Why Complex Analysis Pseudoconvex Domains: Where Holomorphic Functions Live Beautiful theory Applications to Pure Math (PDE’s, Geometry, Number Theory, . . . ) S¨ onmez S ¸ahuto˘ glu Applications to Applied Math (Fourier Analysis, Residue Theorem, Numerical Analysis, . . . ) University of Toledo Applications to other fields (Physics, Engineering, . . . ) Real Differentiable Functions Complex Numbers Complex Numbers: C = { x + iy : x , y ∈ R } where i 2 = − 1. Fundamental Theorem of Algebra: Every polynomial (with f : ( a , b ) → R is (real) differentiable at p ∈ ( a , b ) if the following complex coefficients) of degree n has n roots, counting multiplicity. limit exists f ( p + h ) − f ( p ) For example, z 2 + 1 has two roots ± i yet x 2 + 1 has no real roots. f ′ ( p ) = lim . h R ∋ h → 0 Euler’s Formula: e i θ = cos θ + i sin θ . Then �⇐ f is differentiable on ( a , b ) ⇒ f is continuous on ( a , b ). � cos( θ 1 + θ 2 ) + i sin( θ 1 + θ 2 ) = e i ( θ 1 + θ 2 ) x 2 sin(1 / x ) , x � = 0 x = 0 is differentiable on R but f �∈ C 2 ( R ). f ( x ) = = e i θ 1 e i θ 2 0 , =(cos θ 1 + i sin θ 1 )(cos θ 2 + i sin θ 2 ) In fact, C ( R ) � C 1 ( R ) � C 2 ( R ) � · · · � C ∞ ( R ). =(cos θ 1 cos θ 2 − sin θ 1 sin θ 2 ) + i (cos θ 1 sin θ 2 + cos θ 2 sin θ 1 ) .
Complex Differentiable Functions C -Differentiable Versus R -Differentiable Let F C be complex differentiable and f R be real differentiable. Then Let U ⊂ C be an open set, f : U → C be a function, and p ∈ U . F C is C ∞ -smooth but not necessarily f R . Then f is complex differentiable at p ∈ U if the following limit F C is analytic but not necessarily f R . exists f ( p + h ) − f ( p ) f ′ ( p ) = lim . F C has max modulus principle but not necessarily f R . h C ∋ h → 0 F C has integral representation formula (Cauchy integral f is complex differentiable (holomorphic) on U if it is complex formula) but not such thing exists for f R . differentiable at p for every p ∈ U . F C satisfies Cauchy-Riemann equations (a PDE): Fact: f is holomorphic if and only if f z = 0 (CR-equations). ( F C ) z = 0 ⇔ u x = v y and u y = − v x for F C = u + iv . Why Analysis in C n R -analytic versus C -analytic 1 1 + x 2 is real analytic on R . ∞ Range: “Could anyone seriously argue that it might be sufficient to � 1 ( − 1) k x 2 k converges for | x | < 1 only. 1 + x 2 = train a mathematics major in calculus of functions of one real k =0 variable without expecting him or her to learn at least something ∞ � about partial derivatives, multiple integrals, and some higher 1 ( − 1) k z 2 k converges for | z | < 1 only. 1 + z 2 = dimensional version of the Fundamental Theorem of Calculus? Of k =0 course not, the real world is not one-dimensional! But neither is 1 the complex world ...” 1 + z 2 is not defined at ± i . The obstruction for analyticity is “detectable” in the complex plane but not necessarily in R .
Differentiation in C n f : U ⊂ R n → R is differentiable at p ∈ U if there exists a linear “Aside from questions of applicability, shouldn’t the pure function T : R n → R such that mathematician’s mind wonder about the restriction to functions of only one complex variable? It should not surprise anyone that there | f ( p + h ) − f ( p ) − Th | lim = 0 . is a natural extension of complex analysis to the multivariable � h � R n ∋ h → 0 setting. What is surprising is the many new and intriguing phenomena that appear when one considers more than one Fact: If f and all of its partial derivatives are continuous then f is variable. Indeed, these phenomena presented major challenges to differentiable. any straightforward generalization of familiar theorems... ” Definition: f : U ⊂ C n → C is holomorphic ( C -analytic) if f ∈ C ( U ) and f z j = 0 for j = 1 , 2 , . . . , n . C versus C n Riemann Mapping Theorem � There are two non-conformal simply connected In C : domains: D and C . Holomorphic functions Cauchy integral formula and its consequences � There are infinitely many non-conformal In C n , n ≥ 2 : Identity principle simply connected domains. Riemann mapping theorem Domain on holomorphy � The unit ball and the unit bidisc in C 2 [Poincar´ e] are non-conformal.
