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Global structure of holomorphic webs on surfaces

Vincent Cavalier, Daniel Lehmann (2008)

Banach Center Publications

The webs have been studied mainly locally, near regular points (see a short list of references on the topic in the bibliography). Let d be an integer ≥ 1. A d-web on an open set U of ℂ² is a differential equation F(x,y,y’) = 0 with F ( x , y , y ' ) = ∑ i = 0 d a i ( x , y ) ( y ' ) d - i , where the coefficients a i are holomorphic functions, a₀ being not identically zero. A regular point is a point (x,y) where the d roots in y’ are distinct (near such a point, we have locally d foliations mutually transverse to each other, and caustics appear through...

Gradient estimates and Harnack inequalities for solutions to the minimal surface equation

Mario Miranda (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A gradient estimate for solutions to the minimal surface equation can be proved by Partial Differential Equations methods, as in [2]. In such a case, the oscillation of the solution controls its gradient. In the article presented here, the estimate is derived from the Harnack type inequality established in [1]. In our case, the gradient is controlled by the area of the graph of the solution or by the integral of it. These new results are similar to the one announced by Ennio De Giorgi in [3].

Gradient estimates for inverse curvature flows in hyperbolic space

Julian Scheuer (2015)

Geometric Flows

We prove gradient estimates for hypersurfaces in the hyperbolic space Hn+1, expanding by negative powers of a certain class of homogeneous curvature functions F. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers p > 1 of F-1 and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface....

Gradient horizontal de fonctions polynomiales

Si Tiep Dinh, Krzysztof Kurdyka, Patrice Orro (2009)

Annales de l’institut Fourier

Nous étudions les trajectoires du gradient sous-riemannien (appellé horizontal) de fonctions polynômes. Dans ce cadre l’inégalité de Łojasiewicz n’est pas valide et une trajectoire du gradient horizontal peut être de longueur infinie, et peut même s’accumuler sur une courbe fermée. Nous montrons que ces comportement sont exceptionnels ; et que, pour une fonction générique les trajectoires de son gradient horizontal ont des propriétés similaires au cas du gradient riemannien. Pour obtenir la finitude...

Graph selectors and viscosity solutions on Lagrangian manifolds

David McCaffrey (2006)

ESAIM: Control, Optimisation and Calculus of Variations

Let Λ be a Lagrangian submanifold of T * X for some closed manifold X. Let S ( x , ξ ) be a generating function for Λ which is quadratic at infinity, and let W(x) be the corresponding graph selector for Λ , in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset X 0 ⊂ X of measure zero such that W is Lipschitz continuous on X, smooth on X ∖ X 0 and ( x , ∂ W / ∂ x ( x ) ) ∈ Λ for X ∖ X 0 . Let H(x,p)=0 for ( x , p ) ∈ Λ . Then W is a classical solution to H ( x , ∂ W / ∂ x ( x ) ) = 0 on X ∖ X 0 and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational...

Grassmann manifold V 3 4 in the projective space P 7 with characteristics consisting of a quadric and two planes

Josef Vala (1993)

Mathematica Bohemica

Some results in the geometry of four-parametric manifolds of three-dimensional spaces in the projective space P 7 are found. The properties of such a manifold V 3 4 with characteristics consisting of a quadric and two planes are studied. The properties of the manifold dual to V 3 4 are found. Some results in the geometry of linear spaces from [1],[2],[3],[4] are used. The notation of the quantities is the same as in [4].

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