Plurisubharmonic interpolation and plurisubharmonic geodesics

We give a short survey on plurisubharmonic interpolation, with focus on possibility of connecting two given plurisubharmonic functions by plurisubharmonic geodesic.

Its plurisubharmonic version was considered in [23] as follows.Given two plurisubharmonic functions u 0 , u 1 in a bounded domain D of C n , find a plurisubharmonic function u in D × A ⊂ C n+1 with A = {ζ : 0 < log |ζ| < 1} ⊂ C, whose boundary values on A j = {log |ζ| = j} coincide with u j (j = 0, 1) (the annulus A being used instead of the strip S just to stick to the standard setting of the Dirichlet problem for the complex Monge-Ampère operator in bounded domains.) More precisely, denote W (u 0 , u 1 ) = {v ∈ PSH(D × A) : lim sup When both u 0 and u 1 are bounded above, this set is not empty (it contains u 0 + u 1 − M for a constant M big enough).Note also that for any v ∈ W (u 0 , u 1 ), the function sup{v(z, ξ) : |ξ| = |ζ|} belongs to W (u 0 , u 1 ) as well.
This shows that the plurisubharmonic interpolation problem is equivalent to finding a (sub)geodesic passing through the two given plurisubharmonic functions, which we will call here the geodesic connectivity problem.
We would like especially to refer to [62] and [25] where the connectivity problem for quasi-plurisubharmonic functions on compact Kähler manifolds was for the first time treated in terms of rooftop envelopes; the approach was developed then in [28]- [32], [49] and other recent papers.A nice overview of this activity can be found in [34].
For the local setting of plurisubharmonic functions on bounded domains, the geodesics were considered in the (unpublished) preprint [9] and, independently, in [57] and [1], and then in [58], [42], [23], [59], [60].Comparing with the compact manifold case, two main difficulties arise.First, a plurisubharmonic function can have its singularities on the boundary of the domain, and theory of boundary behavior of such plurisubharmonic functions is at the moment underdeveloped.Another issue is the lack of control over the total Monge-Ampère mass of plurisubharmonic functions and of monotonicity for its non-pluripolar part, the central tools used in the compact case.
In this paper, we survey the local theory results; also, we present new results on solvability of the connectivity problem in the Cegrell class F.
We start with the simplest case of interpolation of bounded plurisubharmonic functions and functions from the Cegrell class E 0 (Sections 1, 2) and applications to set interpolation by means of relative extremal functions (Sections 3-5).In Section 6, we extend these results to the Cegrell class F 1 .The main tool for interpolation of unbounded functions is the rooftop technique, which we present in Section 7. A general setting of the connectivity problem is given in Section 8, and we introduce one of the main objects of the theory, residual plurisubharmonic functions, in Section 9. We formulate a fundamental Rooftop Equality conjecture and confirm it for functions from the class F in Section 10, and we apply it to solving the connectivity problem for certain cases in Section 11.Finally, in Section 12, we mention a few other directions in the field of plurisubharmonic interpolation and geodesics.
The set of all plurisubharmonic (psh for short) functions in a domain Ω will be denoted by PSH(Ω), and its subset of negative functions is PSH − (Ω).For basics on psh functions, we refer to [43], [39].A nice (and comprehensive) presentation of the theory of Cegrell classes can be found in [24].
2 Bounded psh functions and class E 0 Let D be a bounded domain in C n and let u 0 , u 1 ∈ PSH(D) ∩ L ∞ (D).We consider the class W (u 0 , u 1 ) given by (1.1) and define its upper envelope u and geodesic u t as described in Introduction.We define the subgeodesic V t := max{u 0 − M t, u 1 − M (1 − t)}, where M = u 0 − u 1 ∞ , and get (the second inequality being a consequence of convexity of u t in t), which implies that u t → u j , uniformly on D, as t → j ∈ {0, 1}.This means that, in the bounded case, the geodesic always exists and attains the boundary values in a very nice sense.When u j = 0 on ∂D, one has control on the regularity of the upper envelope u of W (u 0 , u 1 ) and, therefore, on u t : Subgeodesics and geodesics of psh functions have some special properties when the functions belong to a specific class -namely, to the Cegrell class F 1 .Here we start however with another Cegrell class, E 0 .
Assume D is a bounded hyperconvex domain, i.e., there exists a negative psh function ρ exhausting D: D c = {z : ρ(z) < c} ⋐ D for all c < 0 and ∪ c<0 D c = D.The Cegrell class E 0 (D) is formed by all bounded psh functions u in D that have zero boundary value on ∂D (u(z) → 0 as z → ∂D) and finite total Monge-Ampère mass: D (dd c u) n < ∞.
Note that if u 0 , u 1 ∈ E 0 (D), then u t ∈ E 0 (D) for any t as well.
Consider the energy functional By integration by parts, The energy functional has the following remarkable properties.
This works also for larger classes of psh functions, however E 0 suffices for an application to plurisubharmonic interpolation of certain measures and sets.

