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The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications Mei-Chi Shaw University of Notre Dame Joint work with Siqi Fu and Christine Laurent-Thi ebaut 2018 Taipei Conference on Geometric Invariance and Partial Differential

The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications Mei-Chi Shaw University of Notre Dame Joint work with Siqi Fu and Christine Laurent-Thi´ ebaut 2018 Taipei Conference on Geometric Invariance and Partial Differential Equations Academia Sinica
Outline The ∂ -problem and Dolbeault cohomology groups 1 The Strong Oka’s Lemma 2 Dolbeault cohomology on annuli 3 Solution to the Chinese Coin Problem 4 The Cauchy-Riemann Equations in Complex Projective Spaces 5 Non-closed Range Property for Some smooth bounded Stein Domain 6 Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 2 / 34
Outline The ∂ -problem and Dolbeault cohomology groups 1 The Strong Oka’s Lemma 2 Dolbeault cohomology on annuli 3 Solution to the Chinese Coin Problem 4 The Cauchy-Riemann Equations in Complex Projective Spaces 5 Non-closed Range Property for Some smooth bounded Stein Domain 6 Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 3 / 34
Inhomogeneous Cauchy-Riemann equations The ∂ -problem Let Ω be a domain in C n (or a complex manifold), n ≥ 2. Given a ( p , q ) -form g such that ∂ g = 0, find a ( p , q − 1 ) -form u such that ∂ u = g . If g is in C ∞ p , q (Ω) (or g ∈ C ∞ p , q (Ω) ), one seeks u ∈ C ∞ p , q − 1 (Ω) (or u ∈ C ∞ p , q − 1 (Ω)) . Dolbeault Cohomology ker { ∂ : C ∞ p , q (Ω) → C ∞ p , q + 1 (Ω) } H p , q (Ω) = ( H p , q (Ω)) range { ∂ : C ∞ p , q − 1 (Ω) → C ∞ p , q (Ω) } Obstruction to solving the ∂ -problem on Ω . echet topologies on ker ( ∂ ) and Natural topology arising as quotients of Fr´ range ( ∂ ) . This topology is Hausdorff iff range ( ∂ ) is closed in C ∞ p , q (Ω) Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 4 / 34
Inhomogeneous Cauchy-Riemann equations The ∂ -problem Let Ω be a domain in C n (or a complex manifold), n ≥ 2. Given a ( p , q ) -form g such that ∂ g = 0, find a ( p , q − 1 ) -form u such that ∂ u = g . If g is in C ∞ p , q (Ω) (or g ∈ C ∞ p , q (Ω) ), one seeks u ∈ C ∞ p , q − 1 (Ω) (or u ∈ C ∞ p , q − 1 (Ω)) . Dolbeault Cohomology ker { ∂ : C ∞ p , q (Ω) → C ∞ p , q + 1 (Ω) } H p , q (Ω) = ( H p , q (Ω)) range { ∂ : C ∞ p , q − 1 (Ω) → C ∞ p , q (Ω) } Obstruction to solving the ∂ -problem on Ω . echet topologies on ker ( ∂ ) and Natural topology arising as quotients of Fr´ range ( ∂ ) . This topology is Hausdorff iff range ( ∂ ) is closed in C ∞ p , q (Ω) Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 4 / 34
Inhomogeneous Cauchy-Riemann equations The ∂ -problem Let Ω be a domain in C n (or a complex manifold), n ≥ 2. Given a ( p , q ) -form g such that ∂ g = 0, find a ( p , q − 1 ) -form u such that ∂ u = g . If g is in C ∞ p , q (Ω) (or g ∈ C ∞ p , q (Ω) ), one seeks u ∈ C ∞ p , q − 1 (Ω) (or u ∈ C ∞ p , q − 1 (Ω)) . Dolbeault Cohomology ker { ∂ : C ∞ p , q (Ω) → C ∞ p , q + 1 (Ω) } H p , q (Ω) = ( H p , q (Ω)) range { ∂ : C ∞ p , q − 1 (Ω) → C ∞ p , q (Ω) } Obstruction to solving the ∂ -problem on Ω . echet topologies on ker ( ∂ ) and Natural topology arising as quotients of Fr´ range ( ∂ ) . This topology is Hausdorff iff range ( ∂ ) is closed in C ∞ p , q (Ω) Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 4 / 34
Inhomogeneous Cauchy-Riemann equations The ∂ -problem Let Ω be a domain in C n (or a complex manifold), n ≥ 2. Given a ( p , q ) -form g such that ∂ g = 0, find a ( p , q − 1 ) -form u such that ∂ u = g . If g is in C ∞ p , q (Ω) (or g ∈ C ∞ p , q (Ω) ), one seeks u ∈ C ∞ p , q − 1 (Ω) (or u ∈ C ∞ p , q − 1 (Ω)) . Dolbeault Cohomology ker { ∂ : C ∞ p , q (Ω) → C ∞ p , q + 1 (Ω) } H p , q (Ω) = ( H p , q (Ω)) range { ∂ : C ∞ p , q − 1 (Ω) → C ∞ p , q (Ω) } Obstruction to solving the ∂ -problem on Ω . echet topologies on ker ( ∂ ) and Natural topology arising as quotients of Fr´ range ( ∂ ) . This topology is Hausdorff iff range ( ∂ ) is closed in C ∞ p , q (Ω) Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 4 / 34
