TOTALLY REAL EMBEDDINGS WITH PRESCRIBED POLYNOMIAL HULLS

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1 TOTALLY REAL EMBEDDINGS WITH PRESCRIBED POLYNOMIAL HULLS LEANDRO AROSIO AND ERLEND FORNÆSS WOLD Abstract. We embed compact C manifolds into n as totally real manifolds with prescribed polynomial hulls. As a consequence we show that any compact C manifold of dimension d admits a totally real embedding into 3d with non-trivial polynomial hull without complex structure. We will prove the following result. 1. Introduction Theorem 1.1. Let X m be a totally real C submanifold of dimension d > 1 and let K X be a compact subset. Let M be a compact C manifold of dimension d (possibly with boundary) and assume that there exists a C embedding φ : X M. Then there exists a totally real C embedding ψ : M m+l, where l = max{ 3d m,0}, such that (i) ψ φ = id on K, and (ii) ψ(m) = ψ(m) K. Notice that by a slight abuse of notation we denote the compact set K {0} m+l also by the letter K. Our motivation for proving this result is the following recent result by Izzo and Stout. Theorem 1. (Izzo Stout [9]). Let M be a compact C surface (possibly with boundary). Then M embeds into 3 as a totally real C surface Σ such that Σ \ Σ is not empty and does not contain any analytic disc. In the case where M is a torus, Theorem 1. was proved by Izzo, Samuelsson Kalm and the second named author [8]. In a recent paper Gupta [4] gave an explicit example of an isotropic torus in 3 whose polynomial hull consists only of an annulus attached to the torus. We note that the torus embedded in [8] is also isotropic, the image being the graph of a real valued function on the torus in. Date: July 31, Mathematics Subject Classification. 3E0. Key words and phrases. Polynomial Convexity, Totally Real Manifolds. Supported by NRC grant no Supported by the SIR grant NEWHOLITE - New methods in holomorphic iteration no. RBSI14CFME. Part of this work was done during the international research program Several Complex Variables and Complex Dynamics at the Center for Advanced Study at the Academy of Science and Letters in Oslo during the academic year 016/017. 1

2 L. AROSIO AND E. F. WOLD As a corollary to Theorem 1.1 we prove the following generalization of Theorem 1. to every dimension d > 1. Corollary 1.3. Let M be a compact C manifold (possibly with boundary) of dimension d > 1. Then M embeds into 3d as a totally real C submanifold Σ such that Σ\Σ is not empty and does not contain any analytic disc. The same result was proved in [8] for manifolds of dimension d >, with dimension of the target space d + 4 (and with a better dimension of the target space for certain specific -manifolds). Notice that by [7, Theorem.1] the dimension 3d is optimal for totally real embeddings of d-dimensional manifolds, which implies that Corollary 1.3 is optimal. We emphasize that Theorem 1.1 is a simple consequence of the general fact that any C manifold of dimension d embeds into 3d as a totally real submanifold, and the following general genericity result regarding polynomial hulls and totally real embeddings (to be proved in Section 4): Theorem 1.4. Let M be a compact C manifold (possibly with boundary) of dimension d < n and let f: M n be a totally real C embedding. Let K n be a compact polynomially convex set. Then for any k 1 and for any ε > 0 there exists a totally real C embedding f ε : M n such that (1) f ε f C k (M) < ε, () f ε f = 0 on f 1 (K), (3) K fε (M) = K f ε (M). In the case K =, Theorem 1.4 was proved by Forstneric and Rosay for d = and n 3, by Forstneric [] for d n 3, and by Løw and the second named author [10] for d < n. To obtain Corollary 1.3 it is crucial that Alexander [1] has constructed a suitable compact set K in the totally real distinguished boundary of the bidisk in (see Section ).. Proof of Theorem 1.1 and Corollary 1.3 In this section we prove Theorem 1.1 and Corollary 1.3 assuming Theorem 1.4 which will be proved in the next section. We recall first the following result [3, Lemma 5.3]. By generic we mean open and dense in the Whitney C -topology. Lemma.1. Let M be a compact C manifold (possibly with boundary) of dimension d. Then a generic C embedding ψ : M 3d is totally real. Proof of Theorem 1.1: Let U be a relatively compact open subset of X containing K. Set U := φ(u ), and let ψ 1 : φ(x) m be the inverse of φ. Let ψ : M m be a C mapping which agrees with ψ 1 on U. Notice that since (m + l) d + 1, by Whitney s theorem (see e.g. [5, Theorem.13]) a generic C mapping from M to m+l is an embedding, and since m + l 3d, by Lemma.1 a generic C embedding of M into m+l is totally real. Let ψ 3 : M m+l be a small perturbation of ψ which is a totally real C embedding.

