The Geometry of Differential Harnack Estimates

Helmensdorfer, Sebastian; Topping, Peter

Séminaire de théorie spectrale et géométrie, Tome 30 (2011-2012), pp. 77-89.

Dans cette note, nous espérons introduire rapidement les non-experts dans le monde des inégalités de Harnack différentielles, qui ont eu tant d’influence en analyse géométrique et en théorie des probabilités durant les dernières décennies. Au niveau le plus grossier, ce sont des inégalités d’apparence souvent mystérieuse, qui valent pour les solutions « positives » de certaines EDP paraboliques, et peuvent se vérifier rapidement en appliquant le principe du maximum. Dans cette note nous insistons sur la géométrie sous-jacente aux inégalités de Harnack, qui se révèlent souvent traduire la convexité d’un objet naturel. En guise d’application, nous expliquons comment l’inégalité de Harnack différentielle due à Hamilton pour le flot de la courbure moyenne d’une sous-variété de dimension $n$ de ${ℝ}^{n+1}$, peut se voir comme une conséquence directe de la préservation bien connue de la convexité par le flot de la courbure moyenne, mais cette fois d’une sous-variété de dimension $n+1$ de ${ℝ}^{n+2}$. Nous passons également brièvement en revue les travaux antérieurs qui nous ont amenés à ces observations.

In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for ‘positive’ solutions of some parabolic PDE, and can be verified quickly by grinding out a computation and applying a maximum principle. In this note we emphasise the geometry behind the Harnack inequalities, which typically turn out to be assertions of the convexity of some natural object. As an application, we explain how Hamilton’s Differential Harnack inequality for mean curvature flow of a $n$-dimensional submanifold of ${ℝ}^{n+1}$ can be viewed as following directly from the well-known preservation of convexity under mean curvature flow, but this time of a $(n+1)$-dimensional submanifold of ${ℝ}^{n+2}$. We also briefly survey the earlier work that led us to these observations.

Classification : 53C44, 35K05, 35K08, 35K10, 35K55
Keywords: differential Harnack estimates, mean curvature flow, heat equation, log convexity, canonical solitons, self-similar solutions.
Mot clés : inégalités de Harnack différentielles, flot de la courbure moyenne, équation de la chaleur, log-convexité, solitons canoniques, solutions autosimilaires ?

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