Discs and boundary uniqueness for psh functions on almost complex manifold Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 10. № 4 (2018). С. 130-137.

УДК 517.55

DISCS AND BOUNDARY UNIQUENESS FOR PSH FUNCTIONS ON ALMOST COMPLEX MANIFOLD

A. SUKHOV

Abstract. This paper is inspired by the work by J.-P. Rosav (2010). In this work, there was sketched a proof of the fact that a totally real submanifold of dimension 2 can not be a pluripolar subset of an almost complex manifold of complex dimension 2. In the present paper we prove a considerably more general result, which can be viewed as a boundary uniqueness theorem for plurisubharmonic functions. It states that a function plurisubharmonic in a wedge with a generic totally real edge is equal to identically if it tends to approaching the edge. Our proof is completely different from the argument by J.-P. Rosav. We develop a method based on construction of a suitable family of J-complex discs. The origin of this approach is due to the well-known work by S. Pinchuk (1974), where the case of the standard complex structure was settled. The required family of complex discs is obtained as a solution to a suitable integral equation generalizing the classical Bishop method. In the almost complex case this equation arises from the Cauchv-Green type formula. We hope that the almost complex version of this construction presented here will have other applications.

Keywords:almost complex structure, plurisubharmonic function, complex disc, totally real manifold.

Mathematics Subject Classification: 32H02, 53C15

1. Introduction

This paper addresses some aspects of pluripotential theory on almost complex manifolds. In the general case of non-integrable almost complex structures, the development of this theory began recently and many natural questions remain open. Our main motivation arises from the paper by J.-P.Rosay [5], where several interesting properties of (non) pluripolar subsets of almost complex manifolds were established. In particular, he proved that a J-complex curve is a pluripolar set. On the other hand, [5] there was sketched a proof of the fact that a totally real submanifold of dimension 2 can not be a pluripolar subset of an almost complex manifold of complex dimension 2 (a hint for a higher dimension argument was also indicated without details).

In the present paper we prove a considerably more general result (Theorem 4.1), which can be viewed as a boundary uniqueness theorem for plurisubharmonic functions. It states that a function plurisubharmonic in a wedge with generic totally real edge is equal to identically if it tends to approaching the edge. Our proof is completely different from the argument of [5]. Our approach is based on construction of a suitable family of J-complex discs and is inspired by the well-known work [3]. We hope that the almost complex version of this construction

A. Sukhov, Discs and boundary uniqueness for psh functions on an almost complex manifold.

© Sukhov A. 2018.

Поступила 19 июня 2018 г.

The author is partially supported by Labex CEMPI.

presented here will have other applications. Another approach to filling a wedge by discs is used in [6] and is based on stability results for the Riemann-Hilbert boundary value problem. The method of present paper is more direct and gives an additional information on geometry of the discs. The required family of complex discs is obtained as a solution to a suitable integral equation generalizing the classical Bishop method. In the almost complex case this equation arises from the Cauehv-Green type formula.

The paper is organized as follows. In Section 2 we recall basic properties of almost complex manifolds (see more in [1]), Section 3 contains the construction of Bishop type discs glued to a totally real manifold along a boundary arc. In section 4 we obtain the main result on boundary uniqueness for psh functions,

2. Almost complex manifolds, discs and plurisubharmonic functions

Throughout this paper we assume that manifolds and structures are C^ smooth although the main results remain true under considerably weaker regularity assumptions.

Let M be a smooth manifold of dimension 2n. An almost complex structure J on M is a smooth map, which associates to every point p E M a linear isomorphism J(p) : TPM M TPM of the tangent space TPM satisfying J (p)2 = —/.Here I denotes the identity map of TPM. Thus, J(p) is a linear complex strueture on TPM. A couple (M, J) is called an almost complex manifold of complex dimension n. Note that every almost complex manifolds admits the canonical orientation.

The standard complex structure Jst = J(2) on M = R2 is given by the matrix

J-> = (0 —1) w

in the canonical coordinates of R2. More generally, the standard complex structure Jst on R2n is represented by the block diagonal matrix dia.g(Jg2\ ..., jjt2)); we skip the notation of dimension because it will be clear from the context. As usually, setting iv := Jv for v E R2n, we identify (R2n, Jst) wit h Cn havin g z = x + iy = x + J у for the standard complex coordinates z=( zl,.'.., Zn) E Cn.

