Logarithmic Sobolev Inequalities, Gaussian Upper Bounds for the Heat Kernel, and the G2-Laplacian Flow

We prove a logarithmic Sobolev inequality along the G 2 -Laplacian flow. A uniform Sololev inequality along the G 2 -Laplacian flow with uniformly bounded scalar curvature is derived from the logarithmic Sobolev inequality. The uniform Sololev inequality implies a κ -noncollapsing estimate for the 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 G 2 -Laplacian flow with uniformly bounded scalar curvature.