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11
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 spacetime M×I. We relate properties of the original Ricci flow to properties of the new higherdimensional Ricci flow equipped with its own timeparameter. 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 spacetime M×I. We relate properties of the original Ricci flow to properties of the new higherdimensional Ricci flow equipped with its own timeparameter. 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 nonexperts 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 mysteriouslooking 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 nonexperts 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 mysteriouslooking 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 ndimensional 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 LiYauHamilton 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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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 LiYauHamilton) estimates for the positive solutions of the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the KählerRicci flow and some interpolating matrix differential Harnacktype estimates for both the KählerRicci flow and the Ricci flow.