Let (M, J) and (M', J') be smooth almost complex manifolds, А С*-тар f : M' ^ M is called (J', J)-complex or (J', J)-holomorphie if it satisfies the Cauchy-Riemann equations

df о J' = J о df. (2)

Of course, a map f : Cn ^ Cm is (Jst, J^-holomorphic if and only if each component of f is a usual holomorphic function.

Every almost complex manifold (M, J) can be viewed locally as a unit ball В in Cn equipped with a small almost complex deformation of Jst. Indeed, we have the following useful statement.

Lemma 2.1. Let ( M, J) be an almost complex manifold. Then for every point p E M, every real a ^ 0 and Ao > 0 there exist a neighborhood U of p and a coordinate diffeomorphism z : U ^ В such that z(p) = 0 dz(p) о J(p) о dz-1(0) = Jst and the direct image z*(J) := dz о J о dz-1 satisfies ||z*(J) — Jst||с«(В) ^ A0.

Доказательство. There exists a diifeomorphism z from a neighborhood U' of p E M onto В satisfying z(p) = 0 and dz(p) о J(p) о dz-1(0) = Jst. For A > 0, we consider the dilation d\ : t M- A-1t in R2n and the composition za = d\ о z. Then Ишл^0 ||(z\)*(J) — Jst||C«(I) = 0 for each real a ^ 0 Setting U = z-1(I) for A > 0 small enough, we obtain the desired statement, □

the special case, when M' has the complex dimension 1. These holomorphic maps are called

J-complex (or J-holomorphic) curves. We use the notation D = {( e C : | < 1} for the unit disc in C always assuming that it is equipped with the standard complex structure Jst. If in the above definition we have M' = D we call such a map fa J-complex disc or a pseudo-holomorphie disc or just a holomorphie disc if J is fixed. Similarly, if M' is the Riemann sphere, f is called a J-complex sphere. These two classes of pseudo-holomorphie curves are particularly useful.

Let (M, J) be an almost complex manifold and E C M be a real submanifold of M. Suppose that a J-complex disc f : D ^ M is continuous on D. With some abuse of terminology, we also call the image f (D) simply by a disc and we call he image f (bD) by the boundary of a disc. If f (bD) C E, then we say that (the boundary of ) the disc f is glued or attached to E or simply that f is attached to E. Sometimes such maps are called Bishop discs for E and we employ this terminology. Of course, if p is a point of E, then the constant map f = p always satisfies this definition.

In this paper we deal with a special class of real submanifolds, A submanifold E of an almost complex n-dimensional (M, J) is called totally real if at every point p e E the tangent space TPE contains no non-trivial complex vectors that is TPE n JTPE = {0}. This is a well-known fact that the (real) dimension of a totally real submanifold of M does not exceed n; we will consider in this paper only n-dimensional totally real submanifolds that is the case of the maximal dimension.

A totally real manifold E can be defined as

E = {p e M : p3 (p) = 0} (3)

where pj : M ^ R are smooth functions with non-vanishing gradients. The condition of total reality means that for every p e E the J-complex linear parts of the differentials dpj are (complex) linearly independent.

A subdomain

W = {p e M : Pj < 0, j = 1,...,n}. (4)

t is called the wedge with the edge E.

2.1. Cauchy-Riemann equations in coordinates. All our consideration are local, so (as above) we are in a neighborhood Q of 0 in Cn with the standard complex coordinates z = (zi,..., zn). We assume that J is an almost complex structure defined on Q and J(0) = Jst. Let

z : D ^ Q,

* : C ^ Z(C)

be a disc. Setting ( = £ + we write (2) in the form zv = J(ZThis equation

can be in turn written as

z^ - A(z)z-C = 0, ( e D. (5)

Here a smooth map A : Q ^ Mat(n, C) is defined by the identity L(z)v = Av for any vector v e Cn and L is an ^^^^^^^ map defined by L = (Jst + J)-i(Jst — J). It is easy to check that the condition J2 = — Id iq equivalent to the fact that L is C-linear. The matrix A(z) is called the complex matrix of J in the local coordinates z. Locally the correspondence between A and J is one-to-one. Note that the condition J (0) = Jst means th at A(0) = 0.

