Logarithmic Sobolev Inequalities, Gaussian Upper Bounds for the Heat Kernel, and the \(\textrm{G}_{2}\)-Laplacian Flow

We prove a logarithmic Sobolev inequality along the \(\textrm{G}_{2}\) ${\text{G}}_2$ -Laplacian flow. A uniform Sololev inequality along the \(\textrm{G}_{2}\) ${\text{G}}_2$ -Laplacian flow with uniformly bounded scalar curvature is derived from the logarithmic Sobolev inequality. The uniform Sololev inequality implies a \(\kappa \) $\kappa$ -noncollapsing estimate for the \(\textrm{G}_{2}\) ${\text{G}}_2$ -Laplacian flow with uniformly bounded scalar curvature. Furthermore, by using the logarithmic Sobolev inequality, we prove Gaussian-type upper bounds for the heat kernel along the \(\textrm{G}_{2}\) ${\text{G}}_2$ -Laplacian flow with uniformly bounded scalar curvature.