3 Let χ C 0 POLYNOMIALLY CONVEX HULLS 3 (U) with χ 1 in a neighborhood of φ(k) and 0 χ 1. Then if the perturbationψ 3 wassmallenough,themappingfromm to m+l definedasψ 4 := χ ψ 1 +(1 χ)ψ 3 is a totally real C immersion injective in U. There exists a small C perturbation ψ: M m+l whichis atotally real C embeddingandsuchthat ψ φ(k) = ψ 4 φ(k) (see e.g. [6, Theorem.4.3] where the approximation is in C 0 -norm but the proof can be easily adapted to our case) which implies that K ψ(m). Notice that ψ 4 φ(k) = ψ 1 φ(k), and thus ψ φ = id on K. Since m+l > d, we can apply Theorem 1.4 with K replaced by K and Theorem 1.1 follows. To prove Corollary 1.3 we need the following result of Alexander: Theorem.. (Alexander, [1]) The standard torus Ì := {(e iϑ,e iψ ): ϑ,ψ Ê} in contains a compact subset K such that K\K is not empty but contains no analytic discs. Such a set can be found in every neighborhood of the diagonal in Ì. Proof of Corollary 1.3: Choose a small annular neighbourhood A of the diagonal in Ì, and let K A be the compact subset given by Theorem.. Define a totally real C submanifold X of d by X = {(z 1,z,z 3,...,z d ) : (z 1,z ) A,Im(z j ) = 0 and ε < Re(z j ) < ε for j = 3,...,d}, where ε > 0. Now X embeds into (any small portion of) the interior of M. By Theorem 1.1, there exists a totally real C embedding ψ: M 3d such that K ψ(m) and ψ(m) = ψ(m) K. Recall [1, Theorem 6.3.] that for a polynomially convex subset H of a totally real C submanifold the uniform algebra P(H), consisting of all the functions on H that can be approximated uniformly by polynomials, coincides with the algebra of continuous functions on H. Hence the hull K cannot be contained in ψ(m), and the result follows from Theorem.. 3. Proof of the main lemma For any compact set K n set h(k) := K \K. We will sometimes denote K by [K ]. The proof of the following main lemma is essentially contained in [10, Proposition 6,7] where it is given in the case of C 1 -regularity. We give a proof for completeness. Lemma 3.1. Let M be a compact C manifold (possibly with boundary) of dimension d < n and let f: M n be a totally real C embedding. Let K n be a compact polynomially convex set, and let U be a neighborhood of K. Then for any k 1 and for any ε > 0 there exists a totally real C embedding f ε : M n such that (1) f ε f C k (M) < ε, () f ε f = 0 in a neighborhood of f 1 (K), (3) K fε (M) U f ε (M). Thefollowingresultisprovedin[10,Proposition4]. RecallthatasmallenoughC 1 -perturbation of an embedding of a compact manifold is still an embedding (see e.g [5, Theorem 1.4]). Proposition 3.. Let M be a compact C 1 manifold (possibly with boundary) and let f: M n be a totally real C 1 embedding. Let U U n be open sets. Then there exists an open neighborhood Ω of f(m) such that

4 4 L. AROSIO AND E. F. WOLD (1) if S M is closed and K U is compact, then h(k f(s)) U Ω h(k f(s)) U, () if f is a sufficiently small C 1 -perturbation of f, then (1) holds with f in place of f. The following lemma is proved in [10, Corollary ] Lemma 3.3. Let K n be a compact subset. Let M be a compact C 1 manifold (possibly with boundary) and let f: M n be a totally real C 1 embedding. Let S M be a closed subset. Let U be an open set such that h(k f(s)) U. Then there exists a constant c > 0 such that if f f C 1 (M) < c, then h(k f(s)) U. Proof of Lemma 3.1: The proof is in two steps. Let I denote the interval [0,1] Ê. First Step: The first step is proving the lemma in the case M = I d. The proof is by induction on d. Fix a strictly positive d Æ, and