(·), j = 0, 1}.

• Definition. vt is a subgeodesic for u0, u1 if vlog |ζ| ∈ W (u0, u1).

The largest subgeodesic, ut, is called geodesic: ut(z) = u(u, et), where

• u = sup{u ∈ W (u1, u2)} ∈ PSH−(D × S).

Alexander Rashkovskii (UiS) Local geodesics 6 / 28

Geodesics for E0

S = {0 < log |ζ| < 1} ⊂ C, Sj = {log |ζ| = j}, log |S| = (0, 1) Given u0, u1 ∈ E0(D), denote W (u0, u1) = {u ∈ PSH−(D × S) : lim sup

ζ→Sj

u(·, ζ) ≤ uj(·), j = 0, 1}.

• Definition. vt is a subgeodesic for u0, u1 if vlog |ζ| ∈ W (u0, u1).

The largest subgeodesic, ut, is called geodesic: ut(z) = u(u, et), where

• u = sup{u ∈ W (u1, u2)} ∈ PSH−(D × S).

Alexander Rashkovskii (UiS) Local geodesics 6 / 28

Properties of geodesics:

• 1. ut ∈ E0(D)
• 2. ut ≤ (1 − t)u0 + tu1
• 3. ut ≥ max{u0 − M1 t, u1 − M0 (1 − t)}, where Mj = uj∞.
• 4. ut ⇒ uj as t → j ∈ {0, 1}

Theorem

1 The energy functional u → E(u) =

• D u(ddcu)n is concave on E0.

2 For any subgeodesic vt, E(vt) is a convex function of t. 3 t → E(ut) is linear iff ut is a geodesic. Alexander Rashkovskii (UiS) Local geodesics 7 / 28

Properties of geodesics:

• 1. ut ∈ E0(D)
• 2. ut ≤ (1 − t)u0 + tu1
• 3. ut ≥ max{u0 − M1 t, u1 − M0 (1 − t)}, where Mj = uj∞.
• 4. ut ⇒ uj as t → j ∈ {0, 1}

Theorem

1 The energy functional u → E(u) =

• D u(ddcu)n is concave on E0.

2 For any subgeodesic vt, E(vt) is a convex function of t. 3 t → E(ut) is linear iff ut is a geodesic. Alexander Rashkovskii (UiS) Local geodesics 7 / 28

Sketch of the proof

Denote v(z, ζ) = vlog |ζ|(z). Convexity of E(vt) is equivalent to subharmonicity of the function

• E = E(

v) =

• D
• v(dzdc

z

v)n, and the linearity of E corresponds to the harmonicity of E. All justifications hidden, dc

ζ

E = (n + 1)

• D

dc

ζ

v ∧ (dzdc

z

v)n and 1 n + 1 dζdc

ζ

E =

• D

dζdc

ζ

v ∧ (dzdc

z

v)n − n

• D

dzdc

ζ

v ∧ dc

z dζ

v ∧ (dzdc

z

v)n−1 = 1 n + 1

• D

(ddc v)n+1.

Alexander Rashkovskii (UiS) Local geodesics 8 / 28

Sketch of the proof

Denote v(z, ζ) = vlog |ζ|(z). Convexity of E(vt) is equivalent to subharmonicity of the function

• E = E(

v) =

• D
• v(dzdc

z

v)n, and the linearity of E corresponds to the harmonicity of E. All justifications hidden, dc

ζ

E = (n + 1)

• D

dc

ζ

v ∧ (dzdc

z

v)n and 1 n + 1 dζdc

ζ

E =

• D

dζdc

ζ

v ∧ (dzdc

z

v)n − n

• D

dzdc

ζ

v ∧ dc

z dζ

v ∧ (dzdc

z

v)n−1 = 1 n + 1

• D

(ddc v)n+1.

