Differential Harnack and logarithmic Sobolev inequalities along Ricci-harmonic map flow

Logarithmic Harnack inequalities∗

Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9, 12]. Recently, Diaconis and Saloff-Coste [8] considered logarithmic Sobolev inequalities for Markov chains. The lower bounds for log-Sobolev constants can be used to improve convergence bounds for ...

Modified logarithmic Sobolev inequalities and transportation inequalities

Uniform Sobolev Inequalities for Manifolds Evolving by Ricci Flow

Let M be a compact n-dimensional manifold, n ≥ 2, with metric g(t) evolving by the Ricci flow ∂gij/∂t = −2Rij in (0, T ) for some T ∈ R + ∪ {∞} with g(0) = g0. Let λ0(g0) be the first eigenvalue of the operator −∆g0 + R(g0) 4 with respect to g0. We extend a recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any n ≥ 2...

Time-dependent Sobolev Inequality along the Ricci Flow

d dt g = −2Rc g(0) = g0 on a closed manifold M. An important ingredient of Perelman’s proof of geometrization conjecture is the non-collapsing theorem of Ricci flow which makes sure that we can get a singularity model of the flow when a singularity exists. In [7], Zhang gave an easier way to prove the noncollapsing theorem of Ricci flow via a uniform Sobolev inequality along the flow. Unfortuna...

Optimal inequalities for the power, harmonic and logarithmic means

For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...