assume that the result holds for embeddings of I d 1. Since K is polynomially convex, it admits fundamental system of open neighborhoods which are Runge and Stein open sets. Thus there exists a Runge and Stein open set U n such that K U U. Since U is Runge and Stein, it admits a normal exhaustion by polynomially convex subsets. Hence there exists a polynomially convex set K U such that K int(k ). Let us fix some notation. (1) for J Æ and α Æ d, 0 α j J 1, we denote by I J α the cube [ Iα J := α1 J, α 1 +1 J ] [ αd J, α d +1 J () for 1 m d, 0 j J, we denote by G J m,j the (d 1)-dimensional cube { } j G J m,j := I I I I, J m-th index (3) we denote by G J the grid which is the union of all (d 1)-dimensional cubes G J := m,jg J m,j, ], (4) we denote by G J (δ) the closed δ-neighborhood of the grid G J, (5) we denote by Q J α (δ) IJ α G J (δ) := {x I d : dist(x,g J ) δ}, the smaller d-dimensional cube Q J α(δ) := I J α \G J (δ), (6) we denote by S J an n-tuple of distinct cubes I J α. We will prove the following statement. There exists a big enough J, a small enough δ and a totally real C embedding f ε : I d n such that (i) f ε f C k (I d ) < ε, (ii) f ε f = 0 in a neighborhood of f 1 (K),

5 POLYNOMIALLY CONVEX HULLS 5 (iii) f ε (Iα J) K f ε(iα J) K, (iv) h(k f ε (S J ) f ε (G J (δ))) U for any choice of n-tuple S J, (v) for all α such that f ε (Q J α(δ)) n \K, there exists an entire function g α O( n ) such that f ε (Q J α (δ)) {g α = 0} =: Z(g α ), (vi) for all choices of n+1 different multi-indices α 1,...,α n+1 such that f ε (Q J α(δ)) n \K we have 1 j n+1 Z(g αj) =. Once this is proved, the first step of the proof goes as follows. Assume that q [K f ε (I d ) ]. Let µ be a representative Jensen measure for the linear functional on P(K f ε (I d )) defined as the evaluation at q, such that for all entire functions g O( n ) we have log g(q)) log g dµ. (3.1) Assume that there exists α such that µ(f ε (Q J α (δ))) > 0 and f ε(q J α (δ)) K =. By (v) we get that log g α dµ =, and by (3.1) we have that q Z(g α ). By (vi), there exists an n-tuple of cubes S J such that the measure µ is concentrated on the subset K f ε (S J ) f ε (G J (δ)). It follows that q [K f ε (S J ) f ε (G J (δ)) ], and by (iv) we obtain that q U f ε (I d ). We proceed to prove the statement above. Claim 1: If J is big enough then h(k f(s J )) U (3.) for any choice of n-tuple S J. Moreover, for a fixed big enough J, if f 0 is a sufficiently small C 1 -perturbation of f, then (3.) holds with f 0 in place of f. Proof of Claim 1: Let U be an open neighborhood of K with U U. Let Ω be given by Proposition 3. with data (f,u,u ). Assume by contradiction that there is a sequence (J t ) t Æ, J t and a sequence S Jt of n-tuples such that h(k f(s Jt )) U. Let I Jt α 1 (t),...,ijt α n (t) be the cubes in SJt. Up to passing to a subsequence we may assume that for any 1 j n, the image f(i Jt α j (t) ) converges to a point q j M as t. Since K {q j } 1 j n is polynomially convex, there exists a Runge and Stein neighborhood V of K {q j } 1 j n such that V U Ω. For large enough t we have that K f(s Jt ) V, which implies h(k f(s Jt )) U Ω, and thus by Proposition 3. we have a contradiction. Since there are only a finite number of n-tuples S J for a fixed J, the perturbation claim follows from Lemma 3.3. End proof of Claim 1. Choose J big enough such that Claim 1 holds and that for all α, f(i J α) K f(i J α) int(k ), and such that for all α the image f(i J α) is polynomially convex (this can be obtained by [11, Proposition 4.]).

