# Complex Variables

## New submissions

[ total of 6 entries: 1-6 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Kähler submanifolds of the symmetrized polydisc
Comments: To appear in Comptes Rendus Mathematique
Subjects: Complex Variables (math.CV); Differential Geometry (math.DG)

This paper proves the non-existence of common K\"ahler submanifolds of the complex Euclidean space and the symmetrized polydisc endowed with their canonical metrics.

[2]
Title: The invariant subspaces of the shift plus integer multiple of Volterra operator on Hardy spaces
Authors: Qingze Lin
Subjects: Complex Variables (math.CV)

\v{C}u\v{c}kovi\'{c} and Paudyal recently characterized the lattice of invariant subspaces of the shift plus a complex Volterra operator on the Hilbert space $H^2$ on the unit disk. Motivated by the idea of Ong, in this paper, we give a complete characterization of the lattice of invariant subspaces of the shift operator plus a positive integer multiple of the Volterra operator on Hardy spaces $H^p$, which essentially extends their works to the more general cases when $1\leq p<\infty$.

[3]
Title: Positive neighborhoods of curves
Authors: M. Falla Luza, P. Sad
Subjects: Complex Variables (math.CV)

In this work we study neighborhoods of curves in surfaces with positive self-intersection that can be embeeded as a germ of neighborhood of a curve on the projective plane.

### Cross-lists for Tue, 20 Mar 18

[4]  arXiv:1803.06472 (cross-list from math.DG) [pdf, ps, other]
Title: Transversely holomorphic branched Cartan geometry
Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)

Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension $d$ admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension $d$ endowed with a transversely flat branched complex projective geometry (equivalently, a ${\mathbb C}P^d$-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective rationally connected varieties and on compact simply connected Calabi-Yau manifolds are always flat (consequently, they are defined by holomorphic maps into homogeneous spaces).

[5]  arXiv:1803.06697 (cross-list from math.DG) [pdf, ps, other]
Title: Higher-order estimates for collapsing Calabi-Yau metrics
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Complex Variables (math.CV)

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In particular, as an intermediate step, we use such arguments to prove sharp new Schauder estimates for the Laplacian on cylinders. If the fibers of the submersion are pairwise biholomorphic, our method yields a uniform C^infinity estimate. We then apply these local results to the case of collapsing Calabi-Yau metrics on compact Calabi-Yau manifolds. In this global setting, the C^0 estimate required as a hypothesis in our new local C^alpha and C^infinity estimates is known to hold thanks to earlier work of the second-named author.

### Replacements for Tue, 20 Mar 18

[6]  arXiv:1711.08024 (replaced) [pdf, ps, other]
Title: New complex analytic methods in the theory of minimal surfaces: a survey
Comments: To appear in J. Aust. Math. Soc
Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)
[ total of 6 entries: 1-6 ]
[ showing up to 2000 entries per page: fewer | more ]

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