Local geodesics for plurisubharmonic functions Alexander Rashkovskii University of Stavanger, Norway Alexander Rashkovskii (UiS) Local geodesics 1 / 28
Motivations 1. General goal: ’good’ transformations u 0 �→ u 1 of psh functions 2. Global setting: metrics on K¨ ahler manifolds ( X , ω ) ω > 0 K¨ ahler form ahler form ω ′ = ω + dd c ϕ ∈ [ ω ], so ω ′ ↔ ϕ : metrics another K¨ Geodesics on the space of metrics : ϕ t that minimize energy functional � 1 � t ( ω + dd c ϕ t ) n dt ϕ 2 ˙ 0 X (Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕ t is a geodesic ⇔ ( ω + dd c Φ) n +1 = 0 on X × S ( n = dim X , Φ( z , ζ ) = ϕ log | ζ | ( z ), and S is an annulus in C ) Moreover, geodesics ϕ t linearize Mabuchi functional t �→ M ( ϕ t ). A curve ψ t is subgeodesic if the corresponding function Ψ satisfies ( ω + dd c Φ) n +1 ≥ 0. Mabuchi functional is convex on subgeodesics. Alexander Rashkovskii (UiS) Local geodesics 2 / 28
Motivations 1. General goal: ’good’ transformations u 0 �→ u 1 of psh functions 2. Global setting: metrics on K¨ ahler manifolds ( X , ω ) ω > 0 K¨ ahler form ahler form ω ′ = ω + dd c ϕ ∈ [ ω ], so ω ′ ↔ ϕ : metrics another K¨ Geodesics on the space of metrics : ϕ t that minimize energy functional � 1 � t ( ω + dd c ϕ t ) n dt ϕ 2 ˙ 0 X (Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕ t is a geodesic ⇔ ( ω + dd c Φ) n +1 = 0 on X × S ( n = dim X , Φ( z , ζ ) = ϕ log | ζ | ( z ), and S is an annulus in C ) Moreover, geodesics ϕ t linearize Mabuchi functional t �→ M ( ϕ t ). A curve ψ t is subgeodesic if the corresponding function Ψ satisfies ( ω + dd c Φ) n +1 ≥ 0. Mabuchi functional is convex on subgeodesics. Alexander Rashkovskii (UiS) Local geodesics 2 / 28
Motivations 1. General goal: ’good’ transformations u 0 �→ u 1 of psh functions 2. Global setting: metrics on K¨ ahler manifolds ( X , ω ) ω > 0 K¨ ahler form ahler form ω ′ = ω + dd c ϕ ∈ [ ω ], so ω ′ ↔ ϕ : metrics another K¨ Geodesics on the space of metrics : ϕ t that minimize energy functional � 1 � t ( ω + dd c ϕ t ) n dt ϕ 2 ˙ 0 X (Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕ t is a geodesic ⇔ ( ω + dd c Φ) n +1 = 0 on X × S ( n = dim X , Φ( z , ζ ) = ϕ log | ζ | ( z ), and S is an annulus in C ) Moreover, geodesics ϕ t linearize Mabuchi functional t �→ M ( ϕ t ). A curve ψ t is subgeodesic if the corresponding function Ψ satisfies ( ω + dd c Φ) n +1 ≥ 0. Mabuchi functional is convex on subgeodesics. Alexander Rashkovskii (UiS) Local geodesics 2 / 28
Motivations 1. General goal: ’good’ transformations u 0 �→ u 1 of psh functions 2. Global setting: metrics on K¨ ahler manifolds ( X , ω ) ω > 0 K¨ ahler form ahler form ω ′ = ω + dd c ϕ ∈ [ ω ], so ω ′ ↔ ϕ : metrics another K¨ Geodesics on the space of metrics : ϕ t that minimize energy functional � 1 � t ( ω + dd c ϕ t ) n dt ϕ 2 ˙ 0 X (Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕ t is a geodesic ⇔ ( ω + dd c Φ) n +1 = 0 on X × S ( n = dim X , Φ( z , ζ ) = ϕ log | ζ | ( z ), and S is an annulus in C ) Moreover, geodesics ϕ t linearize Mabuchi functional t �→ M ( ϕ t ). A curve ψ t is subgeodesic if the corresponding function Ψ satisfies ( ω + dd c Φ) n +1 ≥ 0. Mabuchi functional is convex on subgeodesics. Alexander Rashkovskii (UiS) Local geodesics 2 / 28
Motivations 1. General goal: ’good’ transformations u 0 �→ u 1 of psh functions 2. Global setting: metrics on K¨ ahler manifolds ( X , ω ) ω > 0 K¨ ahler form ahler form ω ′ = ω + dd c ϕ ∈ [ ω ], so ω ′ ↔ ϕ : metrics another K¨ Geodesics on the space of metrics : ϕ t that minimize energy functional � 1 � t ( ω + dd c ϕ t ) n dt ϕ 2 ˙ 0 X (Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕ t is a geodesic ⇔ ( ω + dd c Φ) n +1 = 0 on X × S ( n = dim X , Φ( z , ζ ) = ϕ log | ζ | ( z ), and S is an annulus in C ) Moreover, geodesics ϕ t linearize Mabuchi functional t �→ M ( ϕ t ). A curve ψ t is subgeodesic if the corresponding function Ψ satisfies ( ω + dd c Φ) n +1 ≥ 0. Mabuchi functional is convex on subgeodesics. Alexander Rashkovskii (UiS) Local geodesics 2 / 28
