Some gradient estimates for the heat equation on domains and for an equation by Perelman

By Qi S. Zhang. 2006

135 citations recorded by the source corpus.

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73e3ada1612a17991bf0cbf198d620244c8aa441

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Bing Wang
2. Yu Li
3. Xiuxiong Chen
4. Qi S. Zhang
5. Ben Andrews
6. Bruce Kleiner
7. Xiaodong Cao
8. A. Naber

In Mathematical Physics

1. Bing Wang
2. Ben Andrews
3. Xiaodong Cao
4. Albert Chau
5. Luen-Fai Tam
6. Chengjie Yu
7. Yu Li
8. Qi S. Zhang

Citations in the corpus

• The conjugate heat equation and Ancient solutions of the Ricci flow (2010, methodology): “In the paper [29], this kind of bound depending on the Sobolev constants of the manifolds (M, g(t)) was proven.”
• Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations (2014, background): “Various other authors have observed that the interaction of the heat equation and Ricci flow leads to good gradient estimates, see [24, 65, 78, 79].”
• Heat kernel on Ricci shrinkers (2019, background): “We also need the following gradient estimate from [61].”
• Gradient estimates for the heat equation under the Ricci flow (2009, methodology): “The list of relevant references includes but is not limited to [36, 5, 6] and [10, Chapter 16].”
• New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow (2012, methodology): “We can fix this to some extent by bringing in a pointwise Gaussian lower bound for Hx2 due to Zhang [22]; see Theorem 2.16 for a review.”
• Space of Ricci Flows I (2009, background): “[123], [12]) and the fact dgi(0)(yi, xi) < C implies that | log wi(xi, 0)| is uniformly bounded.”
• Pseudolocality for the Ricci Flow and Applications (2007, methodology): “6 More Gradient Estimates In this section we want to obtain more gradient estimates, which are generalizations of the gradient estimates in [29] and [18] to complete noncompact manifolds.”
• Gradient estimates for positive solutions of the heat equation under geometric flow (2011, unknown): “Since then, there have been many works on gradient estimates along the Ricci flow or the conjugate Ricci flow for the solution of the heat equation or the conjugate heat equation; examples include [Cao 2008; Cao and Hamilton 2009; Kuang and Zhang 2008; Zhang 2006].”
• Notes on Perelman's papers (2006, background): “To see this, one has the uniform estimates [58] needed to take a subsequence of the uk’s that converges to a smooth function u∞ on open intervals contained in (t∞ − 12α, t∞].”
• Gradient estimates for solutions of the heat equation under Ricci flow (2009, unknown): “Similar to the fact that the interpolation inequality follows from this estimate in [Zhang 2006], here we get for any x, y ∈ M and 0 t ≤ T that (2-30) u(x, t)≤ u(y, t)+ √ C 2 d(x, y, t) √ t , where C =maxx∈M u2(x, 0).”