If Z' are other local coordinates and N is the corresponding complex matrix of J', then, as it is easy to check, we have the following transformation rule:

A = (z' z A + z't)(z't + A)-i (6)

(see [7]).

2.2. Plurisubharmonic functions on almost complex manifolds: background. Let

u be a real C2 function on an open subset Q of an almost complex manifold (M, J). Denote by J * dux the differential form acting on a vector field X by J * du(X) := du(JX). Given a point p E M and a tangent vector V E TP(M), we consider a smooth vector field X in a neighborhood of p satisfying X(p) = V. The value of the complex Hessian ( or of the Levi form ) of u with respect to J at p and V is defined by H(u)(p, V) := -(dJ*du)p(X, JX). This definition is independent of the choice of a vector field X. For instance, if J = Jst in C, then —d J * dux = Au A dr]; here A denotes the Laplacian. In particular, HJst (u)(0, ) = Au(0).

Let us recall some basic properties of the complex Hessian (see, for instance, [2]),

Lemma 2.2. Consider a real function u of class C2 in a neighborhood of a point p E M.

(i) Let F : (M',J') —> (M,J) be a (J', J)-holomorphic map, F(p') = p. For each, vector V' E Tp,(M') we have Hj,(u o F)(p', V') = Hj(u)(p, dF(p)(V')).

(ii) If f : D —> M is a J-complex disc satisfying f(0) = p, and df (0)() = V E Tp(M), then Hj (u)(p, V )=A(u o f)(0).

Property (i) expresses the holomorphic invarianee of the complex Hessian, Property (ii) is

V

Let Q be a domain M. An upper semicontinuous function u : Q ^ +<^[ on (M,J) is J-plurisubharmonic (psh) if for every J-complex disc f : D ^ Q the composition u o f is

D C2 u Q

it has a positive semi-definite complex Hessian on Q i.e. HJ(u)(p,V) ^ 0 for any p E Q and V E Tp(M). A real C2 function u : Q ^ R is called strictly J-psh on ^^f HJ(u)(p, V) > 0 for each p E M and V E Tp(M)\{0}. Obviously, these notions are local: an upper semicontinuous C2 Q J Q J

Q

manifold (M, J) coincides with the Levi form with respect to the standard structure Jst of R2ra coordinate systems,

J

Lemma 2.3. There exists a second order polynomial local diffeomorphism $ fixing the origin and with linear part equal to the identity such that in the new coordinates the complex matrix A of J (that is A from the equation (5)) satisfies

A(0) = 0, A, (0) = 0 (7)

matrix A. We stress that in general it is impossible to get rid of first order terms containing ~z

J

The author learned this result from unpublished E, Chirka's notes; see [2] for the proof. It was shown in [7] that in an almost complex manifold of (complex) dimension 2, a similar

J

As a typical consequence, we consider a totally real manifold E defined by (3), Then the function u = YTj=i p2 is strictly J-psh in a neighborhood of E. Indeed, it suffices to choose local coordinates near p E M according to Lemma 2,3, This reduces the verification to the

Js

3. Filling a wedge by complex discs

Here we construct a family of Bishop's type discs filling a wedge with a totally real edge. Each disc is glued to the edge E along the upper semi-circle,

3.1. Case of the standard structure. First consider the model case M = Cn with J = Jst

and E = ¿Rra = {xj = 0,j = 1,..., n}. Denote by W the wedge

W = {z = x + iy : Xj < 0,j = 1,...., n}.

Let

Po<P =±, i 0(u) -

2m% JbD u

denotes the average of a real function 0 over bD, and let also

sm = 2" i ^dU

2m% Jmu — C u be the Schwarz integral. In terms of the Cauchy transform

Kf {0 = ±_ f m^

2m Jbd u — <

we have the following relation: S = 2K — P0. As a consequence, the boundary properties of the Schwarz integral are the same as the classical properties of the Cauchy integral.

For a non-integer r > 1 consider the Banach spaces Cr (6D) and Cr (D) (with the usual Holder

K S

For real function 0 e Cr (6D) the Schwarz integral S0 is a function of class Cr (D) holomorphie in D the trace of its real part on the boundary coincides with 0 and its imaginary part vanishes at the origin. In particular, every holomorphie function f e Cr (D) satisfies the Schwarz formula f = S Ref + iPof.

