Lucio Bedulli, Luigi Vezzoni
We observe that the DeTurck Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of G2-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in [4].
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We give a lower bound on the number of small positive eigenvalues of the p-form Laplacian in a certain type of collapse with curvature bounded below.
Why the results of that article [arXiv:1304.3212] are immediate consequences of known ones.
We show the existence of expanding solitons of the G$_2$-Laplacian flow on non-solvable Lie groups, and we give the first example of a steady soliton that is not an extremally Ricci pinched G$_2$-structure.
Eigenvalue comparison theorems for the Witten-Laplacian and the weighted $p$-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
在具有从下方有界的修正Ricci曲率的完备流形上，建立Witten拉普拉斯算子和加权p-拉普拉斯算子在测地球上的第一狄利克雷特征值的程氏型特征值比较定理。
A remark on the first eigenvalue of the p-Laplacian on compact submanifolds in the unit sphere
本文针对3/2<p<2建立了奇异p-laplacian的积分不等式，并由此推导出单位球面中极小子流形和预定数量曲率子流形p-laplacian第一特征值的下界。
More than forty years ago J. H. Samson has defined the Laplacian $Δ_{sym}$ acting on the space of symmetric covariant $p$-tensors on an $n$-dimensional Riemannian manifold $(M, g)$. This operator is an analogue of the well known Hodge-de Rham Laplacian $Δ$ which acts on the space of exterior differential $p$-forms ($1 \le p \le n$) on $(M, g)$. In the present paper we will prove that for $n > p = 1$ the operator $Δ_{sym}$ is the Yano rough Laplacian and show its spectrum properties on a compact Riemannian manifold.
In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Ba…
We consider $G_{2}$-structures on $7$-manifolds that are warped products of an interval and a six-manifold, which is either a Calabi-Yau manifold, or a nearly Kähler manifold. We show that in these cases the $G_{2}$-structures are determined by their torsion components up to a phase factor. We then study the modified Laplacian coflow $\frac{dψ}{dt}=Δ_{ψ}ψ+2d\left( \left( C-Tr T\right) φ\right) $ of these $G_{2}$-structures, where $φ$ and $ψ$ are the fundamental $3$-form and $4 $-form which define the $G_{2}$-structure and $Δ_{ψ}$ is the Hodge Laplacian associated with the $G_{2}$-structure. Th…
We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-\infty,T)$, with $0<T<+\infty$. However, for the modified Laplacian coflow, we prove that in some cases the solution is defined only on a finite interval while in other cases the solution is ancient or eternal, that is it is defined on $(-\infty, \infty)$.
We modify the Laplacian coflow of co-closed G2-structures - $\frac{d}{dt}ψ=Δψ$ where $ψ$ is the closed dual 4-form of a $G_{2}$-structure $φ$. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified flow.
Real analyticity of the modified Laplacian coflow
针对紧致7维流形上具有闭余G2结构的修正拉普拉斯余流，改进Chen的Shi型估计并证明其关联度量实解析，回答Grigorian提出的问题，进而得到流的唯一延拓结果。
An infinite dimensional Laplacian defined as the Cesáro mean of the second order directional derivatives on manifold is considered. This Laplacian is parameterized by the choice of a curve in the group of orthogonal rotations. It is shown that, under certain conditions on the curve, this operator is related to instantons on a 4-dimensional manifold.
The connection between Yang--Mills gauge fields on $4$-dimensional orientable compact Riemannian manifolds and modified Lévy Laplacians is studied. A modified Lévy Laplacian is obtained from the Lévy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group of the bundle of self-dual 2-forms, the following is proved. There is a modified Lévy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified Lévy Laplacian if and only if …
We give a characterization of relative Ding stable toric Fano manifolds in terms of the behavior of the modified Ding functional. We call the corresponding behavior of the modified Ding functional the pseudo-boundedness from below. We also discuss the pseudo-boundedness of the Ding / Mabuchi functional of general Fano manifolds.
