The entropy formula for linear heat equation

By Lei Ni. 2003

154 citations recorded by the source corpus.

We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c3c977ceb8cd1f29aa6253b9b222c7b8ffadfbf9

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

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.”
A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow (2007, methodology): “to the heat equation, it was proved in [ N3 ] that W(f, τ) is monotone non-increasing in τ. Moreover, it was shown that”
New monotonicity formulas for Ricci curvature and applications. I (2011, background): “scuss Perelman’s F and W functional in the present setting of manifolds with nonnegative Ricci curvature. There are very few new things in this section and most of the results can be found in [P1] or [N1]–[N3]; however the presentation we give emphasize the parallels to the previous sections which i”
Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula (2007, methodology): “This further implies that the metric on N £ I is of the form”
Hamilton's gradient estimate for the heat kernel on complete manifolds (2007, background): “[ N2 ]) is pointwise nonpositive for the fundamental solution to the heat equation.”
Sharp logarithmic Sobolev inequalities on gradient solitons and applications (2008, methodology): “The case of μe is an easy application of a rigidity result in [29].”
Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold (2008, background): “Strong connections exist with studies of entropies for heat equations on a static manifold, see for example Ni [ 34 ] and also the general references [14,15,29].”
Eigenvalues and energy functionals with monotonicity formulae under Ricci flow (2007, unknown): “ture tensor. We use the following convention of Riemann curvature tensor through out this paper: Rm(X,Y )Z ≡ [∇X,∇Y ]Z −∇[X,Y ]Z, where {X,Y,Z} are vector ﬁelds on the manifold. 6 Jun-Fang Li Combing (13), (14), and the contracted second Bianchi identity, we have Z |∇∇ϕ|2 e−ϕdµ = − Z ∇jϕ(∇j∇ i∇iϕ −R”
The local entropy along Ricci flow---Part A: the no-local-collapsing theorems (2017, background): “Ni [32], which was suggested by Perelman at the end of the proof of Theorem 10.”
Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds (2008, background): “In this section we show that Perelman’s entropy formula [28] can be adapted to the porous medium equation, with the help of the computation in Section 3, via the thermodynamical considerations [19,25,26].”
ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS (2013, background): “We thus present a unified formula of various W entropies established by various authors for different flows (including the static case), see [5, 8, 9, 11, 12, 13].”
Monotonicity and its analytic and geometric implications (2012, background): “The first key point is that if A 1⁄4 AðrÞ is defined as above, then A is monotone, A ↓, and we have [17].”
Differential Harnack inequalities and the Ricci flow (2006, methodology): “2 of [25]) Let u = (4πt)−n/2e−f be the positive heat kernel on M , i.”