Alexander Rashkovskii (UiS) Local geodesics 8 / 28

Sketch of the proof

Denote v(z, ζ) = vlog |ζ|(z). Convexity of E(vt) is equivalent to subharmonicity of the function

• E = E(

v) =

• D
• v(dzdc

z

v)n, and the linearity of E corresponds to the harmonicity of E. All justifications hidden, dc

ζ

E = (n + 1)

• D

dc

ζ

v ∧ (dzdc

z

v)n and 1 n + 1 dζdc

ζ

E =

• D

dζdc

ζ

v ∧ (dzdc

z

v)n − n

• D

dzdc

ζ

v ∧ dc

z dζ

v ∧ (dzdc

z

v)n−1 = 1 n + 1

• D

(ddc v)n+1.

Alexander Rashkovskii (UiS) Local geodesics 8 / 28

Uniqueness theorem

Corollary

If u0, u1 ∈ E0(D) satisfy

• D

u0(ddcu0)k ∧ (ddcu1)n−k = E(u1), k = 0, . . . , n, then u0 = u1 in D.

Alexander Rashkovskii (UiS) Local geodesics 9 / 28

Example: relative extremal functions

Let K ⋐ D Relative extremal function ωK = sup∗{u ∈ PSH−(D) : u|K ≤ −1}. We have: ωK ∈ E0(D), E(ωK) = −

• D(ddcωK)n = −Cap (K).

Let uj = uKj, j = 0, 1. Then E(ut) = (t − 1) Cap (K0) − t Cap (K1). Question: What is the geodesic ut? Is ut = ωKt for some Kt? Answer: No.

Alexander Rashkovskii (UiS) Local geodesics 10 / 28

Example: relative extremal functions

Let K ⋐ D Relative extremal function ωK = sup∗{u ∈ PSH−(D) : u|K ≤ −1}. We have: ωK ∈ E0(D), E(ωK) = −

• D(ddcωK)n = −Cap (K).

Let uj = uKj, j = 0, 1. Then E(ut) = (t − 1) Cap (K0) − t Cap (K1). Question: What is the geodesic ut? Is ut = ωKt for some Kt? Answer: No.

Alexander Rashkovskii (UiS) Local geodesics 10 / 28

Example: relative extremal functions

Let K ⋐ D Relative extremal function ωK = sup∗{u ∈ PSH−(D) : u|K ≤ −1}. We have: ωK ∈ E0(D), E(ωK) = −

• D(ddcωK)n = −Cap (K).

Let uj = uKj, j = 0, 1. Then E(ut) = (t − 1) Cap (K0) − t Cap (K1). Question: What is the geodesic ut? Is ut = ωKt for some Kt? Answer: No.

Alexander Rashkovskii (UiS) Local geodesics 10 / 28

Example: relative extremal functions

Let K ⋐ D Relative extremal function ωK = sup∗{u ∈ PSH−(D) : u|K ≤ −1}. We have: ωK ∈ E0(D), E(ωK) = −

• D(ddcωK)n = −Cap (K).

Let uj = uKj, j = 0, 1. Then E(ut) = (t − 1) Cap (K0) − t Cap (K1). Question: What is the geodesic ut? Is ut = ωKt for some Kt? Answer: No.

Alexander Rashkovskii (UiS) Local geodesics 10 / 28

Example: relative extremal functions

Let K ⋐ D Relative extremal function ωK = sup∗{u ∈ PSH−(D) : u|K ≤ −1}. We have: ωK ∈ E0(D), E(ωK) = −

• D(ddcωK)n = −Cap (K).

Let uj = uKj, j = 0, 1. Then E(ut) = (t − 1) Cap (K0) − t Cap (K1). Question: What is the geodesic ut? Is ut = ωKt for some Kt? Answer: No.

Alexander Rashkovskii (UiS) Local geodesics 10 / 28

REF in toric setting

Let D be a bounded complete logarithmically convex Reinhardt domain: 1) y ∈ D provided z ∈ D and |yl| ≤ |zl| for all l, 2) log D = {s ∈ Rn

− : es1, . . . , esn ∈ D} is a convex subset of Rn.

