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The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions

Ursula Ludwig (2010)

Annales de l’institut Fourier

In a previous note the author gave a generalisation of Witten’s proof of the Morse inequalities to the model of a complex singular curve X and a stratified Morse function f . In this note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of f .

The gradient flow of Higgs pairs

Jiayu Li, Xi Zhang (2011)

Journal of the European Mathematical Society

We consider the gradient flow of the Yang–Mills–Higgs functional of Higgs pairs on a Hermitian vector bundle ( E , H 0 ) over a Kähler surface ( M , ω ) , and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition ( A 0 , φ 0 ) converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point ( A ∞ , φ ∞ ) of this functional. We also prove that the limiting Higgs pair ( A ∞ , φ ∞ ) can be extended smoothly to a vector bundle E ∞ over...

The rectifiable distance in the unitary Fredholm group

Esteban Andruchow, Gabriel Larotonda (2010)

Studia Mathematica

Let U c ( ) = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by d ∞ the rectifiable distance induced by the Finsler metric given by the operator norm in U c ( ) . If u ₀ , u ₁ , u ∈ U c ( ) and the geodesic β joining u₀ and u₁ in U c ( ) satisfy d ∞ ( u , β ) < π / 2 , then the map f ( s ) = d ∞ ( u , β ( s ) ) is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in U c ( ) is π/4. The same convexity property holds in the p-Schatten unitary groups U p ( ) = u: u unitary and u-1 in the p-Schatten class...

The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions

Jean Dolbeault, Maria J. Esteban, Gabriella Tarantello (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than - 1 . Without symmetry assumption, it holds if and only if the parameter is in the interval ( - 1 , 0 ] . The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of parameters (asymptotically...

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