An Equivariant Setting of the Morse Theory

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

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Topology of Subvarieties of Complex Semi-abelian Varieties

International Mathematics Research Notices ◽

10.1093/imrn/rnaa242 ◽

2020 ◽

Author(s):

Yongqiang Liu ◽

Laurentiu Maxim ◽

Botong Wang

Keyword(s):

Euler Characteristic ◽

Morse Theory ◽

Characteristic Property ◽

Abelian Varieties ◽

Betti Numbers ◽

Local Systems ◽

Affine Manifolds ◽

Cyclic Covers ◽

Rank One ◽

Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

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Three Solutions for a Neumann Partial Differential Inclusion Via Nonsmooth Morse Theory

Set-Valued and Variational Analysis ◽

10.1007/s11228-016-0387-2 ◽

2016 ◽

Vol 25 (2) ◽

pp. 405-425 ◽

Cited By ~ 2

Author(s):

Francesca Colasuonno ◽

Antonio Iannizzotto ◽

Dimitri Mugnai

Keyword(s):

Differential Inclusion ◽

Morse Theory ◽

Three Solutions ◽

Partial Differential ◽

Partial Differential Inclusion ◽

Nonsmooth Morse Theory

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Periodic Problems with the Scalar p-Laplacian Resonant at any Eigenvalue via Critical Point Methods

Advanced Nonlinear Studies ◽

10.1515/ans-2013-0309 ◽

2013 ◽

Vol 13 (3) ◽

Author(s):

Sophia Th. Kyritsi ◽

Donal O’ Regan ◽

Nikolaos S. Papageorgiou

Keyword(s):

Critical Point ◽

Morse Theory ◽

Critical Point Theory ◽

Smooth Solution ◽

Point Theory ◽

Reaction Term ◽

Variational Techniques ◽

Periodic Problems

AbstractWe consider nonlinear periodic problems driven by the scalar p-Laplacian with a Carathéodory reaction term. Under conditions which permit resonance at infinity with respect to any eigenvalue, we show that the problem has a nontrivial smooth solution. Our approach combines variational techniques based on critical point theory with Morse theory.

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