Let Kj also be compact Reinhardt subsets of D. Then ωKj are toric (multi-circled) and so, the function ut(z) is convex in (log |z1|, . . . , log |zn|, t). For 0 < t < 1, denote Kt = K 1−t K t

1 = {z : |zl| = |ηl|1−t|ξl|t, 1 ≤ l ≤ n, η ∈ K0, ξ ∈ K1}.

Alexander Rashkovskii (UiS) Local geodesics 11 / 28

REF in toric setting

Let D be a bounded complete logarithmically convex Reinhardt domain: 1) y ∈ D provided z ∈ D and |yl| ≤ |zl| for all l, 2) log D = {s ∈ Rn

− : es1, . . . , esn ∈ D} is a convex subset of Rn.

Let Kj also be compact Reinhardt subsets of D. Then ωKj are toric (multi-circled) and so, the function ut(z) is convex in (log |z1|, . . . , log |zn|, t). For 0 < t < 1, denote Kt = K 1−t K t

1 = {z : |zl| = |ηl|1−t|ξl|t, 1 ≤ l ≤ n, η ∈ K0, ξ ∈ K1}.

Alexander Rashkovskii (UiS) Local geodesics 11 / 28

REF in toric setting

Let D be a bounded complete logarithmically convex Reinhardt domain: 1) y ∈ D provided z ∈ D and |yl| ≤ |zl| for all l, 2) log D = {s ∈ Rn

− : es1, . . . , esn ∈ D} is a convex subset of Rn.

Let Kj also be compact Reinhardt subsets of D. Then ωKj are toric (multi-circled) and so, the function ut(z) is convex in (log |z1|, . . . , log |zn|, t). For 0 < t < 1, denote Kt = K 1−t K t

1 = {z : |zl| = |ηl|1−t|ξl|t, 1 ≤ l ≤ n, η ∈ K0, ξ ∈ K1}.

Alexander Rashkovskii (UiS) Local geodesics 11 / 28

In other words, log Kt = (1 − t) log K0 + t log K1.

Alexander Rashkovskii (UiS) Local geodesics 12 / 28

Brunn-Minkowski inequality

Recall: volumes | · | of convex combinations of two bodies Pj ⊂ Rn satisfy |(1 − t)P0 + t P1| ≥ |P0|1−t |P1|t, the Brunn-Minkowski inequality (in multiplicative form). In our case, the sets log Kj typically are of infinite volume. Instead of the volumes, we have a reversed Brunn-Minkowski inequality for the capacities

• f Kt (multiplicative combinations of Kj), in additive form:

Theorem

In the toric case, the capacities of the sets Kt satisfy Cap (Kt) ≤ (1 − t) Cap (K0) + t Cap (K1).

Alexander Rashkovskii (UiS) Local geodesics 13 / 28

Brunn-Minkowski inequality

Recall: volumes | · | of convex combinations of two bodies Pj ⊂ Rn satisfy |(1 − t)P0 + t P1| ≥ |P0|1−t |P1|t, the Brunn-Minkowski inequality (in multiplicative form). In our case, the sets log Kj typically are of infinite volume. Instead of the volumes, we have a reversed Brunn-Minkowski inequality for the capacities

• f Kt (multiplicative combinations of Kj), in additive form:

Theorem

In the toric case, the capacities of the sets Kt satisfy Cap (Kt) ≤ (1 − t) Cap (K0) + t Cap (K1).

Alexander Rashkovskii (UiS) Local geodesics 13 / 28

Example: geodesic of REFs is not REF

Let n = 1, D = D, K0 = {z : |z| ≤ e−1}, Kj = {z : |z| ≤ e−2}. Then Kt = {z : |z| ≤ e−1−t}. The function ωKt(z) = max log |z| 1 + t , −1

• is not convex in (log |z|, t), so ωKt is not geodesic.

E(ωKt) = −Cap (Kt) = −(1 + t)−1 is far from being linear. Actually, ut(z) = max

• log |z|, log |z| + t − 1

2 , −1

• is not a relative extremal function at all.