Relative extremal functions
Let K ⋐ D. Recall that the relative extremal function of K w.r.t.D is (here sup * is the upper semicontinuous regularization of sup).This is a function from E 0 (D), maximal outside K: (dd c ω K ) n = 0 on D \ K.The relative capacity of K (w.r.t.D) can be defined as Let u t be the geodesic between u j = ω K j , j = 0, 1.Consider the sets then One might expect that the geodesic interpolations u t of ω K j are ω Kt , which would replace the inequality in this theorem by equality.It turns out to be false, even in simplest examples (see also Theorem 4.3): is not a relative extremal function.We have supp While the sets K t , as defined, could depend on the choice of the domain D, this is not actually true -at least, in the case when K j are polynomially convex, which means that P being the collection of all polynomials.Namely, they are sections of certain holomorphic hulls of the set Theorem 3.3 [23] If K j are compact and polynomially convex, then, for any ζ ∈ A,

Toric case
More can be said if K j are closures of complete, logarithmically convex, multicircled (Reinhardt) domains, which means that y ∈ K j provided z ∈ K j and |y i | ≤ |z i | for all i, and Log K j := {s ∈ R n : exp s = (e s 1 , . . ., e sn ) ∈ K j } are convex subsets of R n .When, in addition, the domain D is multicircled, the functions ω K j are toric (multicircled), and so are the geodesics u t .
Any toric plurisubharmonic function u(z) on D can be identified with its convex image: the convex function ǔ(s) = u(exp s) on Log D ⊂ R n , increasing in each variable s j .In addition, ǔt is a convex function on Log D × (0, 1) ⊂ R n+1 .
The geodesics between toric psh functions have an easy description: Theorem 4.1 [41], [58] The convex image ǔt of any toric geodesic u t is given by where L is the Legendre transform, We can assume D = D n , the unit polydisk.By [6], where is the support function of L j .Then L[ω K j ] = max{h L j +1, 0}, which gives an explicit formula for the geodesic u t of u j = ω K j : As a consequence, we get Theorem 4.2 [58] Let u t be the geodesic between u j = ω K j for complete, logarithmically convex Reinhardt sets K j ⋐ D n .Then the sets K t defined by (3.1) are the geometric means of K j : One can show that, in the toric case, there can never be an equality in Theorem 3.1, unless the geodesic is constant: Theorem 4.3 [58] In the conditions of Theorem 4.2, if there exists 0 < t 0 < 1 such that u t 0 = ω Kt 0 , then u 0 = u 1 .

Recall that volumes of convex combinations
in other words, volumes of K(t) are logarithmically concave.The same is true for the multiplicative combinations . The geodesic interpolation gives us a reverse estimate: the (usual) convexity of the capacities for logarithmically convex Reinhardt bodies K j .

Weighted extremal functions
One can get a stronger relation between the capacities if the functions u j = ω K j are replaced with weighted ones u j = c j ω K j for some c j > 0.
Let u c t be the corresponding geodesic.The sets K t are now to be defined as , where m t = min{u c t (z) : z ∈ Ω}.It can be shown that m t is a concave function and Theorem 5.1 [59] Let K j ⋐ D be polynomially convex, K 0 ∩ K 1 = ∅, and let the weights c j be chosen such that Cap (K 1 ). Then A little drawback of this result is that the sets K c t depend on the parameters c j .It turns out not to be the case in the toric situation, and the capacity inequality becomes the one on concavity of the function Cap ( K t ) for any weights c j > 0. Furthermore, the geodesic u c τ equals c τ ω Kτ for some τ ∈ (0, 1) if and only if Since the concavity of v −1 (t) implies convexity of v(t) and since the function x → x 1+ 1 n is convex, the conclusion of Theorem 5.2 is stronger than convexity of (Cap ( K t )) 1 n , and the latter is equivalent to logarithmic convexity of Cap ( K t ).This implies Corollary 5.3 [59] In the conditions of Theorem 5.2,

Geodesics in F 1
To get the above results on geodesics, including the linearity of the energy functional E, extended to larger classes of psh functions, one should stick to those where the functional is still finite.This leads to considering Cegrell's energy classes, of which the simplest one is the class F 1 .
Let D be a bounded hyperconvex domain in C n .Definition 6.1 [16], [17] The class F(D) is formed by all u ∈ PSH − (D) that are limits of decreasing sequences u N ∈ E 0 (D) such that , and E(u) = lim N →∞ E(u N ).Then, given u j ∈ F 1 (D), we approximate them by u j,N ∈ E 0 (D), find the geodesics u t,N , and then we can look at u t = lim u N,t as N → ∞.
A piece missing from the bounded case is an argument for u t to converge to u j as t → j, since one cannot have L ∞ -bounds now.Instead, a rooftop technique can be used.Since it will be playing a central role in the rest of the exposition, we will present it in the next section, while here we just state a result on F 1 -geodesics based on that technique.