6 6 L. AROSIO AND E. F. WOLD Once J is fixed, by [10, Corollary 1] there exists θ > 0 small enough such that for all α and for every totally real C embedding g: I J α n satisfying g f C k (I J α) < θ we have that g(i J α) stays polynomially convex. Claim : There exists a totally real C embedding fε : Id n such that (1) fε f C k (I d ) < max(θ, ε ), () fε = f in a neighborhood of f 1 (K ), (3) h(k fε (SJ ) fε (GJ )) U for any choice of n-tuple S J. Moreover, if f 0 is a sufficiently small C 1 -perturbation of fε, then (3) holds with f 0 in place of fε. Proof of Claim : Fix an n-tuple S J. Choose an ordering of the (d 1)-dimensional cubes (G J m,j ) m,j and denote them by G 1,...,G l. Fix 0 < η << 1 (to be determined later). We will construct inductively a family of totally real C embeddings (f j : I d n ) 0 j l such that (a) f j f j 1 C k (I d ) η, (b) f j = f j 1 in a neighborhood of f 1 (K ) S J ( 1 i j 1 G i, (c) h K f j (S J ) ) 1 i j f j(g i ) U. If we set f 0 := f, then (c) is true by Claim 1. Assume we constructed f j, where 0 j l 1. Then by (c) the set h(k f j (S J ) 0 i j f j(g i )) is compact in U, so there exists a set U 1 such that K U 1 U satisfying h(k f j (S J ) 1 i j f j(g i )) U 1. Let Ω 1 be given by Proposition 3. with data (f j,u,u 1 ). Consider thetotally real C embeddingf j Gj+1 : G j+1 n. Since [K f j (S J ) f j (G i ) ] U 1 Ω 1, 0 i j by the inductive assumption (the result holds for embeddings of I d 1 ) there exists a totally real C embedding f j Gj+1 : G j+1 n such that (1) f j f j C k (G j+1 ) is small, () f j = f j in a neighborhood of fj 1 (K ) S J ( 1 i j G i, (3) h K f j (S J ) 1 i j f j(g i ) f ) j (G j+1 ) U 1 Ω 1. If f j f j C k (G j+1 ) is small enough, then we may extend f j to a totally real C embedding f j+1 : I d n such that f j+1 f j C k (I d ) η, which coincides with f j on f 1 (K ) S J 1 i j G i. By the choice of Ω 1 it follows that h(k f j+1 (S J ) f j+1 (G i )) U. 1 i j+1 The mapping f l satisfies Claim for the chosen n-tuple S J. We then iterate this argument for every n-tuple S J (choosing η small enough), and we obtain. The perturbation claim follows from Lemma 3.3. fε

7 POLYNOMIALLY CONVEX HULLS 7 End proof of Claim. Let Ω be an open neighborhood of fε (Id ) given by Proposition 3. with data (fε,u,u ). Claim 3: There exists a δ > 0 such that h(k fε (SJ ) fε (GJ (δ))) U (3.3) for any choice of n-tuple S J. Moreover, if f 0 is a sufficiently small C 1 -perturbation of fε, then (3.3) holds with f 0 in place of fε. Proof of Claim 3: Fix an n-tuple S J. Since h(k fε (SJ ) fε (GJ )) U, there exists a Runge and Stein neighborhood V Ω U of K fε (SJ ) fε (GJ ). If δ > 0 is small enough we have that K fε (SJ ) fε (GJ (δ)) V, and thus h(k fε (SJ ) fε (GJ (δ))) U. Since the number of n-tuples is finite, we can choose a δ which works for all of them. The perturbation claim follows from Lemma 3.3. End proof of Claim 3. We end Step 1 by pushing each fε (QJ α (δ)) with f ε (QJ α (δ)) K = into an analytic hypersurface of n. Fix a multi-index α such that fε (QJ α (δ)) K =. We identify the cube IJ α with the cube centred at the origin in Ê d with the decomposition Ê iê d d n d of n. After an affine linear change of coordinates we may assume that f : Iα J n satisfies f(0) = 0, with the tangent space of f(iα J) at the origin equal toêd. Moreover we can assume that Df(0): Ê Ê d d is orientation preserving. We now claim that we may approximate fε arbitrarily well in Ck -norm on I J α by holomorphic automorphisms G α of n. Granted this for a moment, we proceed as follows. Let χ be a cutoff function with compact support in int(i