Alexander Rashkovskii (UiS) Local geodesics 14 / 28

Example: geodesic of REFs is not REF

Let n = 1, D = D, K0 = {z : |z| ≤ e−1}, Kj = {z : |z| ≤ e−2}. Then Kt = {z : |z| ≤ e−1−t}. The function ωKt(z) = max log |z| 1 + t , −1

• is not convex in (log |z|, t), so ωKt is not geodesic.

E(ωKt) = −Cap (Kt) = −(1 + t)−1 is far from being linear. Actually, ut(z) = max

• log |z|, log |z| + t − 1

2 , −1

• is not a relative extremal function at all.

Alexander Rashkovskii (UiS) Local geodesics 14 / 28

Example: geodesic of REFs is not REF

Let n = 1, D = D, K0 = {z : |z| ≤ e−1}, Kj = {z : |z| ≤ e−2}. Then Kt = {z : |z| ≤ e−1−t}. The function ωKt(z) = max log |z| 1 + t , −1

• is not convex in (log |z|, t), so ωKt is not geodesic.

E(ωKt) = −Cap (Kt) = −(1 + t)−1 is far from being linear. Actually, ut(z) = max

• log |z|, log |z| + t − 1

2 , −1

• is not a relative extremal function at all.

Alexander Rashkovskii (UiS) Local geodesics 14 / 28

Example: geodesic of REFs is not REF, cont’d

Kt = {z : |z| ≤ e−1−t}

Alexander Rashkovskii (UiS) Local geodesics 15 / 28

Example: geodesic of REFs is not REF, cont’d

Kt = {z : |z| ≤ e−1−t}, K ∗

t = {z : |z| ≤ et−1}

Alexander Rashkovskii (UiS) Local geodesics 16 / 28

Example: geodesic of REFs is not REF, cont’d

Alexander Rashkovskii (UiS) Local geodesics 17 / 28

Example: geodesic of REFs is not REF, cont’d

Alexander Rashkovskii (UiS) Local geodesics 18 / 28

Example: geodesic of REFs is not REF, cont’d

In this example, the geodesic ut still pertains some ’extremality’ property: it is the extremal function for the multi-plate condenser (due to Poletsky) with the plates Kt ⊂ K ∗

t ⊂ K = D.

Namely, ut does not exceed preassigned constants on the plates and is a maximal psh function between the plates. It would be nice to know if anything similar holds in the general case of geodesics of relative extremal functions.

Alexander Rashkovskii (UiS) Local geodesics 19 / 28

Example: geodesic of REFs is not REF, cont’d

In this example, the geodesic ut still pertains some ’extremality’ property: it is the extremal function for the multi-plate condenser (due to Poletsky) with the plates Kt ⊂ K ∗

t ⊂ K = D.

Namely, ut does not exceed preassigned constants on the plates and is a maximal psh function between the plates. It would be nice to know if anything similar holds in the general case of geodesics of relative extremal functions.

Alexander Rashkovskii (UiS) Local geodesics 19 / 28

Singular case

What happens if uj are not bounded and can have singularities? The construction can be extended to other classes of psh functions. If we still want to use the energy functional, we have to stick with Cegrell’s energy classes - and then the whole picture (existence, linearity, uniqueness property) remains nearly the same. The only new feature is that the end points uj of the geodesic are attained in a weaker sense (convergence in capacity). In larger classes, strong singularity can occur. The main message here is that essentially different singularities cannot be connected with (sub)geodesics.

Alexander Rashkovskii (UiS) Local geodesics 20 / 28

Singular case

What happens if uj are not bounded and can have singularities? The construction can be extended to other classes of psh functions. If we still want to use the energy functional, we have to stick with Cegrell’s energy classes - and then the whole picture (existence, linearity, uniqueness property) remains nearly the same. The only new feature is that the end points uj of the geodesic are attained in a weaker sense (convergence in capacity). In larger classes, strong singularity can occur. The main message here is that essentially different singularities cannot be connected with (sub)geodesics.