We survey recent progress in the study of $G_{2}$-structure Laplacian coflows, that is, heat flows of co-closed $G_{2}$-structures. We introduce the properties of the original Laplacian coflow of $G_{2}$-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified coflow and well as recent Shi-type estimates that apply to a more general class of $G_{2}$-structure flows.
Remarks on the spectra of minimal hypersurfaces in the hyperbolic space
本文计算了双曲空间中具有给定渐近数据的奇异面积最小化超曲面的Laplacian谱，推广至高余维数，并探讨了谱底部的极值性质。
Laplacian coflows of $G_2$-structures on contact Calabi--Yau 7-manifolds
本文研究圆纤维化上 $G_2$-结构的三种 Laplacian 余流，通过维数约化证明它们对应于基底 Kähler 几何中 Kähler–Ricci 流的自然修正。
We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acoustic scattering in layered media.
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning the scalar curvature of a smooth limit space arising in a bounded curvature collapse.
An important open question in G$_{2}$ geometry concerns whether or not a compact seven-manifold can support an exact G$_{2}$-Structure. Given the significance of this question we initiate a study of exact G$_{2}$-Structures on compact manifolds. We focus on exact G$_2$-Structures subject to no additional constraints but we also consider the relationship between the exact condition and other conditions for closed G$_{2}$-Structures such as the Extremally Ricci-Pinched and Laplacian Soliton conditions.
In this paper the rate relations of Riemann, conformal, conharmonic and Weyl curvature tensors under Yamabe flow are studied. Modified Riemann extensions under Yamabe flow is discussed. The paper ends with remarks on some standard metrics.
Remark on Laplacians and Riemannian Submersions with Totally Geodesic Fibers
本文研究具有全测地纤维的黎曼淹没的典型变分中拉普拉斯算子的第一正特征值，在纤维为Einstein且Ricci曲率满足条件时得到特征值界，并应用于Yamabe泛函临界点稳定性及四元数Kähler流形的twistor纤维化。
Eigenvalue lower bounds and splitting for modified Ricci flow
该研究针对修正Ricci流，证明了滑移拉普拉斯算子特征值的精确下界，并给出了等号成立时的分裂定理，丰富了Ricci流相关理论。
The hypersymplectic flow descended from the $G_2$-Laplacian coflow
本文基于正三元体定义超辛和超kähler结构，并通过修改的$G_2$-Laplacian coflow演化得到正三元体流，从而实现从超辛结构到超kähler结构的流形变换。
In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G$_2$ structures including the modified Laplacian co-flow. Then we prove a version of $κ$-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G$_2$ structures.
In this paper, we consider the principal eigenvalue problem for Hormander's laplacian on $R^n$. We also study a related semi-linear sub-elliptic equation in the whole $R^n$ and prove that under a suitable condition, we have infinite many positive solutions of the problem.
Nearly $G_2$-manifolds and $G_2$-Laplacian co-flows
该研究探讨了近$G_2$-结构在修正$G_2$-拉普拉斯共流下的稳定性，发现许多自然来源于3-萨西亚几何的结构不稳定，特别是标准的近$G_2$-结构在7维球面上的高索引不稳定临界点。
On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-linear operators. These inequalities seem not available in the literature; And our proof, solely based on the ABP estimate, does not involve any Sobolev inequalities nor gradient estimate. We also propose a question regarding the characterization of Ricci lower bound by the Harnack …
Comparison Geometry on Manifolds with Density via Modified Hessians
本文通过Wylie等人提出的加权截面曲率框架下的修正Hessian，研究了密度流形上的径向比较几何，在非负加权截面曲率和适当密度控制下，得到了径向函数u=1/2 r^2的修正Hessian估计，并导出了Hessian比较、形状算子比较、加权Laplacian比较、渐近径向体积密度估计和多项式加权体积增长界，同时研究了刚性现象。
We prove a general result about the stability of geometric flows of "closed" sections of vector bundles on compact manifolds. Our theorem allows to prove a stability result for the modified Laplacian coflow in G2-geometry introduced by Grigorian and for the balanced flow introduced by the authors in a previous paper.
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