We are going to fill W by complex discs glued to ¿Rra along the (closed) upper semi-circle 6D+ = {eie :d e [0, m]}; let also 6D- := 6D \ 6D+.

Fix a smooth real function 0 : 6D ^ R such that 6D+ = ^d 6D- < 0, Consider now a real 2 n-parameter family of holomorphie discs z° = (z°,..., z°) : D ^ Cn with components

z0( c, t)(() = xJ (0 + i y, (0 = t3sm + i Cj ,j = 1,...,n (8)

Here tj > 0 and Cj e R ^re parameters, t= (ti,..., t),c= (ci,..., cn). The following properties of this family are obvious:

(i) for every j one has Xj16D+ = 0 and Xj(() < 0 when ( e D (by the maximum principle for harmonic functions),

(ii) the evaluation map Ev0 : (t, c) ^ z0(c, t)(0) is one-to-one from {(c, t) : Cj > 0, tj e R} on W.

Our goal is to construct a local analog of this family in the general case,

3.2. General case. In order to write an integral equation defining a required family of discs, we need to employ an analog of the Schwarz formula and to choose suitable local coordinates. We proceed in several steps.

Step 1. Recall that for any complex function f e Cr (D) the Cauchy-Green transform is defined by

i rr m<L, a^

2^ j Jo u — C

This is a classical fact that T : C (D) M Cr+1(D) is a bounded linear operator for every noninteger r > 0, Furthermore, (Tf)-^ = f, i.e., T solves the 5-equation in the unit disc. Recall also that the function Tf is holomorphie on C \ D,

We have the following Green-Schwarz formula (see the proof, for example, in [7], although, of course, it can be found in the vaste list of classical works). Let f = f + iip : D m C be a function in the class Cr(D). Then for each ( E D one has

f(() = Sf(() + tPa 4 + T ft (() — T ft (1/() (9)

T

D

Step 2. Now let (M, J) be an almost complex manifold of complex dimension n and E be a totally real n-dimensional submanifold of M. We assume that E and W are given by (3) and (4), respectively.

First, according to Section 2 we choose local coordinates z such that p = 0 and the complex A J > 0 C > 0

n

Pi := Pi— r + CJ2 p2k

k=j k=1

J

WT = {pj < 0, j = 1,..., n^ ^^ ^^^teined in W. After a ^^^^^^^ (with respect to Jst) change of coordinates one can assume that pj = Xj + o(lzl). Consider now a local diffeomorphism

$ : z = Xj + iyj M- z' = xj + iyj = pj + iyj

Then $(0) = 0 d$(0) = 0 and in the new coordinates pj = Xj (we drop the primes), E = ¿Rn and WT = {xj < 0,j = 1,..., n}. We keep the notation J for the direct image $*(J). Then in

A(0) = 0

Xj J

Finally, similarly to the proof of Lemma 2.1, for A > 0, we consider the isotropic dilations d\ : z M A-1z and the direct images J\ := (d\)*(J). Denote by A(z, A) the complex matrix of Jx-

Step 3. For A > 0 small enough, we are looking for the solutions z : D M Cn to a Bishop's type integral equation

z(() = h(z((),t,c,A) (10)

with

h(z(0,t, c, A) = tS<f>(0 + ic + TA(z, A)z^(() — TA(z, A)z^(1/()

where t = (t 1,...,tn), tj > ^d c E Rn are real parameters. Note that the first and the

D

the Cauchy-Riemann equations (5) i.e. is a J-complex disc. Furthemore, Xj (() vanishes on 6D+ (that is, z(6D+) C E) and is negative on 6D-, Since the function M Xj J

the maximum principle the image z(D) is contained in WT.

A = 0

(10) admits solution (8), We consider the smooth map of Banach spaces

H : Cr(D) x Rn x Rn x R —> Cri H : ( , , , A) M h( ( ), , , A).

Obviously the partial derivative of H in z vanishes: (DZH)(z0,c,t, 0) = 0, where z° is a disc given by ( 8), By the implicit function theorem, for every (c,t,A) close enough to the origin the equation ( 10) admits a unique solution z(c, t)(() of class Cr(D) depending smoothly on parameters (c, t) (as well as A, of course).