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F1

D ⊂ Cn: bounded hyperconvex domain. Cegrell’s class F1(D): u ∈ PSH(D) that are limits of decreasing sequences uN ∈ E0(D) such that sup

N

• D

|uN|(ddcuN)n < ∞ and sup

N

• D

(ddcuN)n < ∞. For u ∈ F1(D), (ddcu)n = lim

N→∞(ddcuN)n,

u(ddcu)n = lim

N→∞ uN(ddcuN)n

are independent of the choice of the approximating sequence uN, and E(uN) → E(u).

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F1

D ⊂ Cn: bounded hyperconvex domain. Cegrell’s class F1(D): u ∈ PSH(D) that are limits of decreasing sequences uN ∈ E0(D) such that sup

N

• D

|uN|(ddcuN)n < ∞ and sup

N

• D

(ddcuN)n < ∞. For u ∈ F1(D), (ddcu)n = lim

N→∞(ddcuN)n,

u(ddcu)n = lim

N→∞ uN(ddcuN)n

are independent of the choice of the approximating sequence uN, and E(uN) → E(u).

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F1 (cont’d)

Like for E0, we still have the identity E(u) − E(v) =

• D

(u − v)

n

• k=0

(ddcu)k ∧ (ddcv)n−k for u, v ∈ F1(D) and the properties u ≤ v ⇔ E(u) ≤ E(v), and {u ≤ v} & {E(u) = E(v)} ⇔ u = v.

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Geodesics on F1

And the main result is valid on uj ∈ F1(D) as well:

Theorem

For any pair u0, u1 ∈ F1(D) there exists a geodesic ut ⊂ F1(D), 0 < t < 1, such that ut converge in capacity to uj as t approaches j = 0 and j = 1. The energy functional v → E(v) is concave on F1(D), while the function t → E(ut) is linear on geodesics ut and convex on subgeodesics vt ∈ F1(D). The uniqueness result

• D

u0(ddcu0)k ∧ (ddcu1)n−k = E(u1) ∀k ⇒ u0 = u1 remains true for u0, u1 ∈ F1(D).

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Geodesics on F1

And the main result is valid on uj ∈ F1(D) as well:

Theorem

For any pair u0, u1 ∈ F1(D) there exists a geodesic ut ⊂ F1(D), 0 < t < 1, such that ut converge in capacity to uj as t approaches j = 0 and j = 1. The energy functional v → E(v) is concave on F1(D), while the function t → E(ut) is linear on geodesics ut and convex on subgeodesics vt ∈ F1(D). The uniqueness result

• D

u0(ddcu0)k ∧ (ddcu1)n−k = E(u1) ∀k ⇒ u0 = u1 remains true for u0, u1 ∈ F1(D).

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Geodesics on PSH−

Any u ∈ PSH−(D) is the limit of a decreasing sequence uN ∈ E0(D). Let uj ∈ F1(D), j = 0, 1, and let uj,N ∈ E0(D) decrease to uj as N → ∞. Then their geodesics ut,N ∈ E0(D) decrease to some functions vt such that vlog |ζ|(z) ∈ PSH−(D × S). Question: How are vt related to uj? Easy to see: lim supt→j vt ≤ uj. What about equality?

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Geodesics on PSH−

Any u ∈ PSH−(D) is the limit of a decreasing sequence uN ∈ E0(D). Let uj ∈ F1(D), j = 0, 1, and let uj,N ∈ E0(D) decrease to uj as N → ∞. Then their geodesics ut,N ∈ E0(D) decrease to some functions vt such that vlog |ζ|(z) ∈ PSH−(D × S). Question: How are vt related to uj? Easy to see: lim supt→j vt ≤ uj. What about equality?

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Geodesics on PSH−

Any u ∈ PSH−(D) is the limit of a decreasing sequence uN ∈ E0(D). Let uj ∈ F1(D), j = 0, 1, and let uj,N ∈ E0(D) decrease to uj as N → ∞. Then their geodesics ut,N ∈ E0(D) decrease to some functions vt such that vlog |ζ|(z) ∈ PSH−(D × S). Question: How are vt related to uj? Easy to see: lim supt→j vt ≤ uj. What about equality?