Rooftop envelopes
Rooftop envelopes were explicitly introduced in [62] for quasi-psh functions on compact Kähler manifolds, and then the technique was developed in [25], [27], [28], [40], [35], [63] and others.In the local context, they were considered first in [57] for functions in the Cegrell class F 1 and then in [60] for arbitrary psh functions, bounded from above.
The rooftop envelope P (u, v) of bounded above functions u and v is the largest psh minorant of min{u, v}.Since P (u, v) ≥ u + v − M for some M ≥ 0, P (u, v) ≡ −∞.
While P (u, v j ) decreases to P (u, v) when v j decreases to v, its behavior for increasing v j can be more complicated, provided v j are unbounded from below.
Then min{u, v j } increase, as j → ∞, to the function ĥ equal to 0 outside the origin and ĥ(0) = −∞, while P (u, v j ) = v 0 for all j.
This observation is a particular case of how the rooftop envelopes P (u, v + C) behave when C → ∞.Denote sup the asymptotic rooftop, or asymptotic envelope, of u with respect to the singularity of v.
Remark.This actually implies the connectivity of any u 0 , u 1 ∈ F 1 (D), see Theorem 8.3.
One can ask if this works for any negative psh functions.The rest of the paper will be actually devoted to this question.

Geodesics on PSH − (D)
Any u ∈ PSH − (D) is the limit of a decreasing sequence u N ∈ E 0 (D) [19].So, for any pair u j ∈ PSH − (D), j = 0, 1, we repeat what we did for F 1 : approximate u j by u j,N , connect them by the geodesics u t,N , and then get the 'geodesic' u t as the limit of u t,N as N → ∞.
The crucial question is if u t connects u j .More precisely: Does u t converge to u j , in any sense, as t → j ∈ {0, 1}?
For any N > 0, the function u N,t = max{u 1 , −N t} is the geodesic between u 0 and u 1,N = max{u 1 , −N }.Therefore, u t ≡ u 1 = log |z| is not passing through u 0 .
The same works for u 0 = 0 and Even more striking is Example 8.2 Let u j = G a j be the (pluricomplex) Green functions with different poles.Then u t does not exceed the geodesics between G a 0 and 0, that is, by Example 8.1, G a 0 , and, by the same argument, it does not exceed G a 1 .Therefore, it does not exceed (actually, equals) P (G a 0 , G a 0 ) = G a 0 ,a 1 , the Green functions with two logarithmic poles.
In this case, we get a 'geodesic' that does not pass through any of the endpoints.
So, there are functions that cannot be connected by geodesics, the obstacle being that they have different 'strong' singularities.This sets the following Geodesic connectivity problem: What pairs u 0 , u 1 ∈ PSH − (D) can be connected by a psh (sub)geodesic?
The problem in the compact setting (quasi-psh functions on compact Kähler manifolds) was handled in [25] in terms of asymptotic envelopes, and this easily adapts to the local case: Theorem 8.3 [60] Let u 0 , u 1 ∈ PSH − (D), then the geodesic u t converges to u 0 in L 1 loc (D) (and in capacity) as t → 0 if and only if As we have already seen, the possible obstacles can arise only from the singularities of u 0 and u 1 .When the singularities are equivalent, u 0 ≍ u 1 , which means for some A > 0 and all z ∈ D, we have both P [u 1 ](u 0 ) = u 0 and P [u 0 ](u 1 ) = u 1 and so, the geodesic connects the data functions.This however does not cover Theorem 6.2 because functions from F 1 need not have equivalent singularities.Then one should look for a coarser equivalency of singularities.
In [42], it was proved that two toric psh functions in D n with isolated singularities at 0 can be connected if and only if all their directional Lelong numbers coincide.This means that the main terms of their singularities are the same.The proof was based on relating toric psh functions to convex ones (as in Section 4) and then using technique of convex analysis.This will not work for arbitrary psh functions, so one should find another way to single out such 'main terms' of the singularities.

