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Award ID contains: 2054989

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  1. (, Transactions of the American Mathematical Society)
    We study regularity of solutions u u to ∂<#comment/> ¯<#comment/> u = f \overline \partial u=f on a relatively compact C 2 C^2 domain D D in a complex manifold of dimension n n , where f f is a ( 0 , q ) (0,q) form. Assume that there are either ( q + 1 ) (q+1) negative or ( n −<#comment/> q ) (n-q) positive Levi eigenvalues at each point of boundary ∂<#comment/> D \partial D . Under the necessary condition that a locally L 2 L^2 solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain 1 / 2 1/2 derivative when q = 1 q=1 and f f is in the Hölder–Zygmund space Λ<#comment/> r ( D ) \Lambda ^r( D) with r > 1 r>1 . For q > 1 q>1 , the same regularity for the solutions is achieved when ∂<#comment/> D \partial D is either sufficiently smooth or of ( n −<#comment/> q ) (n-q) positive Levi eigenvalues everywhere on ∂<#comment/> D \partial D . 
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    Free, publicly-accessible full text available March 1, 2026
  2. ; (, Mathematische Annalen)
    Free, publicly-accessible full text available February 1, 2026
  3. (, Journal of Mathematical Analysis and Applications)
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  4. ; (, The Journal of Geometric Analysis)
    We construct an injective map from the set of holomorphic equivalence classes of neighborhoods M of a compact complex manifold C into a finite dimension complex Euclidean space when the normal bundle of C in M is fixed and is either weakly negative or 2-positive. 
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  5. ; (, Michigan Mathematical Journal)
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  6. ; (, Arnold mathematical journal)
    We consider an embedded n-dimensional compact complex manifold in n+d dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of C in M is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in Mn+d having C as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors conditions arising from solutions to their cohomological equations. 
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