Fixing A > 0, we consider the smooth evaluation map

Evx : (c, t)^ z(c, t)(0)

A = 0

we obtain the linear mapping Ev0(c, t) = (tS0(0) + ic) appeared already in the model case of the standard structure. More precisely, for every A > 0 and every c e Rra, the equation (10) has the unique constant solution z = ic when t = 0, Hence, Ev\ admits the expansion Evx(c, t) = Evo(c, t) + o(l(c,t,A)l).

Denote by V the wedge V = {(c, t) : c e Rra, tj > 0,j = 1,...,n} with the edge L = {(c, 0) : ce Rra}. Then Ev0(V) coincides with WT in a neighborhood of the origin. Furthermore, Ev\(L) = E for A > 0. Also, for a > 0 the "truncated" wedge Wa = {xj — k=j Xk < 0} with the edge E is contained in WT. The faces of the boundary of Wa are transverse to the face of WT. Since this property is stable under small perturbations, we conclude that Wa C Ev\ (V) for all A small enough. In terms of the initial defining functions Pj we have {z : pj — 5 Pk < 0} C W£ when t + e <5.

We summarize the said above in the following theorem.

Theorem 3.1. Let E be a totally real submanifold (3) of an almost complex 'manifold (M, J) and W be a wedge (4)- For every 5 > 0 there exists a family of J-complex discs smoothly 2 n

(i) the boundary of every disc is glued along bD+ to E and every disc is contained in W;

(ii) every point of the truncated wedge {z : pj — 5 Pk < 0} belongs to some disc.

Although we do not need this in the paper, note that every point of the wedge E belongs to the boundary of some disc. Concerning regularity of manifolds and almost complex structures, it suffices to require the class Ca with real a > 2. We skip the details,

4. Boundary uniqueness

Now we can prove easily the main result of the present paper.

Theorem 4.1. Let E be a totally real submanif old (3) of an almost co mplex 'manifold (M, J) and W be a wedge (4). Assume that u is a plurisubharmonic function in W such, that

lim sup u(z) = —<x> (11)

W 3z^E

Then u = —to.

Indeed, for every disc f constructed in Theorem 3,1 the function (u o f)(() is subharmonie on D and tends to —to as ( ^ 6D+; therefore, u of = —to (by classical properties of subharmonie functions, see [4]), Now by (ii) Theorem 3,1 we conclude that u = —to.

Corollary 4.2. Let E be a closed subset of an almost complex n-dimensional 'manifold (M,J). Suppose that E contains a germ of a totally real 'manifold of dimension n. Then E is not a pluripolar set.

REFERENCES

1. M.Audin, J.Lafontaine. Holomorphic curves in symplectic geometry. Progress in Mathematics. 117 Birkhauser, Basel (1994).

2. K. Diederich, A. Sukhov. Plurisubharmonic exhaustion functions and almost complex Stein structures // Mich. Math. J. 56:2, 331-355 (2008).

3. S. Pinchuk. A boundary uniqueness theorem for holomorphic functions of several complex variables // Mat. Zametki. 15:2, 205-212 (1974). [Math. Notes. 15:2, 116-120 (1974).]

4. Th. Ransford. Potential theory on the complex plane. Cambridge Univ. Press, Cambridge (1995).

5. J.-P. Rosav. Pluri-polarity in almost complex structures // Math. Z. 265:1, 133-149 (2010).

6. B. Coupet, H. Gaussier, A. Sukhov. Fefferman's mapping theorem on almost complex manifolds of complex dimension 2 11 Math. Z. 250:1, 59-90 (2005).

7. A. Sukhov, A.Tumanov. Filling hypersurfaces by discs in almost complex manifolds of dimension 2 11 Indiana Univ. Math. J. 57:1, 509-544 (2008).

Alexandre Sukhov,

Université des Sciences et Technologies de Lille,

Laboratoire Paul Painlevé,

U.F.R. de Mathématiques,

59655 Villeneuve d'Ascq, Cedex, France

Institute of Mathematics,

Ufa Federal Research Center, RAS,

450008, Ufa, Russia

E-mail: sukhov@math.univ-lillel.fr