arXiv subjects
Publications and source records attributed to Jia-Yong Wu.
In this paper we study some splitting properties on complete noncompact manifolds with smooth measures when $\infty$-dimensional Bakry-Émery Ricci curvature is bounded from below by some negative constant and spectrum of the weighted Laplacian has a positive lower bound. These results extend the cases of Ricci curvature and $m$-dimensional Bakry-Émery Ricci curvature.
In this paper we study complete manifolds equipped with smooth measures whose spectrum of the weighted Laplacian has an optimal positive lower bound and the $m$-dimensional Bakry-Émery Ricci curvature is bounded from below by some negative constant. In particular, we prove a splitting type theorem for complete smooth measure manifolds that have a finite weighted volume end. This result is regarded as a study of the equality case of an author's theorem (J. Math. Anal. Appl. 361 (2010) 10-18).
In this short note we establish an integral geometric inequality in a smooth metric measure space of the nonnegative Bakry-Émery Ricci curvature. This result can be regarded as a mild generalization of the almost Schur theorem due to De Lellis and Topping (Calc. Var., DOI: 10.1007/s00526-011-0413-z).
Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we estimate the diameter of $M$ in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of Topping in the Euclidean case (Comment. Math. Helv., 83 (2008), 539--546).
Given a family of smooth immersions $F_t: M^n\to N^{n+1}$ of closed hypersurfaces in a locally symmetric Riemannian manifold $N^{n+1}$ with bounded geometry, moving by the mean curvature flow, we show that at the first finite singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a recent result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of N. Le in the Euclidean case (arXiv: math.DG/1002.4669v2).
Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \[ u_t=Δu-\nablaϕ\cdot\nabla u-au\log u-bu, \] where $ϕ$ is a $C^2$ function, and $a\neq0$ and $b$ are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).
In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first $p$-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a $p$-eigenvalue comparison-type theorem when its Euler characteristic is negative.