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A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities
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The canonical shrinking soliton associated to a Ricci flow
- Calc. Var
"... Abstract To every Ricci flow on a manifold M over a time interval I ⊂ R−, we associate a shrinking Ricci soliton on the space-time M×I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric constru ..."
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Cited by 8 (2 self)
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Abstract To every Ricci flow on a manifold M over a time interval I ⊂ R−, we associate a shrinking Ricci soliton on the space-time M×I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author
The Geometry of Differential Harnack Estimates
, 2013
"... 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 tha ..."
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Cited by 4 (1 self)
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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 wellknown 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. 1
Sharp differential estimates of Li-Yau-Hamilton type for positive (p, p)-forms on Kähler manifolds
- COMMUN. PURE APPL. MATH
, 2011
"... In this paper we study the heat equation (of Hodge Laplacian) deformation of.p; p/-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a.p; p/-form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the ..."
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Cited by 1 (1 self)
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In this paper we study the heat equation (of Hodge Laplacian) deformation of.p; p/-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a.p; p/-form solution is preserved under such an invariant condition, we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow and some interpolating matrix differential Harnack-type estimates for both the Kähler-Ricci flow and the Ricci flow.
