scholarly journals Singular sets of holonomy maps for algebraic foliations

2013 ◽  
Vol 15 (3) ◽  
pp. 1067-1099 ◽  
Author(s):  
Gabriel Calsamiglia ◽  
Bertrand Deroin ◽  
Sidney Frankel ◽  
Adolfo Guillot
Keyword(s):  
Singular Sets ◽  
Algebraic Foliations

scholarly journals Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

2021 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Map ◽  
Local Regularity ◽  
Yang Mills ◽  
Heat Flows ◽  
Singular Sets ◽  
Regularity Results ◽  
Local Monotonicity ◽  
Partial Differ ◽  
Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

scholarly journals Some remarks on regular foliations with numerically trivial canonical class

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Stéphane Druel
Keyword(s):  
Small Rank ◽  
Codimension Two ◽  
Canonical Class ◽  
Special Cases ◽  
Algebraicity Criterion ◽  
Projective Manifolds ◽  
Complex Projective ◽  
Algebraic Foliations

In this article, we first describe codimension two regular foliations with numerically trivial canonical class on complex projective manifolds whose canonical class is not numerically effective. Building on a recent algebraicity criterion for leaves of algebraic foliations, we then address regular foliations of small rank with numerically trivial canonical class on complex projective manifolds whose canonical class is pseudo-effective. Finally, we confirm the generalized Bondal conjecture formulated by Beauville in some special cases. Comment: 20 pages

Analytic Functions with an Irregular Linearly Measurable Set of Singular Points

1952 ◽  
Vol 4 ◽  
pp. 424-435 ◽  
Author(s):  
I. E. Glover
Keyword(s):  
Imaginary Part ◽  
Analytic Functions ◽  
Singular Points ◽  
Definite Integrals ◽  
Singular Sets ◽  
Dense Point ◽  
Point Set ◽  
Dense Point Set ◽  
Nowhere Dense Set ◽  
Dense Set

V. V. Golubev, in his study [6], has constructed, by using definite integrals, various examples of analytic functions having a perfect nowhere-dense set of singular points. These functions were shown to be single-valued with a bounded imaginary part. In attempting to extend his work to the problem of constructing analytic functions having perfect, nowhere-dense singular sets under quite general conditions, he posed the following question: Given an arbitrary, perfect, nowhere-dense point-set E of positive measure in the complex plane, is it possible to construct, by passing a Jordan curve through E and by using definite integrals, an example of a single-valued analytic function, which has E as its singular set, with its imaginary part bounded.

Non-differentiable functionals and singular sets of minima

2005 ◽  
Vol 340 (1) ◽  
pp. 93-98 ◽  
Author(s):  
Jan Kristensen ◽  
Giuseppe Mingione
Keyword(s):  
Singular Sets

Nodal sets and horizontal singular sets of ℍ-harmonic functions on the Heisenberg group

2014 ◽  
Vol 16 (04) ◽  
pp. 1350049 ◽  
Author(s):  
Long Tian ◽  
Xiaoping Yang
Keyword(s):  
Heisenberg Group ◽  
Geometric Structure ◽  
Harmonic Functions ◽  
Nodal Sets ◽  
Singular Sets ◽  
Definition Of

In this paper, we give measure estimates of nodal sets of ℍ-harmonic functions on the Heisenberg group ℍn. We also introduce a definition of horizontal singular sets and show the geometric structure of the horizontal singular sets of ℍ-harmonic functions.

scholarly journals Singular Sets of Some Kleinian Groups (II)

1967 ◽  
Vol 29 ◽  
pp. 145-162 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Kleinian Groups ◽  
Singular Set ◽  
Fundamental Domains ◽  
Dimensional Measure ◽  
Singular Sets ◽  
Discontinuous Groups ◽  
Positive Capacity ◽  
Automorphic Functions

In the theory of automorphic functions it is important to investigate the properties of the singular sets of the properly discontinuous groups. But we seem to know nothing about the size or structure of the singular sets of Kleinian groups except the results due to Myrberg and Akaza [1], which state that the singular set has positive capacity and there exist Kleinian groups whose singular sets have positive 1-dimensional measure. In our recent paper [2], we proved the existence of Kleinian groups with fundamental domains bounded by five circles whose singular sets have positive 1-dimensional measure and presented the problem whether there exist or not such groups in the case of four circles. The purpose of this paper is to solve this problem. Here we note that, by Schottky’s condition [4], the 1-dimensional measure of the singular set is always zero in the case of three circles.

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