# The Geometry of Differential Harnack Estimates

Sebastian Helmensdorfer^{[1]}; Peter Topping^{[2]}

- [1] Sintzenichstr. 11, 81479 München, Germany
- [2] Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

Séminaire de théorie spectrale et géométrie (2011-2012)

- Volume: 30, page 77-89
- ISSN: 1624-5458

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topHelmensdorfer, Sebastian, and Topping, Peter. "The Geometry of Differential Harnack Estimates." Séminaire de théorie spectrale et géométrie 30 (2011-2012): 77-89. <http://eudml.org/doc/275753>.

@article{Helmensdorfer2011-2012,

abstract = {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 $\mathbb\{R\}^\{n+1\}$ 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 $\mathbb\{R\}^\{n+2\}$. We also briefly survey the earlier work that led us to these observations.},

affiliation = {Sintzenichstr. 11, 81479 München, Germany; Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK},

author = {Helmensdorfer, Sebastian, Topping, Peter},

journal = {Séminaire de théorie spectrale et géométrie},

keywords = {differential Harnack estimates; mean curvature flow; heat equation; log convexity; canonical solitons; self-similar solutions},

language = {eng},

pages = {77-89},

publisher = {Institut Fourier},

title = {The Geometry of Differential Harnack Estimates},

url = {http://eudml.org/doc/275753},

volume = {30},

year = {2011-2012},

}

TY - JOUR

AU - Helmensdorfer, Sebastian

AU - Topping, Peter

TI - The Geometry of Differential Harnack Estimates

JO - Séminaire de théorie spectrale et géométrie

PY - 2011-2012

PB - Institut Fourier

VL - 30

SP - 77

EP - 89

AB - 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 $\mathbb{R}^{n+1}$ 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 $\mathbb{R}^{n+2}$. We also briefly survey the earlier work that led us to these observations.

LA - eng

KW - differential Harnack estimates; mean curvature flow; heat equation; log convexity; canonical solitons; self-similar solutions

UR - http://eudml.org/doc/275753

ER -

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