599 research outputs found

    Gradient estimate for eigenforms of Hodge Laplacian

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    In this paper, we derive a gradient estimate for the linear combinations of eigenforms of the Hodge Laplacian on a closed manifold. The estimate is given in terms of the dimension, volume, diameter and curvature bound of the manifold. As an application, we obtain directly a sharp estimate for the heat kernel of the Hodge Laplacian.Comment: Some comments are added and references are revise

    Analysis of weighted Laplacian and applications to Ricci solitons

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    We study both function theoretic and spectral properties of the weighted Laplacian Ξ”f\Delta_f on complete smooth metric measure space (M,g,eβˆ’fdv)(M,g,e^{-f}dv) with its Bakry-\'{E}mery curvature RicfRic_f bounded from below by a constant. In particular, we establish a gradient estimate for positive fβˆ’f-harmonic functions and a sharp upper bound of the bottom spectrum of Ξ”f\Delta_f in terms of the lower bound of RicfRic_{f} and the linear growth rate of f.f. We also address the rigidity issue when the bottom spectrum achieves its optimal upper bound under a slightly stronger assumption that the gradient of ff is bounded. Applications to the study of the geometry and topology of gradient Ricci solitons are also considered. Among other things, it is shown that the volume of a noncompact shrinking Ricci soliton must be of at least linear growth. It is also shown that a nontrivial expanding Ricci soliton must be connected at infinity provided its scalar curvature satisfies a suitable lower bound.Comment: Will appear in Comm. Anal. Geo

    Geometry of shrinking Ricci solitons

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    The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton MM with bounded scalar curvature SS, it is shown that the curvature operator Rm\mathrm{Rm} of MM satisfies the estimate ∣Rmβˆ£β‰€c S|\mathrm{Rm}|\le c\,S for some constant cc. Moreover, the curvature operator Rm\mathrm{Rm} is asymptotically nonnegative at infinity and admits a lower bound Rmβ‰₯βˆ’c (ln⁑r)βˆ’1/4,\mathrm{Rm}\geq -c\,\left(\ln r\right)^{-1/4}, where rr is the distance function to a fixed point in MM. As application, we prove that if the scalar curvature converges to zero at infinity, then the manifold must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.Comment: 28 pages, submitted, v2 has a new section about the conical structure of soliton

    Structure at infinity for shrinking Ricci solitons

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    The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder RΓ—Snβˆ’1/Ξ“\mathbb{R}\times \mathbb{S}^{n-1}/\Gamma occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity. The result is then applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It is previously known that such solitons with scalar curvature approaching zero at infinity must be smoothly asymptotic to a cone. For the case that the scalar curvature is bounded from below by a positive constant, we conclude that along each end the soliton is asymptotic to a quotient of RΓ—S3\mathbb{R}\times \mathbb{S}^{3} or converges to a quotient of R2Γ—S2\mathbb{R}^{2}\times \mathbb{S}^{2} along each integral curve of the gradient vector field of the potential function. For four dimensional K\"{a}hler Ricci solitons, stronger conclusion holds. Namely, they either are smoothly asymptotic to a cone or converge to a quotient of R2Γ—S2\mathbb{R}^{2}\times \mathbb{S}^{2} at infinity

    Kahler manifolds with real holomorphic vector fields

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    For a K\"{a}hler manifold endowed with a weighted measure eβˆ’f dv,e^{-f}\,dv, the associated weighted Hodge Laplacian Ξ”f\Delta _{f} maps the space of (p,q)(p,q)-forms to itself if and only if the (1,0)(1,0)-part of the gradient vector field βˆ‡f\nabla f is holomorphic. We use this fact to prove that for such ff, a finite energy ff harmonic function must be pluriharmonic. Motivated by this result, we verify that the same also holds true for ff-harmonic maps into a strongly negatively curved manifold. Furthermore, we demonstrate that such ff-harmonic maps must be constant if ff has an isolated minimum point. In particular, this implies that for a compact K\"{a}hler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.Comment: 16 pages, submitte

    Positively curved shrinking Ricci solitons are compact

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    We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.Comment: submitte

    Conical structure for shrinking Ricci solitons

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    For a shrinking Ricci soliton with Ricci curvature convergent to zero at infinity, it is proved that it must be asymptotically conical.Comment: submitte

    Counting dimensions of L-harmonic functions

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    In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially of degree d. More precisely, in terms of a rectangular coordinate system {x_1,...,x_n}, a second order uniformly elliptic operator of divergence form, L, acting on a function f in H^1_loc(R^n) is given by Lf = sum_{ij} d/dx_i (a^{ij}(x) df/dx_j) where (a^{ij}(x)) is an n x n symmetric matrix satisfying the ellipticity bounds \lambda I <= (a^{ij}) <= Lambda I for some constants 0 < lambda <= Lambda < \infty. Other than the ellipticity bounds, we only assume that the coefficients (a_{ij}) are merely measurable functions.Comment: 14 pages, published version, abstract added in migratio

    Connectedness at infinity of complete K\"ahler manifolds and locally symmetric spaces

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    One of the main purposes of this paper is to prove that on a complete K\"ahler manifold of dimension mm, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum Ξ»1(M)β‰₯m2\lambda_1(M) \ge m^2, then it must either be connected at infinity or diffeomorphic to RΓ—N\Bbb R \times N, where NN is a compact quotient of the Heisenberg group. Similar type results are also proven for irreducible, locally symmetric spaces of noncompact type. Generalizations to complete K\"ahler manifolds satisfying a weighted Poincar\'e inequality are also being considere

    Weighted Poincar\'e inequality and rigidity of complete manifolds

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    We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincar\'e inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincar\'e inequality, and yields a criterion of parabolicity of connected components at infinity in terms of the weight function
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