We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li–Yau’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equalit...

This is the first of two papers where we address and partially confirm a conjecture of Deser and Schwimmer, originally postulated in high energy physics, [10]. The objects of study are scalar Riemannian quantities constructed out of the curvature and its covariant derivatives, whose integrals over compact manifolds are invariant under conformal changes of the underlying metric. Our main conclus...

Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where newtonian motions are seen as geodesics of suitable “mechanical” manifolds. The existence of constants of motion that entail integrability is associated with the existence of Killing tensor fields on the mechanical manifolds. Such tens...

We show that the expected limit distribution of the real zero set of a Gaussian random linear combination of eigenfunctions with frequencies from a short interval (‘asymptotically fixed frequency’) is uniform with respect to the volume form of a compact Riemannian manifold (M, g). We further show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauer...

We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called horizontal area functional associated to the canonical horizontal area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimen...

on a Riemannian manifold (M n , g), where f is a symmetric function of λ ∈ R , κ is a constant, ∇2u denotes the Hessian of a function u on M and, for a (0, 2) tensor h on M , λ(h) = (λ1, · · · , λn ) denotes the eigenvalues of h with respect to the metric g. The Dirichlet problem for equations of type (1.1) in R , with κ = 0, under various hypothesis, is studied by Caffarelli, Nirenberg and Spr...

The quantum states which satisfy the equality in the generalised uncertainty relation are called intelligent states. We prove the existence of intelligent states for the Anandan-Aharonov uncertainty relation based on the geometry of the quantum state space for arbitrary parametric evolutions of quantum states when the initial and final states are non-orthogonal. email:[email protected] I...

We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.

In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M t →֒ (N, h) are characterised by the blow up of the mean curvature.

In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. We define the characteristic degree of a subgroup H in a finite group G as the ratio of the number of all pairs (h,α) ∈ H×Aut(G) such that h^α∈H, by the order of H × Aut(G), where Aut(G) is the automorphisms group of G. This quantity measures the probability that H can be characteristic ...