Inhomogeneous Cauchy-Riemann equations The ∂ -problem Let Ω be a domain in C n (or a complex manifold), n ≥ 2. Given a ( p , q ) -form g such that ∂ g = 0, find a ( p , q − 1 ) -form u such that ∂ u = g . If g is in C ∞ p , q (Ω) (or g ∈ C ∞ p , q (Ω) ), one seeks u ∈ C ∞ p , q − 1 (Ω) (or u ∈ C ∞ p , q − 1 (Ω)) . Dolbeault Cohomology ker { ∂ : C ∞ p , q (Ω) → C ∞ p , q + 1 (Ω) } H p , q (Ω) = ( H p , q (Ω)) range { ∂ : C ∞ p , q − 1 (Ω) → C ∞ p , q (Ω) } Obstruction to solving the ∂ -problem on Ω . echet topologies on ker ( ∂ ) and Natural topology arising as quotients of Fr´ range ( ∂ ) . This topology is Hausdorff iff range ( ∂ ) is closed in C ∞ p , q (Ω) Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 4 / 34
Inhomogeneous Cauchy-Riemann equations The ∂ -problem Let Ω be a domain in C n (or a complex manifold), n ≥ 2. Given a ( p , q ) -form g such that ∂ g = 0, find a ( p , q − 1 ) -form u such that ∂ u = g . If g is in C ∞ p , q (Ω) (or g ∈ C ∞ p , q (Ω) ), one seeks u ∈ C ∞ p , q − 1 (Ω) (or u ∈ C ∞ p , q − 1 (Ω)) . Dolbeault Cohomology ker { ∂ : C ∞ p , q (Ω) → C ∞ p , q + 1 (Ω) } H p , q (Ω) = ( H p , q (Ω)) range { ∂ : C ∞ p , q − 1 (Ω) → C ∞ p , q (Ω) } Obstruction to solving the ∂ -problem on Ω . echet topologies on ker ( ∂ ) and Natural topology arising as quotients of Fr´ range ( ∂ ) . This topology is Hausdorff iff range ( ∂ ) is closed in C ∞ p , q (Ω) Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 4 / 34
Inhomogeneous Cauchy-Riemann equations The ∂ -problem Let Ω be a domain in C n (or a complex manifold), n ≥ 2. Given a ( p , q ) -form g such that ∂ g = 0, find a ( p , q − 1 ) -form u such that ∂ u = g . If g is in C ∞ p , q (Ω) (or g ∈ C ∞ p , q (Ω) ), one seeks u ∈ C ∞ p , q − 1 (Ω) (or u ∈ C ∞ p , q − 1 (Ω)) . Dolbeault Cohomology ker { ∂ : C ∞ p , q (Ω) → C ∞ p , q + 1 (Ω) } H p , q (Ω) = ( H p , q (Ω)) range { ∂ : C ∞ p , q − 1 (Ω) → C ∞ p , q (Ω) } Obstruction to solving the ∂ -problem on Ω . echet topologies on ker ( ∂ ) and Natural topology arising as quotients of Fr´ range ( ∂ ) . This topology is Hausdorff iff range ( ∂ ) is closed in C ∞ p , q (Ω) Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 4 / 34
L 2 -approach to ∂ Two ways to close an unbounded operator in L 2 (1) The (weak) maximal closure of ∂ : Realize ∂ as a closed densely defined (maximal) operator ∂ : L 2 p , q (Ω) → L 2 p , q + 1 (Ω) . The L 2 -Dolbeault Coholomolgy is defined by ker { ∂ : L 2 p , q (Ω) → L 2 p , q + 1 (Ω) } H p , q L 2 (Ω) = range { ∂ : L 2 p , q − 1 (Ω) → L 2 p , q (Ω) } (2) The (strong) minimal closure of ∂ : Let ∂ c be the (strong) minimal closed L 2 extension of ∂ . ∂ c : L 2 p , q (Ω) → L 2 p , q + 1 (Ω) . By this we mean that f ∈ Dom ( ∂ c ) if and only if there exists a sequence of compactly supported smooth forms f ν such that f ν → f and ∂ f ν → ∂ f . Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 5 / 34
L 2 -approach to ∂ Two ways to close an unbounded operator in L 2 (1) The (weak) maximal closure of ∂ : Realize ∂ as a closed densely defined (maximal) operator ∂ : L 2 p , q (Ω) → L 2 p , q + 1 (Ω) . The L 2 -Dolbeault Coholomolgy is defined by ker { ∂ : L 2 p , q (Ω) → L 2 p , q + 1 (Ω) } H p , q L 2 (Ω) = range { ∂ : L 2 p , q − 1 (Ω) → L 2 p , q (Ω) } (2) The (strong) minimal closure of ∂ : Let ∂ c be the (strong) minimal closed L 2 extension of ∂ . ∂ c : L 2 p , q (Ω) → L 2 p , q + 1 (Ω) . By this we mean that f ∈ Dom ( ∂ c ) if and only if there exists a sequence of compactly supported smooth forms f ν such that f ν → f and ∂ f ν → ∂ f . Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 5 / 34
Closed range property for pseudoconvex domains in C n H¨ ormander 1965 If Ω ⊂⊂ C n is bounded and pseudoconvex, then H p , q L 2 (Ω) = 0 , q � = 0 . (Kohn) Sobolev estimates for the ∂ -problem Let Ω be a bounded pseudoconvex domain in C n with smooth boundary. Then H p , q W s (Ω) = 0 , s ∈ N . Kohn 1963, 1974 If Ω ⊂⊂ C n is bounded and pseudoconvex with smooth boundary, then H p , q (Ω) = 0 , q � = 0 . Mei-Chi Shaw (Notre Dame) The Cauchy-Riemann Equation on Pseudoconcave Domains with Applications January 17-20, 2018 6 / 34

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