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Geodesics on PSH−

Any u ∈ PSH−(D) is the limit of a decreasing sequence uN ∈ E0(D). Let uj ∈ F1(D), j = 0, 1, and let uj,N ∈ E0(D) decrease to uj as N → ∞. Then their geodesics ut,N ∈ E0(D) decrease to some functions vt such that vlog |ζ|(z) ∈ PSH−(D × S). Question: How are vt related to uj? Easy to see: lim supt→j vt ≤ uj. What about equality?

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Geodesics on PSH−, cont’d

• Example. D = D, u0 = 0, u1 = log |z|.

For any N > 0, the function uN,t = max{u1, −Nt} is the geodesic between u0 and uN,1 = max{u1, −N}. Therefore, vt = u1 = log |z| for any t. More generally: D ⊂ Cn, uj are the multi-pole Green functions of D with weights mj,k ≥ 0 at ak of a finite set A ⊂ D. Then vt is the multi-pole Green function of A with weights Mk = maxj mj,k at ak ∈ A. So: arbitrary pairs (u0, u1) need not be endpoints of (sub)geodesics. Moreover:

• Theorem. Let u0, u1 ∈ PSH−(D) have zero boundary value on ∂D and

satisfy (ddcuj)n = 0 on D \ K for a pluripolar set K ⋐ D. Then there exists a (sub)geodesic connecting u0 and u1 if and only if u0 = u1.

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Geodesics on PSH−, cont’d

• Example. D = D, u0 = 0, u1 = log |z|.

For any N > 0, the function uN,t = max{u1, −Nt} is the geodesic between u0 and uN,1 = max{u1, −N}. Therefore, vt = u1 = log |z| for any t. More generally: D ⊂ Cn, uj are the multi-pole Green functions of D with weights mj,k ≥ 0 at ak of a finite set A ⊂ D. Then vt is the multi-pole Green function of A with weights Mk = maxj mj,k at ak ∈ A. So: arbitrary pairs (u0, u1) need not be endpoints of (sub)geodesics. Moreover:

• Theorem. Let u0, u1 ∈ PSH−(D) have zero boundary value on ∂D and

satisfy (ddcuj)n = 0 on D \ K for a pluripolar set K ⋐ D. Then there exists a (sub)geodesic connecting u0 and u1 if and only if u0 = u1.

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Geodesics on PSH−, cont’d

• Example. D = D, u0 = 0, u1 = log |z|.

For any N > 0, the function uN,t = max{u1, −Nt} is the geodesic between u0 and uN,1 = max{u1, −N}. Therefore, vt = u1 = log |z| for any t. More generally: D ⊂ Cn, uj are the multi-pole Green functions of D with weights mj,k ≥ 0 at ak of a finite set A ⊂ D. Then vt is the multi-pole Green function of A with weights Mk = maxj mj,k at ak ∈ A. So: arbitrary pairs (u0, u1) need not be endpoints of (sub)geodesics. Moreover:

• Theorem. Let u0, u1 ∈ PSH−(D) have zero boundary value on ∂D and

satisfy (ddcuj)n = 0 on D \ K for a pluripolar set K ⋐ D. Then there exists a (sub)geodesic connecting u0 and u1 if and only if u0 = u1.

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Geodesics on PSH−, cont’d

• Example. D = D, u0 = 0, u1 = log |z|.

For any N > 0, the function uN,t = max{u1, −Nt} is the geodesic between u0 and uN,1 = max{u1, −N}. Therefore, vt = u1 = log |z| for any t. More generally: D ⊂ Cn, uj are the multi-pole Green functions of D with weights mj,k ≥ 0 at ak of a finite set A ⊂ D. Then vt is the multi-pole Green function of A with weights Mk = maxj mj,k at ak ∈ A. So: arbitrary pairs (u0, u1) need not be endpoints of (sub)geodesics. Moreover:

• Theorem. Let u0, u1 ∈ PSH−(D) have zero boundary value on ∂D and

satisfy (ddcuj)n = 0 on D \ K for a pluripolar set K ⋐ D. Then there exists a (sub)geodesic connecting u0 and u1 if and only if u0 = u1.

