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International audience; We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.

We prove conjectures of René Thom and Vladimir Arnold for C 2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even when they are not. If so, this would explain various conjectured phenomena., National Science Foundation (U.S.) (Grants DMS 1404540), National Science Foundation (U.S.) (Grants DMS 1206827), National Science Foundation (U.S.) (Grants DMS 1707270)

Thom's Gradient Conjecture states that a solution γ of an analytic gradient vector field X approaching to a singularity P of X has a tangent at P. A stronger version asserts that γ does not meet an analytic hypersurface an infinite number of times (it is non-oscillating). We prove, in dimension 3, that if γ is “infinitely near” an analytic curve Γ not composed of singularities of X, then γ is non-oscillating and, moreover, it does not spiral around Γ in a precise sense.

We show that every subset of $\mathbb{R}^n$ definable in an o-minimal structure can be decomposed into a finite number of definable sets that are quasi-convex i.e. have comparable, up to a constant, the intrinsic distance and the distance induced from the embedding. We apply this result to study the limits of secants of the trajectories of gradient vector field $\nabla f$ of a $C^1$ definable function $f$ defined in an open subset of $\mathbb{R}^n$. We show that if the o-minimal structure is polynomially bounded then the limit of such secants exists, that is an analog of the gradient conjecture of R. Thom holds. Moreover we prove that for $n = 2$ the result is true in any o-minimal structure.

A theorem of Łojasiewicz asserts that any relatively compact solution of a real analytic gradient vector field has finite length. We show here a generalization of this result for relatively compact solutions of an analytic vector field X with a smooth invariant hypersurface, transversally hyperbolic for X, where the restriction of the field is a gradient. This solves some instances of R. Thom's Gradient Conjecture. Furthermore, if the dimension of the ambient space is three, these solutions do not oscillate (in the sense that they cut an analytic set only finitely many times); this can also be applied to some gradient vector fields.

Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x_0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t) at x_0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x_0 onto the unit sphere has finite length., Comment: 30 pages, published version