Domain of Holomorphy Hartogs Phenomena A domain Ω ⊂ C n is a domain of holomorphy if for all p ∈ ∂ Ω [Hartogs] C 2 \ B (0 , 1) is not a domain of holomorphy. there exists F ∈ H (Ω) such that F has no holomorphic extension Sketch of Proof: Let f be holomorphic on Ω = C 2 \ B (0 , 1). through p . Define � Example: C \ { 0 } is a domain of holomorphy. 1 f ( z , ξ ) F ( z , w ) = w − ξ d ξ 2 π i Example: D = { z ∈ C : | z | < 1 } is a domain of holomorphy. In | ξ | =10 fact, the series, Then F is holomorphic on {| z | < ∞ , | w | < 10 } . ∞ � z 2 n Cauchy Integral Formula ⇒ f = F for {| w | < 5 , 2 < | z | < 3 } ⊂ Ω. 2 n n =1 Identity Principle ⇒ f = F where defined. has no extension through any boundary point of D . Therefore, F extends f as holomorphic onto B (0 , 1). Fact: In C every open set is a domain of holomorphy. Examples in C 2 The Levi Problem � Is there a geometric characterization Levi Problem: Example 1: D 2 is a domain of holomorphy. of domain of holomorphy? 1 1 f p ( z ) = if | p 1 | = 1 and f p ( z ) = if | p 2 | = 1 . z 1 − p 1 z 2 − p 2 [Oka, Norguet, Bremermann] Yes. It is pseudoconvexity. Example 2: B (0 , 1) is a domain of holomorphy. A smooth domain Ω ⊂ C n is said to be pseudoconvex if its Levi 1 f p ( z ) = z 1 p 1 + z 2 p 2 − 1 . form is nonnegative on complex tangential directions on boundary points, b Ω, of Ω.
Convexity Pseudoconvexity L A pseudoconvex domain is “convex L with respect to holomorphic image of complex discs”. (L is holomorphic U image of the unit disc) U U L Convexity ⇒ Pseudoconvexity [Kohn-Nirenberg] Pseudoconvex domains may not be convexifiable. A convex domain A non-convex domain pseudoconvex domain L is a linear image of the unit interval, [0,1]. Convex ⇒ Pseudoconvex Properties of Pseudoconvexity Intersection Ω 1 , Ω 2 are pseudoconvex ⇒ Ω 1 ∩ Ω 2 is pseudoconvex. Example 3: Convex Domains in C 2 . Let Ω be a convex domain and p ∈ ∂ Ω. Then Increasing Union [Behnke-Stein] S1: Ω is pseudoconvex ⇔ Ω p = Ω − p is pseudoconvex. � Ω θ ∞ � p = { ( z 1 e i θ 1 , z 2 e i θ 2 ) : z ∈ Ω p } Ω j ⊂ Ω j +1 are pseudoconvex for all j ⇒ Ω j is pseudoconvex. S2: Ω p is pseudoconvex ⇔ is pseudoconvex. j =1 S3: Choose θ so that the (real) normal for Ω θ p at 0 is along y 2 -axis. Product S4: Choose f ( z ) = 1 / z 2 . Then Ω θ p is pseudoconvex at 0 ⇒ Ω is pseudoconvex at p . Ω 1 , Ω 2 are pseudoconvex ⇒ Ω 1 × Ω 2 is pseudoconvex.
Equivalent Conditions Locality for every p ∈ ∂ Ω there exists Ω is pseudoconvex ⇔ r > 0 such that Ω ∩ B ( p , r ) Let Ω ⊂ C n be a domain. TFAE is pseudoconvex. Ω is a domain on holomorphy, 1 Ω is pseudoconvex, 2 there exists F ∈ H (Ω) with no Ω has a continuous plurisubharmonic exhaustion function: 3 Ω is pseudoconvex ⇔ holomorphic extension through { φ < c } ⋐ Ω where φ is continuous plurisubharmonic, any boundary point. Ω has an exhaustion of smooth pseudoconvex domains. 4 Hull Condition The ∂ -problem Let 1 ≤ q ≤ n . PSH (Ω): continuous plurisubharmonic functions on Ω We say ∂ is solvable on (0 , q )-forms if Given f ∈ C ∞ (0 , q ) (Ω) with ∂ f = 0 there exists u ∈ C ∞ (0 , q − 1) (Ω) with K : compact set in Ω � � � K = z ∈ Ω : φ ( z ) ≤ sup { φ ( w ) : w ∈ K } for any φ ∈ PSH (Ω) . ∂ u = f . Example: If K = S 1 then � K = D . Ω satisfies the hull condition: � � ∂ is solvable on (0 , q )-forms K ⋐ Ω whenever K ⋐ Ω. Ω is pseudoconvex ⇔ for all 1 ≤ q ≤ n . Ω is pseudoconvex ⇔ Ω satisfies the hull condition.
Conclusions Further Reading S. Krantz, Function Theory of Several Complex Variables. Reprint Several Complex Variables is very different from complex of the 1992 edition. AMS Chelsea Publishing, Providence, RI, 1 2001. analysis in one variable. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables has strong connections to PDE’s, 2 potential theory, geometry, and analysis. Several Complex Variables, vol. 108. Springer, New York (1986) S. Krantz, What is several complex variables , Amer. Math. Pseudoconvexity (the “home of holomorphic functions”) is a 3 Monthly 94 (1987) 236256 fundamental notion in Several Complex Variables and has many interesting properties. R. M. Range, Complex analysis: A brief tour into higher dimensions , Amer. Math. Monthly 110 (2003), 89–108.
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