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Relations to the K¨ ahler case

Let (X, ω) be a compact K¨ ahler manifold. An upper semicontinuous function ϕ on X is called ω-plurisubharmonic if ω + ddcϕ ≥ 0. Cegrell’s classes were generalized to such functions by Guedj and Zeriahi. A corresponding class E1(X, ω) was introduced, and it has turned to be a natural frame for studying the Mabuchi functional (Berman, Boucksom, Guedj; Zeriahi). Some of problems studied recently Darvas with co-authors in the K¨ ahler setting are close to those treated here. For proving convergence in capacity, we have borrowed the envelope technique due to Ross and Witt Nystr¨

• m.

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Relations to the K¨ ahler case

Let (X, ω) be a compact K¨ ahler manifold. An upper semicontinuous function ϕ on X is called ω-plurisubharmonic if ω + ddcϕ ≥ 0. Cegrell’s classes were generalized to such functions by Guedj and Zeriahi. A corresponding class E1(X, ω) was introduced, and it has turned to be a natural frame for studying the Mabuchi functional (Berman, Boucksom, Guedj; Zeriahi). Some of problems studied recently Darvas with co-authors in the K¨ ahler setting are close to those treated here. For proving convergence in capacity, we have borrowed the envelope technique due to Ross and Witt Nystr¨

• m.

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Literature

[1] R. Berman and S. Boucksom, Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math. 181 (2010), no. 2, 337–394. [2] R. Berman, S. Boucksom, V. Guedj, A. Zeriahi, A variational approach to complex Monge-Amp` ere equations, Publ. Math. Inst. Hautes ´ Etudes

• Sci. 117 (2013), 179–245.

[3] R. Berman, T. Darvas, Chinh H. Lu, Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow, arXiv:1510.01260. [4] B. Berndtsson, A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in K¨ ahler geometry, Invent. Math. 200 (2015), no. 1, 149–200. [5] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), no. 2, 187–217. [6] X.X. Chen, The space of K¨ ahler metrics, J. Diff. Geom. 56 (2000), no. 2, 189–234. [7] T. Darvas, The Mabuchi Completion of the Space of K¨ ahler Potentials, arXiv:1401.7318. [8] T. Darvas, The Mabuchi Geometry of Finite Energy Classes, arXiv:1409.2072.

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[9] T. Darvas and Y. Rubinstein, Kiselman’s principle, the Dirichlet problem for the Monge-Amp` ere equation, and rooftop obstacle problems, arXiv:1405.6548. [10] S.K. Donaldson, Symmetric spaces, K¨ ahler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, 13–33, Amer.

• Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999.

[11] Complex Monge-Amp` ere equations and geodesics in the space of K¨ ahler

• metrics. Edited by V. Guedj. Lecture Notes in Math., 2038, Springer, 2012.

[12] V. Guedj, The metric completion of the Riemannian space of K¨ ahler metrics, arXiv:1401.7857. [13] P. Guan, The extremal functions associated to intrinsic metrics, Ann.

• Math. (2) 156 (2002), 197–211.

[14] T. Mabuchi, Some symplectic geometry on compact K¨ ahler manifolds. I, Osaka J. Math. 24 (1987), no. 2, 227–252. [15] E. Poletsky, Approximation of plurisubharmonic functions by multipole Green functions, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1579–1591. [16] J. Ross and D. Witt Nystr¨

• m, Analytic test configurations and

geodesic rays, J. Symplectic Geom. 12 (2014), no. 1, 125–169. [17] S. Semmes, Complex Monge-Amp` ere and symplectic manifolds, Amer. J.

• Math. 114:3, (1992), 495–550.

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