In this paper, we classify unweighted graphs satisfying the curvature dimension condition $\CD (0,\infty)$ whose girth are at least five.

Given an acyclic representation  of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to an acyclic unitary representation, we define a refinement T of the Ray-Singer torsion associated to , which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by Turaev. T is equal to the graded determinant of the odd signature operator up to a correction term, the metric anomaly, needed to make it independent of the choice of the Riemannian metric.T is a holomorphic function on the space of such representations of the fundamental group. When  is a unitary representation, the absolute value of T is equal to the Ray-Singer torsion and the phase of T is proportional to the -invariant of the odd signature operator. The fact that the Ray-Singer torsion and the-invariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants. In particular, using these methods we compute the quotient of the refined analytic torsion and Turaev’s refinement of the combinatorial torsion generalizing in this way the classical Cheeger-M¨uller theorem. As an application, we extend and improve a result of Farber about the relationship between the Farber-Turaev absolute torsion and the -invariant. As part of our construction of T we prove several new results about determinants and -invariants of non self-adjoint elliptic operators.

In this paper, we derive two subgradient estimates of the CR heat equation in a closed pseudohermitian 3-manifold which are served as the CR version of the Li-Yau gradient estimate. With its applications, we first get a subgradient estimate of the logarithm of the positive solution of the CR heat equation. Secondly, we have the Harnack inequality and upper bound estimate for the heat kernel. Finally, we obtain Perelman-type entropy formulae for the CR heat equation.

We provide a refinement of the Poincaré inequality on the torus $\mathbb{T}^{d}$ : there exists a set $\mathcal{B} \subset \mathbb{T} ^{d}$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha } > 0$ with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative $\langle \nabla f, \alpha \rangle $ does not detect any oscillation in directions orthogonal to $\alpha $ , however, for certain $\alpha $ the geodesic flow in direction $\alpha $ is sufficiently mixing to compensate for that defect. On the two-dimensional torus $\mathbb{T}^{2}$ the inequality holds for $\alpha = (1, \sqrt{2})$ but is not true for $\alpha = (1,e)$ . Similar results should hold at a great level of generality on very general domains.

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