Residual function
Most of the contents of this section is taken from [60].
Equivalently, g φ is the (u.s.c.regularization) of the upper envelope of the class of all functions u ∈ PSH − (D) with singularities at least as strong as that of φ, meaning that u ≤ φ + C for some C ∈ R.
The function g φ is determined by the asymptotic behavior of φ near its singularities, and it is a candidate for the main term of the asymptotic of the singularity of φ.Example 9.4 If φ is toric and D = D n , then g φ coincides with its indicator Ψ u [56] defined in [45], [55] as the toric psh function in D n whose convex image ψ u (s) := Ψ u (exp s) in R n − is given by the directional Lelong numbers ν u,a of u at 0 in directions a ∈ R n + : The next two examples deal with functions with non-isolated singularities.
These two are particular cases of Green functions with poles along complex spaces: given an ideal I = f 1 , . . ., f N ⊂ O(D) with bounded generators, G I is the upper envelope of u ∈ PSH − (D) such that u ≤ log |f |+O(1) [61].Note that in this definition, the asymptotics of u and log |f | are related only locally, not uniformly in D, so their equality to the corresponding residual functions is not trivial facts.[13], [14].
We have P (Ω ζ + C, 0) = Ω ζ for any C > 0 and so, g In the general case, the picture can be much more complicated.Since the singularities can lie both inside the domain and on its boundary, we call g φ the Green-Poisson residual function of φ for the domain D.
The boundary values of g φ need not be zero (outside the unbounded locus of φ), see Example 9.5.However they are zero there if the domain is B-regular (i.e., each boundary point possesses a strong psh barrier [65]).
By the unbounded locus of u ∈ PSH − (D) we mean the set L u of all points z ∈ D such that u is not bounded in D ∩ U z for any neighborhood U z of z.
A very important property we believe the residual functions have is their idempotency: At the moment, however, it is known to hold only under some assumptions on u.To present them, we need the following Remark.The uniqueness part here follows from [3].
Functions from F(D) can have very large unbounded locus, it can even coincide with D. Nevertheless, the boundary value of any u ∈ F(D), in the sense of Cegrell, is zero, which means that the least psh majorant H of u in D satisfying (dd c H) n = 0, is H ≡ 0.
More results on boundary behaviour for functions from other Cegrell classes and their residual functions can be found in [60] (see also last section of this paper).

Residual functions and asymptotic rooftops
In the compact setting, the corresponding analog of the residual function is an ultimate tool for checking the connectivity of quasi-psh functions [29].This was proved by using machinery of non-pluripolar Monge-Ampère operator, including the monotonicity property [12], [66].Unfortunately, that technique does not work in the local setting.In addition, functions from PSH − (D) can have their singularities on the boundary, and theory of boundary behavior of such psh functions is still underdeveloped.That is why the results in the local theory are at the moment not that complete.
By the same reason, Corollary 10.2 proves the Rooftop Equality for the Cegrell class F a (D) of functions φ ∈ F(D) such that (dd c φ) n do not charge pluripolar sets.

Other results
We just mention some other directions related to psh geodesics on domains of C n .
1.One can consider separately residual functions g o u and g b u determined by the singularities of u inside the domain and on its boundary, respectively [60].For example, the Green function G I with poles along a complex space given by an ideal sheaf I = f 1 , . . ., f k , mentioned in Section 8, equals actually g o log |f | ; under some additional conditions on I, it coincides with g log |f | , and in some situations, with g b log |f | .2. Residual functions g u for functions u ∈ PSH − (D) with well-defined (dd c u) n and possessing boundary values ũ in the sense of Cegrell were considered in [60].It was shown there that if D (dd c u) n < ∞ and ũ has small unbounded locus, then g u is idempotent and equal to P (g o u , g b u ). 3. Cegrell's energy classes can be considered for m-subharmonic functions on domains of C n , 1 ≤ m < n [46], [47].Geodesics for such functions, including the linearity of the corresponding energy functional, were studied in [4].4. Dirichlet problem for unbounded psh functions and its relation to the asymptotic rooftop construction were recently considered in [51]- [53].

Example 9 . 2
If φ(z) ≍ log |z − a|, a ∈ D, then g φ = G a , the pluricomplex Green function with pole at a. Example 9.3 More generally, if φ(z) ≍ c k log |z − a k |, then g φ is the weighted multipole pluricomplex Green function.

A
bit different are the next two examples dealing with boundary singularities.Example 9.7 If D = D ⊂ C and φ = −P a , the negative Poisson kernel with pole at a ∈ ∂D, then g φ = −P a .Example 9.8 More generally, if D ⊂ C n is strongly pseudoconvex with smooth boundary, then for any ζ ∈ ∂D there exists the pluricomplex Poisson kernel Ω ζ ∈ PSH − (D) which satisfies (dd c Ω ζ ) n = 0 in D, is continuous in D \ {ζ}, equal to 0 on ∂D \ {ζ}, and such that Ω ζ (z) ≈ −|z − ζ| −1 as z → ζ nontangentially