[Paper Review] The logarithmic Sobolev inequality along the Ricci flow

Rugang Ye|ArXiv.org|Jul 17, 2007

Geometric Analysis and Curvature Flows6 references68 citations

TL;DR

This paper establishes uniform logarithmic Sobolev inequalities along the Ricci flow on compact Riemannian manifolds of dimension $ n \geq 3 $, showing that under the condition $ \lambda_0(g_0) > 0 $, the logarithmic Sobolev constant remains uniformly bounded in time. The key result is a time-uniform inequality of the form $ \int_M u^2 \ln u^2 \, d\text{vol} \leq \sigma \int_M (|\nabla u|^2 + \frac{R}{4}u^2)\, d\text{vol} - \frac{n}{2}\ln\sigma + C $, which implies uniform Sobolev and volume bounds, and extends to $ L^p $-Sobolev inequalities via spectral theory and heat kernel estimates.

ABSTRACT

We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any restriction on time. One application of it is a uniform kappa-noncollapsing estimate which holds true for all time. We also obtain similar results for bounded time without assuming the eigenvalue condition. The results extend to the Ricci flow with surgeries.

Motivation & Objective

• To establish a uniform logarithmic Sobolev inequality along the Ricci flow that is independent of time, under geometric conditions on the initial metric.
• To characterize the dependence of the logarithmic Sobolev constant on geometric invariants such as the modified Sobolev constant, scalar curvature, and first eigenvalue of $ -\Delta + \frac{R}{4} $.
• To derive uniform Sobolev and volume estimates for the Ricci flow, particularly in the large-time regime.
• To extend the logarithmic Sobolev inequality to $ L^p $-Sobolev embeddings via spectral theory of Schr"odinger-type operators.

Proposed method

• Derives a time-dependent logarithmic Sobolev inequality (1.2) using the modified Sobolev constant $ \tilde{C}_S(M,g_0) $, initial volume, and scalar curvature $ R_{g_0} $, with explicit dependence on time $ t $ and parameter $ \sigma $.
• Introduces a time-independent version (1.8) under the condition $ \lambda_0(g_0) > 0 $, using the first eigenvalue of $ -\Delta + \frac{R}{4} $ to control long-time behavior.
• Applies heat kernel estimates and spectral theory to bound the operator $ e^{-tH} $, establishing $ L^p $-boundedness of $ H^{-1/2} $ and $ H^{1/2} $, which links to Sobolev embeddings.
• Uses the equivalence of $ L^p $-Sobolev inequalities to $ L^p $-boundedness of $ H^{-1/2} $, derived via Marcinkiewicz interpolation and weak-type estimates.
• Establishes $ L^p $-Sobolev embeddings (3.36) and (3.37) by relating $ \|u\|_{np/(n-p)} $ to $ \|(-\Delta + R/4)^{1/2}u\|_p $, with constants depending on $ n $, $ \lambda_0(g_0) $, $ C_S(M,g_0) $, and volume bounds.
• Employs approximation arguments and pseudo-differential operator theory to extend inequalities from smooth functions to $ W^{1,p}(M) $, ensuring robustness.

Experimental results

Research questions

• RQ1 Can a uniform logarithmic Sobolev inequality be established along the Ricci flow for all time, independent of $ t \in [0,T) $, under geometric conditions on the initial metric?
• RQ2 What role does the first eigenvalue $ \lambda_0(g_0) $ of the operator $ -\Delta + \frac{R}{4} $ play in ensuring long-time uniformity of the logarithmic Sobolev inequality?
• RQ3 How do heat kernel estimates and spectral theory of Schr"odinger-type operators $ H $ relate to the derivation of $ L^p $-Sobolev inequalities along the flow?
• RQ4 To what extent can the modified Sobolev constant $ \tilde{C}_S(M,g_0) $ and the initial metric's geometry control the time evolution of the logarithmic Sobolev constant?
• RQ5 What are the sharp quantitative bounds on volume and Sobolev constants along the Ricci flow, and how do they depend on $ \lambda_0(g_0) $ and curvature bounds?

Key findings

• The logarithmic Sobolev inequality (1.2) holds uniformly in time with explicit dependence on $ \sigma $, $ t $, and geometric invariants of the initial metric $ g_0 $, including $ \tilde{C}_S(M,g_0) $, $ \text{vol}_{g_0}(M) $, and $ R_{g_0} $.
• Under the condition $ \lambda_0(g_0) > 0 $, a time-uniform logarithmic Sobolev inequality (1.8) is established with a universal constant $ C $ depending only on $ n $, $ \lambda_0(g_0) $, $ \text{vol}_{g_0}(M) $, $ C_S(M,g_0) $, and bounds on $ \frac{1}{p-1} $, $ \frac{1}{n-p} $.
• The volume of the manifold satisfies $ \text{vol}_{g(t)}(M) \geq e^{-1/4 - C} $ when $ \hat{R}(t) \leq 0 $, and $ \geq e^{-1/4 - C} \hat{R}(t)^{-n/2} $ when $ \hat{R}(t) > 0 $, with $ C $ depending on $ \lambda_0(g_0) $ and other geometric invariants.
• An $ L^p $-Sobolev inequality (3.36) is proven: $ \|u\|_{np/(n-p)} \leq C \|(-\Delta + R/4)^{1/2}u\|_p $ for $ 1 < p < n $, with $ C $ depending on $ n $, $ \lambda_0(g_0) $, $ \text{vol}_{g_0}(M) $, $ C_S(M,g_0) $, and bounds on $ \frac{1}{p-1} $, $ \frac{1}{n-p} $.
• For finite-time flows, an $ L^p $-Sobolev inequality (3.37) is established with a modified operator $ H_0 = -\Delta + R/4 - \frac{\min R_{g_0}^{-}}{4} + 1 $, ensuring uniformity under bounded curvature and time.
• The proof relies on heat kernel estimates (3.33)–(3.34), weak-type bounds for $ H^{-1/2} $, and interpolation techniques, showing that $ H^{-1/2} $ is bounded from $ L^p(M) $ to $ W^{1,p}(M) $ for $ 1 < p < \infty $.