Let [Formula: see text] be a closed oriented hyperbolic three-manifold normalized so that [Formula: see text]. We prove a sharp lower bound for the volume of hypersurfaces in [Formula: see text] representing the slice class [Formula: see text], and we classify the equality case. If [Formula: see text] is a smooth Riemannian metric on [Formula: see text] with scalar curvature [Formula: see text], then every closed embedded hypersurface [Formula: see text] representing the slice class [Formula: see text] satisfies [Formula: see text]. The bound is attained by the product metric [Formula: see text], with [Formula: see text] any metric on [Formula: see text]. Conversely, if equality holds for some [Formula: see text], then up to a diffeomorphism preserving the slice class, [Formula: see text] and [Formula: see text].
In this study, we investigate the extremal length of simple closed curves and develop a sharp asymptotic estimate for the extremal length of a simple closed curve along a grafting ray determined by a rational lamination. Using the estimate of the extremal length for a rational lamination, we determine the endpoint of the grafting ray on the Gardiner–Masur boundary. Moreover, we derive an explicit formula for the limiting distance between two distinct grafting rays in the Teichmüller space with the Teichmüller metric.
Cornulier and Tessera established that the Dehn function of a solvable Lie group is either exponential or polynomially bounded, and gave criteria to decide between these two outcomes. In this paper, we make their result more explicit regarding the polynomial order of the Dehn function, and improve their lower bound in some cases. Along the way, we devise a practical method to compute distortion estimates in solvable Lie groups, of independent interest. Using Cornulier and Tessera’s methods we give exact estimates on the Dehn functions of solvable Lie groups of dimension at most 5, and as an application, we prove the QI rigidity of the group [Formula: see text] and its lattices. The latter QI rigidity result also uses works of Peng.
We generalize the construction of the Bismut superconnection to the non-integrable setting of subRiemannian manifolds equipped with a transverse distribution. The non-integrability produces singularities within the superconnection, and it is shown that if one extracts the finite part then the resulting operator is hypoelliptic on two-step subRiemannian manifolds. Moreover, it is shown that a perturbation produces a hypoelliptic operator on any subRiemannian manifold. The explicit form of the operator on principal circle bundles is computed and its kernel is determined for principal circle bundles over the [Formula: see text]-sphere with conformal curvature. A discussion of possible directions for generalizations and open questions follows; in the direction of index theory we introduce matrix twisted Clifford relations in the transverse directions to produce asymptotic superconnections with non-vanishing Fredholm index. We conclude with a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model.
We use the equivariant [Formula: see text]-bubble technique to prove that for any compact connected orientable manifold [Formula: see text] with non-empty boundary, [Formula: see text], the Yamabe invariant of [Formula: see text] is positive if and only if the Yamabe invariant of [Formula: see text] is positive. This generalizes the [Formula: see text]-stability conjecture of Rosenberg to compact manifolds with boundary.
