599 research outputs found

### Gradient estimate for eigenforms of Hodge Laplacian

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

We study both function theoretic and spectral properties of the weighted
Laplacian $\Delta_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$
with its Bakry-\'{E}mery curvature $Ric_f$ bounded from below by a constant. In
particular, we establish a gradient estimate for positive $f-$harmonic
functions and a sharp upper bound of the bottom spectrum of $\Delta_f$ in terms
of the lower bound of $Ric_{f}$ and the linear growth rate of $f.$ We also
address the rigidity issue when the bottom spectrum achieves its optimal upper
bound under a slightly stronger assumption that the gradient of $f$ 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

The main purpose of this paper is to investigate the curvature behavior of
four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with
bounded scalar curvature $S$, it is shown that the curvature operator
$\mathrm{Rm}$ of $M$ satisfies the estimate $|\mathrm{Rm}|\le c\,S$ for some
constant $c$. Moreover, the curvature operator $\mathrm{Rm}$ is asymptotically
nonnegative at infinity and admits a lower bound $\mathrm{Rm}\geq -c\,\left(\ln
r\right)^{-1/4},$ where $r$ is the distance function to a fixed point in $M$.
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

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 $\mathbb{R}\times \mathbb{S}^{n-1}/\Gamma$
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 $\mathbb{R}\times \mathbb{S}^{3}$ or converges to a quotient of
$\mathbb{R}^{2}\times \mathbb{S}^{2}$ 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 $\mathbb{R}^{2}\times
\mathbb{S}^{2}$ at infinity

### Kahler manifolds with real holomorphic vector fields

For a K\"{a}hler manifold endowed with a weighted measure $e^{-f}\,dv,$ the
associated weighted Hodge Laplacian $\Delta _{f}$ maps the space of
$(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector
field $\nabla f$ is holomorphic. We use this fact to prove that for such $f$, a
finite energy $f$ harmonic function must be pluriharmonic. Motivated by this
result, we verify that the same also holds true for $f$-harmonic maps into a
strongly negatively curved manifold. Furthermore, we demonstrate that such
$f$-harmonic maps must be constant if $f$ 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

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

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

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

One of the main purposes of this paper is to prove that on a complete
K\"ahler manifold of dimension $m$, if the holomorphic bisectional curvature is
bounded from below by -1 and the minimum spectrum $\lambda_1(M) \ge m^2$, then
it must either be connected at infinity or diffeomorphic to $\Bbb R \times N$,
where $N$ 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

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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