| 1. |
- Andrist B., Rafael, et al.
(author)
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Holomorphic Automorphisms of Danielewski Surfaces II: Structure of the Overshear Group
- 2015
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 25:3, s. 1859-1889
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Journal article (peer-reviewed)abstract
- We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group, which is known to be dense in the identity component of the holomorphic automorphism group, is a free product.
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| 2. |
- Aldana, C. L., et al.
(author)
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A Polyakov Formula for Sectors
- 2018
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 28:2, s. 1773-1839
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Journal article (peer-reviewed)abstract
- We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.
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| 3. |
- Aldana, Clara L., et al.
(author)
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Correction to: A Polyakov Formula for Sectors
- 2020
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30:3, s. 3371-3372
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Journal article (other academic/artistic)abstract
- Let ?? denote a finite circular sector of opening angle ?∈(0,?) and radius one, and let ?−?Δ? denote the heat operator associated to the Dirichlet extension of the Laplacian. Based on recent joint work [2] and [3], we discovered an extra contribution to the variational Polyakov formula in [1] coming from the curved boundary component of the sector. Theorems 3 and 4 of [1] should have an added term +14?. This calculation will appear in [2]. The corrected statements of these theorems are given below.
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| 4. |
- Aldana, Clara L., et al.
(author)
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Correction to: A Polyakov Formula for Sectors (The Journal of Geometric Analysis, (2018), 28, 2, (1773-1839), 10.1007/s12220-017-9888-y)
- 2020
-
In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30, s. 3371-3372
-
Journal article (peer-reviewed)abstract
- Let ?? denote a finite circular sector of opening angle ?∈(0,?) and radius one, and let ?−?Δ? denote the heat operator associated to the Dirichlet extension of the Laplacian. Based on recent joint work [2] and [3], we discovered an extra contribution to the variational Polyakov formula in [1] coming from the curved boundary component of the sector. Theorems 3 and 4 of [1] should have an added term +14?. This calculation will appear in [2]. The corrected statements of these theorems are given below.
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| 5. |
- Ameur, Yacin
(author)
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Near-boundary asymptotics for correlation kernels
- 2013
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 23:1, s. 73-95
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Journal article (peer-reviewed)abstract
- We discuss asymptotic formulas for the correlation kernel corresponding to a (more or less) general potential Q in the plane. If Kn is such a kernel, there is a known asymptotic formula Kn(z, z)=nΔQ(z)+B1(z)+n-1B2(z)+⋯ valid for z in the interior of a certain compact set known as the "droplet" corresponding to Q (on which ΔQ>0). On the other hand, if z is in the exterior of the droplet, Kn(z, z) converges to zero exponentially in n. Results of this type are useful in random matrix theory and conformal field theory; they have recently been used to prove the Gaussian field convergence in the interior of the droplet for fluctuations of eigenvalues of random normal matrices. To be able to extend such results beyond the interior, it becomes necessary to have a certain uniformity of estimates as z approaches the boundary either from the interior or from the exterior. Such estimates have to our knowledge hitherto not been known on a rigorous level. This note intends to fill this gap. We will consider applications in later publications. Our treatment of the (interior) asymptotics relies in an essential way on previous work due to Berman, Berndtsson, and Sjöstrand (Ark. Mat. 46, 2008), and Berman (Indiana Univ. Math. J. 58, 2009). We hope that this note can to some extent be regarded as a contribution to that work.
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| 6. |
- Andersson, Mats, 1957, et al.
(author)
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Estimates for the ∂¯ -Equation on Canonical Surfaces
- 2020
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30:3, s. 2974-3001
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Journal article (peer-reviewed)abstract
- We study the solvability in Lp of the ∂¯ -equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case p= 2 for two natural closed extensions ∂¯ s and ∂¯ w of ∂¯. For ∂¯ s we have solvability, whereas for ∂¯ w there is solvability if and only if a certain boundary condition (∗) is fulfilled at the singularity. Our main tool is certain integral operators for solving ∂¯ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.
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| 7. |
- Andersson, Mats, 1957
(author)
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Global Koppelman Formulas on (Singular) Projective Varieties
- 2019
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 29:3, s. 2290-2317
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Journal article (peer-reviewed)abstract
- Let i: X→ PN be a projective manifold of dimension n embedded in projective space PN, and let L be the pullback to X of the line bundle OPN(1). We construct global explicit Koppelman formulas on X for smooth (0 , ∗) -forms with values in Ls for any s. The same construction works for singular, even non-reduced, X of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves AX∗ of (0 , ∗) -currents with mild singularities at Xsing. In particular, if s≥regX-1, where regX is the Castelnuovo–Mumford regularity, we get an explicit representation of the well-known vanishing of H,q(X, Ls-q) , q≥ 1. Also some other applications are indicated.
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| 8. |
- Aspenberg, Magnus, et al.
(author)
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Control of cancellations that restrain the growth of a binomial recursion
- 2015
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1559-002X .- 1050-6926. ; 25:3, s. 1666-1700
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Journal article (peer-reviewed)abstract
- We study a recursion that generates real sequences depending on a parameter x. Given a negative x the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a family of integral operators, and prove that for every parameter value except -1, the growth of the sequence is factorial. In the combinatorial part of the proof we show that when x=-1 the resulting recurrence yields the sequence of alternating Catalan numbers, and thus has exponential growth. We expect our methods to be useful in a variety of similar situations.
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| 9. |
- Backlund, Ulf, et al.
(author)
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Semi-Bloch Functions in Several Complex Variables
- 2016
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 26:1, s. 463-473
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Journal article (peer-reviewed)abstract
- Let M be an n-dimensional complex manifold. A holomorphic function f : M -> C is said to be semi-Bloch if for every lambda is an element of C the function g(lambda) = exp(lambda f(z)) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.
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| 10. |
- Bakas, Odysseas
(author)
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On a Problem of Pichorides
- 2021
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 31:7, s. 7455-7512
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Journal article (peer-reviewed)abstract
- Let S(Λ) denote the classical Littlewood–Paley operator formed with respect to a lacunary sequence Λ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of S(Λ) from the analytic Hardy space HAp(T) to Lp(T) and of the behaviour of the Lp(T) → Lp(T) operator norm of S(Λ) (1 < p< 2) in terms of the ratio of the lacunary sequence Λ. Namely, if ρΛ denotes the ratio of Λ , then we prove that sup‖f‖Lp(T)=1f∈HAp(T)‖S(Λ)(f)‖Lp(T)≲1p-1(ρΛ-1)-1/2(1
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