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By the integral method, we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\mathbb{R}^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in [9] and [1] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in [1].

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far.

A geometric classification of the compact four-dimensional Einstein-Weyl manifolds with at least four-dimensional symmetry group is given. Our results also sharpen previous results on four-dimensional Einstein metrics and correct Parker's topological classification of cohomogeneity-one four-manifolds.

In this paper we study the analytic torsion and the L2-torsion of compact locally symmetric manifolds. We consider the analytic torsion with respect to representations of the fundamental group which are obtained by restriction of irreducible representations of the group of isometries of the underlying symmetric space. The main purpose is to study the asymptotic behavior of the analytic torsion with respect to sequences of representations associated to rays of highest weights.