J α) such that χ 1 in a neighborhood of Q J α(δ). Define f α (x) := fε (x)+χ(x)(g α(x) fε(x)). (3.4) Since d < n, the n-th coordinate of every point in I J α is 0. Hence, if we define g α as the n-th coordinate of G 1 α, then for all x Q J α(δ), g α (f α (x)) = 0. (Notice that it is here crucial that d < n. If d = n, there is no way to push cubes into analytic hypersurfaces of n.) Since f α = fε on Id \int(i J α ), the maps f α patch up as α varies among all multi-indices α such that fε (QJ α(δ)) K =, and we obtain an embedding f ε : I d n satisfying the properties (i)-(v). Finally, by the following sublemma, we may assume, possibly having to perturb the automorphisms G α slightly, that the intersection 1 j n+1 Z(g αj) of any collection of n+1 zero sets is empty. Thus property (vi) is satisfied. Sublemma 3.4. Let {g j } 1 j N O( n ) be a finite collection of non-constant holomorphic functions, and for each a j set g a j j := g j a j. Then there exists a dense G δ set A N such that for each a A the following holds: for any collection {g a i 1 i 1,...,g a i n+1 i n+1 } with i j i k if j k,

8 8 L. AROSIO AND E. F. WOLD we have that n+1 l=1 Z(g a i l i l ) =. (3.5) Proof. Denote by δ (p) the disc of center p and radius δ, and by n R n the ball of radius R centered at the origin. Fix R > 0 and fix I := (i 1,...,i n+1 ) where i j {1,...,N} and i j i k if j k. The result immediately follows by the Baire lemma if we prove that the set A I,R := {a N : n+1 l=1 Z(g a i l i l ) n R = } is dense (it is obviously open). Let thus a = (a 1,...,a N ) N be an arbitrary point. The set consists of a finite number of irreducible components, each of dimension n 1. Z(g a i 1 i 1 ) n R Choosing a i δ (a i ) such that g a i i is not identically zero on the regular part of any of these components, we get that Z(g a i 1 i 1 ) Z(g a i i ) n R (3.6) has a finite number of irreducible components, each of dimension at most n, where we have set a i 1 = a i1. Choosing a i 3 δ (a i3 ) such that g a i 3 i 3 is not identically zero on any of these components, and continuing in this fashion, we obtain a collection of points {a i l },l = 1,...,n+1, such that n+1 l=1 Z(g a i l i l ) n R =, (3.7) and such that a i l δ (a il ). We are thus left to show that we may approximate fε arbitrarily well in Ck -norm on I J α by holomorphic automorphisms G α of n. We say that two smooth totally real polynomially convex embeddings f,g: Iα J n are connected if there exists a smooth isotopy of embeddings (G t : Iα J n ) t [0,1] such that G 0 = f, G 1 = g and such that G t (Iα J ) is totally real and polynomially convex for all t. If (h k : Iα J n ) 0 k M is a finite family of smooth totally real polynomially convex embeddings such that h 0 is the identity map id I J α and h k is connected to h k+1 for all 0 k M 1, then by the proof of [, Main Theorem] it follows that h M is approximable by automorphisms. There exists η > 0 small enough such that the embedding x fε (ηx) defined on IJ α can be written as f ǫ(ηx) = ρ(β(x)), where β : I J α Êd is the orientation preserving diffemorphism β(x) := π Ê d(f ǫ (ηx)) and where ρ is a graph map ρ(x) := (x,γ(x)), where Dγ(0) = 0 and γ is c-lipschitz with c > 0 small enough to obtain that for all t [0,1] the image of β(i J α) via the graph map ρ t (x) := (x,tγ(x)) is totally real and polynomially convex (we use [11, Proposition 4.]). We have that

