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Register a new userSharp logarithmic Sobolev inequalities on gradient solitons and applications
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We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the {it asymptotic volume ratio} of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.
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I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Lebedev-Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Lebedev-Milin conjecture. Recently, O.~Roth proved an interesting sharp inequality for the logarithmic coefficients based on the proof by de Branges. In this paper, following Roths ideas, we will show more general sharp inequalities with convex sequences as weight functions and then establish several consequences of them. We also consider the inequality with the help of de Branges system of linear ODE for non-convex sequences where the proof is partly assisted by computer. Also, we apply some of those inequalities to improve previously known results.
In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional gradient shrinking solitons to dimension four.
The nonlinear Schrodinger equation NLSE(p, beta), -iu_t=-u_{xx}+beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space Lp^2(mT). For pleq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure mu^{beta}_N on balls Omega_N= {phi in Lp^2(mT) : | phi |^2_{Lp^2} leq N} in phase space such that the Cauchy problem for NLSE(p,beta) is well posed on the support of mu^{beta}_N, and that mu^{beta}_N is invariant under the flow. This paper shows that mu^{beta}_N satisfies a logarithmic Sobolev inequality for the focussing case beta <0 and 2leq pleq 4 on Omega_N for all N>0; also mu^{beta} satisfies a restricted LSI for 4leq pleq 6 on compact subsets of Omega_N determined by Holder norms. Hence for p=4, the spectral data of the periodic Dirac operator in Lp^2(mT; mC^2) with random potential phi subject to mu^{beta}_N are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hills equation when the potential is random and subject to the Gibbs measure of KdV.
The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $kappa$-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor, which includes the rotationally symmetric ones.
We establish an improved Sobolev trace inequality of order two in the Euclidean unit ball under the vanishing of higher order moments of the boundary volume element, and construct precise test functions to show that such inequalities are almost optimal. Our arguments can be adapted to the fourth order Sobolev trace inequalities in higher dimensional unit ball.
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