Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non-zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.     

Some estimates of fundamental solutions on noncompact manifolds with time-dependent metrics

In this article, we obtain some estimates about the gradient and Laplacian of the fundamental solution of a heat equation with time-dependent Riemannian metrics and give some applications of the estimates on the asymptotic behaviors of fundamental solutions.     

Some gradient estimates and Harnack inequalities for nonlinear parabolic equations on Riemannian manifolds

In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491 ), we shall consider the nonlinear parabolic equation on Riemannian manifolds with , . First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when , our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang.     

流形上的微分Harnack 估计

In the thesis, we study the differential Harnack estimate for the heat equation of the Hodge Laplacian deformation of (p, p)-forms on both fixed and evolving (by Kähler-Ricci flow) Kähler manifolds, which generalize the known differential Harnack estimates for (1, 1)-forms. On a Kähler manifold, we define a new curvature cone C_p and prove that the cone is invariant under Kähler-Ricci flow and that the cone ensures the preservation of the nonnegativity of the solutions to Hodge Laplacian heat equation. After identifying the curvature conditions, we prove the sharp differential Harnack estimates for the positive solution to the Hodge Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow after obtaining some interpolating matrix differential Harnack type estimates for curvature operators between Hamilton’s and Cao’s matrix Harnack estimates. Similarly, we define another new curvature cone, which is invariant under Ricci flow, and prove another interpolating matrix differential Harnack estimates for curvature operators on Riemannian manifolds.     

Local approximations on manifolds and weighted estimates

An approach proposed by the author for representing spaces of local functions (s.l.f.) by means of an abstract function with values in the dual space is developed; thanks to this generalization of s.l.f. to the case of a nonuniform net and manifolds are obtained. Moreover, some weighted estimates of approximation by regular s.l.f. are established, and simple proofs are given of theorems of approximation by regular s.l.f. in the spaces of S. L. Sobolev and O. V. Besov.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 125–138, 1984.     

Center manifolds and optimal estimates

For sufficiently small perturbations of a nonuniform exponential trichotomy, we establish the existence of $C^k$ invariant center manifolds. We consider the general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. In particular, we obtain optimal estimates for the decay of all derivatives along the trajectories on the center manifolds.     

Gradient estimates and Jackson’s theorem in spaces related to measures

Jackson’s theorem is established in a new kind of holomorphic function space Q_μ related to measures in any starlike circular domain in . Particularly, the result covers many spaces including BMOA, Q_p, Q_K, and F(p,q,s) spaces in the unit ball of . Moreover, we construct integral operators which give pointwise estimates for the gradient of the difference in terms of the gradient on the boundary. The gradient estimates are independent of the measures in question and give rise to Jackson’s theorem.     

Spectral gap estimates on compact manifolds

For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities.

     

Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians

In this Note, we extend the Reilly formula for drifting Laplacian operator and apply it to study eigenvalue estimate for drifting Laplacian operators on compact Riemannian manifolds' boundary. Our results on eigenvalue estimates extend previous results of Reilly and Choi and Wang.     

Some results on the signless Laplacians of graphs

In this paper we give two results concerning the signless Laplacian spectra of simple graphs. Firstly, we give a combinatorial expression for the fourth coefficient of the (signless Laplacian) characteristic polynomial of a graph. Secondly, we consider limit points for the (signless Laplacian) eigenvalues and we prove that each non-negative real number is a limit point for (signless Laplacian) eigenvalue of graphs.     

Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived.     

Eigenvalue estimates and L 1 energy on closed manifolds

In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and the decay rates of L1 and L2 energy for drifting heat equation on closed Riemannian manifolds with weighted measure.     

Integral curvature bounds and diameter estimates on Finsler manifolds

In this paper, we study the integrals of the Ricci curvature over metric balls in a Finsler manifold,which can be viewed as an L~q-norm of the Ricci curvature. By bounding such integrals from above, we obtain several Myers type theorems.