The fundamental solution on manifolds with time-dependent metrics

By Christine Guenther. 2002

88 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ae986614e76a714b738f0384d093a64ccbe11a11

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

In Geometry and Topology

Citations in the corpus

Pseudolocality for the Ricci Flow and Applications (2007, background): “(ii) By the asymptotic behavior of the heat kernel, see [11] for example, there is an open neighborhood U of p, some positive constants τ0 and C6, and a positive function u0 ∈ C∞(U × [0, τ0]) with u0(p, 0) = 1, such that ∣∣u − 1 (4πτ ) n 2 e− r2 t (x,p) 4τ u0(x, τ ) ∣∣ ≤ C6τ 1− n 2”
The conjugate heat equation and Ancient solutions of the Ricci flow (2010, methodology): “This equation, coupled with the initial value uτ=0 = u0 is well posed if M is compact or if curvature is bounded and u0 is bounded [11].”
Differential Harnack Estimates for Time-Dependent Heat Equations with Potentials (2008, unknown): “The fundamental solution and Harnack inequality of time-dependent heat equation have also been studied by C. Guenther [ Gue02 ].”
Gradient estimates for solutions of the heat equation under Ricci flow (2009, unknown): “In fact, the Theorem 2 improves the gradient inequality in [Guenther 2002], which requires the boundedness of the gradient of scalar curvature in addition to the boundedness of the Ricci curvature.”
A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow (2006, unknown): “Ricci curvature, i.e. |∇u|2 u2 − ut u ≤ n 2t for positive solutions of ∆u−∂tu= 0. Remark 2.2. Some related gradient estimates with various dependence on the Ricci and other curvatures can be found in [G] and [N3]. Proof. of Theorem 2.1. By a standard approximation argument, we can assume without los”
Some gradient estimates for the heat equation on domains and for an equation by Perelman (2006, background): “related results on local lower and upper bounds for fundamental solutions of (3.2) and for a global lower bound for the conjugate of (3.2) in the spirit of Perelman, please see the interesting papers [G], [Ni1] and [Ni2]. In order to state our theorem, we need to recall two concepts. One is the Yama”
A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow (2007, methodology): “Since one can apply the localization techniques as in [LY, page 161] (see also [NT1, page 647], [Sh1] ) one can easily generalize the gradient estimate, thus the above Harnack estimate, proved in [ Gu ] to complete noncompact case.”
Gradient estimates for positive solutions of the heat equation under geometric flow (2011, unknown): “Integrating the gradient estimate in space-time as in [Li and Yau 1986] or [Guenther 2002], we can derive the following Harnack-type inequality.”
Gradient estimates for the heat equation under the Ricci flow (2009, background): “ider the case where the manifold M is compact. We will present a global estimate on u(x,t) demanding that the Ricci curvature of M be nonnegative. A related inequality for (2.1)–(2.2) may be found in [14]. Theorem2.9. Suppose the manifold M is compact and M,g(x,t) t∈[0,T] is a solution to the Ricci ”
Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow (2008, background): “Guenther [10] studied the fundamental solution and Harnack inequality of time-dependent heat equation.”
Notes on Perelman's papers (2006, background): “We have uniform estimates on the functions {hk}k=1 and their spatial derivatives; compare [26, 45].”