Motivations 1. General goal: ’good’ transformations u 0 �→ u 1 of psh functions 2. Global setting: metrics on K¨ ahler manifolds ( X , ω ) ω > 0 K¨ ahler form ahler form ω ′ = ω + dd c ϕ ∈ [ ω ], so ω ′ ↔ ϕ : metrics another K¨ Geodesics on the space of metrics : ϕ t that minimize energy functional � 1 � t ( ω + dd c ϕ t ) n dt ϕ 2 ˙ 0 X (Mabuchi 1987, Semmes 1992, Donaldson 1997, Chen 2000...) Characterization: ϕ t is a geodesic ⇔ ( ω + dd c Φ) n +1 = 0 on X × S ( n = dim X , Φ( z , ζ ) = ϕ log | ζ | ( z ), and S is an annulus in C ) Moreover, geodesics ϕ t linearize Mabuchi functional t �→ M ( ϕ t ). A curve ψ t is subgeodesic if the corresponding function Ψ satisfies ( ω + dd c Φ) 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¨ om...) 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¨ om...) 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 → C n ; 2. u = ψ (log | z 1 | , . . . , log | z n | ) for a convex function ψ in S ⊂ R n . Basic properties: 1. u k ∈ PSH( M ) , 1 ≤ k ≤ N ⇒ u = max k u k ∈ PSH( M ); 2. u k ∈ PSH( M ) , u k ց u ⇒ u ∈ PSH( M ); 3. u α ∈ PSH( M ) , u α < C ∀ α ⇒ u = sup ∗ α u α ∈ PSH( M ). Alexander Rashkovskii (UiS) Local geodesics 4 / 28
Energy functional on Cegrell classes M = D ⊂ C n : bounded hyperconvex domain. Cegrell’s class E 0 ( D ): bounded plurisubharmonic functions u in D , � D ( dd c u ) n < ∞ . u | ∂ D = 0 with finite total Monge-Amp` ere mass Energy functional on E 0 : � u ( dd c u ) n . E ( u ) = D Identity: � n � ( dd c u ) k ∧ ( dd c v ) n − k . E ( u ) − E ( v ) = ( u − v ) D k =0 Corollary: If u , v ∈ E 0 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 ⊂ C n : bounded hyperconvex domain. Cegrell’s class E 0 ( D ): bounded plurisubharmonic functions u in D , � D ( dd c u ) n < ∞ . u | ∂ D = 0 with finite total Monge-Amp` ere mass Energy functional on E 0 : � u ( dd c u ) n . E ( u ) = D Identity: � n � ( dd c u ) k ∧ ( dd c v ) n − k . E ( u ) − E ( v ) = ( u − v ) D k =0 Corollary: If u , v ∈ E 0 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 ⊂ C n : bounded hyperconvex domain. Cegrell’s class E 0 ( D ): bounded plurisubharmonic functions u in D , � D ( dd c u ) n < ∞ . u | ∂ D = 0 with finite total Monge-Amp` ere mass Energy functional on E 0 : � u ( dd c u ) n . E ( u ) = D Identity: � n � ( dd c u ) k ∧ ( dd c v ) n − k . E ( u ) − E ( v ) = ( u − v ) D k =0 Corollary: If u , v ∈ E 0 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 ⊂ C n : bounded hyperconvex domain. Cegrell’s class E 0 ( D ): bounded plurisubharmonic functions u in D , � D ( dd c u ) n < ∞ . u | ∂ D = 0 with finite total Monge-Amp` ere mass Energy functional on E 0 : � u ( dd c u ) n . E ( u ) = D Identity: � n � ( dd c u ) k ∧ ( dd c v ) n − k . E ( u ) − E ( v ) = ( u − v ) D k =0 Corollary: If u , v ∈ E 0 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 E 0 S = { 0 < log | ζ | < 1 } ⊂ C , S j = { log | ζ | = j } , log | S | = (0 , 1) Given u 0 , u 1 ∈ E 0 ( D ), denote W ( u 0 , u 1 ) = { u ∈ PSH − ( D × S ) : lim sup u ( · , ζ ) ≤ u j ( · ) , j = 0 , 1 } . ζ → S j Definition. v t is a subgeodesic for u 0 , u 1 if v log | ζ | ∈ W ( u 0 , u 1 ). u ( u , e t ), where The largest subgeodesic, u t , is called geodesic : u t ( z ) = � u = sup { u ∈ W ( u 1 , u 2 ) } ∈ PSH − ( D × S ). � Alexander Rashkovskii (UiS) Local geodesics 6 / 28
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