9 POLYNOMIALLY CONVEX HULLS 9 (1) since β is orientation preserving, the identity map id I J α is connected to β via an isotopy with values in Ê d, () the isotopy ρ t (β(x)) connects β to the embedding x f ǫ (ηx), (3) the embedding f ǫ (ηx) is connected to f ε (x) via the isotopy f ε((η +t(1 η))x). Second Step: For simplicity we assume that M is without boundary. Let I 1,...,I N be a cover of M by closed cubes, such that there is a collection of subcubes Ii 0 int(i i ) which also cover M. For each i let χ i be a cutoff function compactly supported in int(i i ) with χ i 1 in a neighborhood of Ii 0. Let U j U j+1 U be open sets for j = 1,...,N, and let Ω be given by Proposition 3. for all the data (f(m),u j+1,u j ) with j = 1,...,N. We will proceed to perturb the image f(m) cube by cube. Assume that we have constructed a small C k -perturbation f j : M n of f such that h(k f j ( j i=1 I0 i )) U j (it will be clear from the construction how to construct f 1 ). By Step 1 we let g j+1 : I j+1 n be a small C k -perturbation of f j such that h(k f j ( j i=1 I0 i ) g j+1(i j+1 )) Ω U j, and g j+1 coincides with f j in a neighbourhood of j i=1 I0 i f 1 j (K). Defining f j+1 by f j+1 := f j +χ j+1 (g j+1 f j ), we obtain h(k f j+1 ( j+1 i=1 I0 i)) U j+1. If all the perturbations were small enough, defining f ε := f N yields the result. 4. Proof of Theorem 1.4 Let ε > 0. Let (U j ) j 1 be a sequence of open neighborhoods of K such that U j+1 U j for all j 1 and K = j 1 U j. Set f 0 := f. We perturb f 0 inductively using Lemma 3.1, obtaining a family of totally real C embeddings (f j : M n ) j Æ such that, (1) for all j 1 we have K fj (M) U j f j (M), () for all j Æ we have f j+1 f j C k+j (M) < η j, where the sequence (η j ) satisfies for all l Æ, η j < δ l, j=l where 0 < δ 0 ε is to be chosen and for all l 1, δ l is the constant c(f l,k,u l ) given by Lemma 3.3, (3) for all j Æ we have f j+1 f j on f 1 (K).

10 10 L. AROSIO AND E. F. WOLD The sequence (f j ) clearly converges to a C map f ε : M n which coincides with f on f 1 (K), and such that f f ε C k (M) < δ 0. If δ 0 is small enough, then f ε is a totally real C embedding. We claim that for all l 1, K f ε (M) U l f ε (M). This follows from Lemma 3.3, since f ε f l C 1 (M) η j < δ l. j=l References 1. H. Alexander, Hulls of subsets of the torus in, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, F. Forstnerič, Approximation by automorphisms on smooth submanifolds of n, Math. Ann. 300 (1994), F. Forstnerič, J.-P. Rosay, Approximation of biholomorphic mappings by automorphisms of n, Invent. Math. 11 (1993), no., P. Gupta, A nonpolynomially convex isotropic two-torus with no attached discs, arxiv: M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33 (1976). 6. A. Mukherjee, Differential Topology, Birkhäuser Basel (015). 7. P. T. Ho, H. Jacobowitz, P. Landweber, Optimality for totally real immersions and independent mappings of manifolds into N, New York J. Math. 18 (01) A. Izzo, H. Samuelsson Kalm, E. F. Wold, Presence or absence of analytic structure in maximal ideal spaces, Math. Ann. 366 (016), no. 1-, A. Izzo, E. L. Stout, Hulls of surfaces, Indiana U. Math. J. (to appear). 10. E. Løw, E. F. Wold, Polynomial convexity and totally real manifolds, Complex Var. Elliptic 54 (009), Nos. 3 4, P. E. Manne, E. F. Wold, N. Øvrelid, Holomorphic convexity and Carleman approximation by entire functions on Stein manifolds, Math. Ann 351 (011), E. L. Stout, Polynomial convexity, Progress in Mathematics, 61 Birkhäuser Boston, Inc., Boston, MA, 007. L. Arosio: Dipartimento Di Matematica, Università di Roma Tor Vergata, Via Della Ricerca Scientifica 1, 00133, Roma, Italy address: arosio@mat.uniroma.it E. F. Wold: Department of Mathematics, University of Oslo, Postboks 1053 Blindern, NO-0316 Oslo, Norway address: erlendfw@math.uio.no