Abstract We obtain a comparison formula for integrals of mean curvatures Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities convex hypersurface Γ \Gamma in Cartan-Hadamard manifold M M . In particular, show that the first curvatu...

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning scalar curvatures of smooth lim...

We show that Brownian motion on any unbounded submanifold P in an ambient manifold N with a pole p is transient if the following conditions are satisfied: The p-radial mean curvatures of P are sufficiently small outside a compact set and the p-radial sectional curvatures of N are sufficiently negative. The ‘sufficiency’ conditions are obtained via comparison with explicit transience criteria fo...

Every point in Teichmüller space is a hyperbolic metric on a given Riemann surface, therefore, a Weil-Petersson geodesic in Teichmüller space can be viewed as a 3-manifold. We investigate the sectional curvatures of this 3-manifold, with a natural metric. We obtain explicit formulas for the curvature tensors of this metric, and show that the “average”s of them are zero, and hence the geometry o...

The shape of a liquid surface bounded by an acute or obtuse wedge is considered by using classical analysis methods. For acute wedges the two principal curvatures are of the order of the (fixed) mean curvature. But for obtuse wedges, the principal curvatures both diverge as the vertex is approached. The power-law divergence becomes stronger with increasing opening angle. Possible implications o...

We investigate the interplay between the existence of fat triangulations, PL approximations of Lipschitz–Killing curvatures and the existence of quasiconformal mappings. In particular we prove that if there exists a quasiconformal mapping between two PL or smooth n-manifolds, then their Lipschitz–Killing curvatures are bilipschitz equivalent. An extension to the case of almost Riemannian manifo...

We discuss recursive formulas for Branson’sQ-curvatures. The formulas present Q-curvatures of any order in terms of lower order Qcurvatures and lower order GJMS-operators. These presentations are universal in the sense that the recursive structure does not depend on the dimension of the underlying space. We give proofs for Q4 and Q6 for general metrics, and for Q8 for conformally flat metrics. ...

We formulate and discuss two conjectures concerning recursive formulae for Branson’s Q-curvatures. The proposed formulae describe all Qcurvatures on manifolds of all even dimensions in terms of respective lower order Q-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlying space. The recursive formulae are generated by an algorithm which rests on the th...