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Monge-Ampère equations and surfaces with negative Gaussian curvature

Mikio Tsuji (1997)

Banach Center Publications

In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?...

Monge-Ampère Equations, Geodesics and Geometric Invariant Theory

D.H. Phong, Jacob Sturm (2005)

Journées Équations aux dérivées partielles

Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....

Monopole metrics and the orbifold Yamabe problem

Jeff A. Viaclovsky (2010)

Annales de l’institut Fourier

We consider the self-dual conformal classes on n # ℂℙ 2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3 -space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds)....

Morales-Ramis Theorems via Malgrange pseudogroup

Guy Casale (2009)

Annales de l’institut Fourier

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

More on deformed oscillator algebras and extended umbral calculus

Kwaśniewski, A. K., Grądzka, E. (2003)

Proceedings of the 22nd Winter School "Geometry and Physics"

This paper deals with ϕ ( q ) calculus which is an extension of finite operator calculus due to Rota, and leading results of Rota’s calculus are easily ϕ -extendable. The particular case ϕ n ( q ) = [ n q 1 ] - 1 is known to be relevant for quantum group investigations. It is shown here that such ϕ ( q ) umbral calculus leads to infinitely many new ϕ -deformed quantum like oscillator algebra representations. The authors point to several references dealing with new applications of q umbral and ϕ ( q ) calculus in which new families of ϕ ( q ) extensions...

Morse index and bifurcation of p-geodesics on semi Riemannian manifolds

Monica Musso, Jacobo Pejsachowicz, Alessandro Portaluri (2007)

ESAIM: Control, Optimisation and Calculus of Variations

Given a one-parameter family { g λ : λ ∈ [ a , b ] } of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials { V λ : λ ∈ [ a , b ] } and a family { σ λ : λ ∈ [ a , b ] } of trajectories connecting two points of the mechanical system defined by ( g λ , V λ ) , we show that there are trajectories bifurcating from the trivial branch σ λ if the generalized Morse indices μ ( σ a ) and μ ( σ b ) are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate...

Motion by curvature of planar networks

Carlo Mantegazza, Matteo Novaga, Vincenzo Maria Tortorelli (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two–dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which...

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