scholarly journals A Classification of Isolated Singularities of Elliptic Monge-Ampère Equations in Dimension Two

2014 ◽  
pp. n/a-n/a
Author(s):  
José A. Gálvez ◽  
Asun Jiménez ◽  
Pablo Mira
Keyword(s):  
Isolated Singularities ◽  
Monge Ampere

scholarly journals A Classification of Isolated Singularities of Elliptic Monge-Ampére Equations in Dimension Two

2015 ◽  
Vol 68 (12) ◽  
pp. 2085-2107 ◽  
Author(s):  
José A. Gálvez ◽  
Asun Jiménez ◽  
Pablo Mira
Keyword(s):  
Isolated Singularities ◽  
Monge Ampere

scholarly journals Isolated singularities of graphs in warped products and Monge–Ampère equations

2016 ◽  
Vol 260 (3) ◽  
pp. 2163-2189 ◽  
Author(s):  
José A. Gálvez ◽  
Asun Jiménez ◽  
Pablo Mira
Keyword(s):  
Warped Products ◽  
Isolated Singularities ◽  
Monge Ampere

Isolated Singularities of Monge-Ampere Equations

1995 ◽  
Vol 123 (12) ◽  
pp. 3705 ◽  
Author(s):  
Friedmar Schulz ◽  
Lihe Wang
Keyword(s):  
Isolated Singularities ◽  
Monge Ampere

Integrability of Dispersionless Hirota-Type Equations and the Symplectic Monge–Ampère Property

2020 ◽  
Author(s):  
E V Ferapontov ◽  
B Kruglikov ◽  
V Novikov
Keyword(s):  
Type Equation ◽  
Lax Pair ◽  
Complete Classification ◽  
Conformal Structure ◽  
Characteristic Variety ◽  
Self Duality ◽  
Symmetry Properties ◽  
Monge Ampere ◽  
Involutive System

Abstract We prove that integrability of a dispersionless Hirota-type equation implies the symplectic Monge–Ampère property in any dimension $\geq 4$. In 4D, this yields a complete classification of integrable dispersionless partial differential equations (PDEs) of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach, we derive an involutive system of relations characterizing symplectic Monge–Ampère equations in any dimension. Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linearizability of a Hirota-type equation via flatness of the corresponding conformal structure and study symmetry properties of integrable equations.

Stable mappings of discriminant varieties

1988 ◽  
Vol 103 (1) ◽  
pp. 69-82
Author(s):  
J. W. Bruce
Keyword(s):  
Wave Front ◽  
Smooth Surfaces ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Low Dimensions

Smooth mappings defined on discriminant varieties of -versal unfoldings of isolated singularities arise in many interesting geometrical contexts, for example when classifying outlines of smooth surfaces in ℝ3 and their duals, or wave-front evolution [1, 2, 5]. In three previous papers we have classified various stable mappings on discriminants. When the isolated singularity is weighted homogeneous the discriminant is not a local smooth product, and this makes the classification of stable germs considerably easier than in general. Moreover, discriminants arising from weighted homogeneous singularities predominate in low dimensions, so such classifications are very useful for applications.

scholarly journals On Monge–Ampère Volumes of Direct Images

2021 ◽  
Author(s):  
Siarhei Finski
Keyword(s):  
Vector Bundles ◽  
Ample Line Bundle ◽  
Symmetric Powers ◽  
Projectively Flat ◽  
Leading Term ◽  
Tensor Powers ◽  
Monge Ampere ◽  
Hermitian Structures

Abstract This paper is devoted to the study of the asymptotics of Monge–Ampère volumes of direct images associated with high tensor powers of an ample line bundle. We study the leading term of this asymptotics and provide a classification of bundles saturating the topological bound of Demailly. In the special case of high symmetric powers of ample vector bundles, this provides a characterization of those admitting projectively flat Hermitian structures.

scholarly journals ALGORITHMS FOR SINGULARITIES AND REAL STRUCTURES OF WEAK DEL PEZZO SURFACES

2014 ◽  
Vol 13 (05) ◽  
pp. 1350158
Author(s):  
NIELS LUBBES
Keyword(s):  
Equivalence Classes ◽  
Point Of View ◽  
Alternative Proof ◽  
Del Pezzo Surfaces ◽  
Isolated Singularities ◽  
Classification Of Singularities ◽  
Del Pezzo ◽  
Real Forms

In this paper, we consider the classification of singularities [P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction. I, II, III, Proc. Camb. Philos. Soc.30 (1934) 453–491] and real structures [C. T. C. Wall, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math.1987(375/376) (1987) 47–66, ISSN 0075-4102] of weak Del Pezzo surfaces from an algorithmic point of view. It is well-known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.

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